7.1: Angles
Learning Objectives
- Draw angles in standard position.
- Convert between degrees and radians.
- Find coterminal angles.
- Find the length of a circular arc.
- Use linear and angular speed to describe motion on a circular path.
A golfer swings to hit a ball over a sand trap and onto the green. An airline pilot maneuvers a plane toward a narrow runway. A dress designer creates the latest fashion. What do they all have in common? They all work with angles, and so do all of us at one time or another. Sometimes we need to measure angles exactly with instruments. Other times we estimate them or judge them by eye. Either way, the proper angle can make the difference between success and failure in many undertakings. In this section, we will examine properties of angles.
Drawing Angles in Standard Position
Properly defining an angle first requires that we define a ray . A ray consists of one point on a line and all points extending in one direction from that point. The first point is called the endpoint of the ray. We can refer to a specific ray by stating its endpoint and any other point on it. The ray in Figure \(\PageIndex{1}\) can be named as ray EF, or in symbol form \(\overrightarrow{EF}\).
An angle is the union of two rays having a common endpoint. The endpoint is called the vertex of the angle, and the two rays are the sides of the angle. The angle in Figure \(\PageIndex{2}\) is formed from \(\overrightarrow{ED}\) and \(\overrightarrow{EF}\). Angles can be named using a point on each ray and the vertex, such as angle DEF , or in symbol form \(∠DEF.\)
Greek letters are often used as variables for the measure of an angle. Table \(\PageIndex{1}\) is a list of Greek letters commonly used to represent angles, and a sample angle is shown in Figure \(\PageIndex{3}\) .
| \(θ\) | \(φ \text{ or }ϕ\) | \(α\) | \(β\) | \(γ\) |
| theta | phi | alpha | beta | gamma |
Figure \(\PageIndex{3}\) : Angle theta, shown as \(∠θ\)
Angle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the initial side , and the rotated ray is the terminal side . In order to identify the different sides, we indicate the rotation with a small arc and arrow close to the vertex as in Figure \(\PageIndex{4}\) .
As we discussed at the beginning of the section, there are many applications for angles, but in order to use them correctly, we must be able to measure them. The measure of an angle is the amount of rotation from the initial side to the terminal side. Probably the most familiar unit of angle measurement is the degree. One degree is \(\frac{1}{360}\) of a circular rotation, so a complete circular rotation contains 360 degrees. An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol °. For example, 90 degrees = 90°.
To formalize our work, we will begin by drawing angles on an x - y coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in standard position if its vertex is located at the origin, and its initial side extends along the positive x -axis. See Figure \(\PageIndex{5}\) .
If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle . If the angle is measured in a clockwise direction, the angle is said to be a negative angle .
Drawing an angle in standard position always starts the same way—draw the initial side along the positive x -axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do that by dividing the angle measure in degrees by 360°. For example, to draw a 90° angle, we calculate that \(\frac{90°}{360°}=\frac{1}{4}\). So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive x -axis. To draw a 360° angle, we calculate that \(\frac{360°}{360°}=1\). So the terminal side will be 1 complete rotation around the circle, moving counterclockwise from the positive x -axis. In this case, the initial side and the terminal side overlap. See Figure \(\PageIndex{6}\) .
Since we define an angle in standard position by its initial side, we have a special type of angle whose terminal side lies on an axis, a quadrantal angle . This type of angle can have a measure of 0°, 90°, 180°, 270° or 360°. See Figure \(\PageIndex{7}\) .
QUADRANTAL ANGLES
Quadrantal angles are angles in standard position whose terminal side lies on an axis, including 0°, 90°, 180°, 270°, or 360°.
Given an angle measure in degrees, draw the angle in standard position
- Express the angle measure as a fraction of 360°.
- Reduce the fraction to simplest form.
- Draw an angle that contains that same fraction of the circle, beginning on the positive x -axis and moving counterclockwise for positive angles and clockwise for negative angles.
Example \(\PageIndex{1}\): Drawing an Angle in Standard Position Measured in Degrees
- Sketch an angle of 30° in standard position.
- Sketch an angle of −135° in standard position.
Solution
-
Divide the angle measure by 360°.
\[\dfrac{30°}{360°}=\dfrac{1}{12}\]
To rewrite the fraction in a more familiar fraction, we can recognize that
\[\dfrac{1}{12}=\dfrac{1}{3}(\frac{1}{4})\]
One-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at 30° as in Figure \(\PageIndex{8}\) .
Figure \(\PageIndex{8}\) -
Divide the angle measure by 360°.
\[\dfrac{−135°}{360°}=−\dfrac{3}{8}\]
In this case, we can recognize that
\[−\dfrac{3}{8}=−\dfrac{3}{2}(\dfrac{1}{4})\]
Negative three-eighths is one and one-half times a quarter, so we place a line by moving clockwise one full quarter and one-half of another quarter, as in Figure \(\PageIndex{9}\) .
Figure \(\PageIndex{9}\)
\(\PageIndex{1}\):
Show an angle of 240° on a circle in standard position.
Solution
Converting Between Degrees and Radians
Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle.
The circumference of a circle is \(C=2πr\). If we divide both sides of this equation by \(r\), we create the ratio of the circumference to the radius, which is always \(2π\) regardless of the length of the radius. So the circumference of any circle is \(2π≈6.28\) times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in Figure \(\PageIndex{11}\) .
This brings us to our new angle measure. One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals \(2π\) times the radius, a full circular rotation is \(2π\) radians. So
\[\begin{align} 2π \text{ radians } & =360^∘ \\ π \text{ radians } & =\dfrac{360^∘}{2}=180^∘ \\ 1 \text{ radian } & =\dfrac{180^∘}{π}≈57.3^∘ \end{align}\]
See Figure \(\PageIndex{12}\) . Note that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel out.
Relating Arc Lengths to Radius
An arc length \(s\) is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius.
This ratio, called the radian measure , is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length s s to the radius \(r\). See Figure \(\PageIndex{13}\) .
\[\begin{align} s &=rθ \\ θ &=\dfrac{s}{r} \end{align}\]
If \( s=r\), then \(θ=\frac{r}{r}= 1 \text{ radian.}\)
To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is \(C=2πr\),where \(r\) is the radius. The smaller circle then has circumference \(2π(2)=4π\) and the larger has circumference \(2π(3)=6π\). Now we draw a 45° angle on the two circles, as in Figure \(\PageIndex{14}\) .
Notice what happens if we find the ratio of the arc length divided by the radius of the circle.
\[\begin{align} \text{Smaller circle: } \dfrac{\frac{1}{2}π}{2} & =\dfrac{1}{4}π \\[2mm] \text{Larger circle: } \dfrac{\frac{3}{4}π}{3} & =\dfrac{1}{4}π \end{align}\]
Since both ratios are \(\frac{1}{4}π\), the angle measures of both circles are the same, even though the arc length and radius differ.
RADIANS
One radian is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution (360°) equals \(2\pi\) radians. A half revolution (180°) is equivalent to \(\pi\) radians.
The radian measure of an angle is the ratio of the length of the arc subtended by the angle to the radius of the circle. In other words, if \(s\) is the length of an arc of a circle, and \(r\) is the radius of the circle, then the central angle containing that arc measures \(\frac{s}{r}\) radians. In a circle of radius 1, the radian measure corresponds to the length of the arc.
A measure of 1 radian looks to be about 60°. Is that correct?
Yes. It is approximately 57.3°. Because \(2π\) radians equals 360°, \(1\) radian equals \(\frac{360°}{2π}≈57.3°\).
Using Radians
Because radian measure is the ratio of two lengths, it is a unitless measure. For example, in Figure \(\PageIndex{14}\) , suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the “inches” cancel, and we have a result without units. Therefore, it is not necessary to write the label “radians” after a radian measure, and if we see an angle that is not labeled with “degrees” or the degree symbol, we can assume that it is a radian measure.
Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360°. We can also track one rotation around a circle by finding the circumference, \(C=2πr\), and for the unit circle \(C=2π.\) These two different ways to rotate around a circle give us a way to convert from degrees to radians.
\[\begin{array}{clll} 1 \text{rotation } & =360° & =2π & \text{radians} \\[2mm] \dfrac{1}{2} \text{rotation } & =180° & =π & \text{radians} \\[2mm] \dfrac{1}{4} \text{rotation } & =90° & =\dfrac{π}{2} & \text{radians} \end{array}\]
Identifying Special Angles Measured in Radians
In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in Figure \(\PageIndex{15}\) . Memorizing these angles will be very useful as we study the properties associated with angles.
Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in Figure \(\PageIndex{15}\) , which are shown in Figure \(\PageIndex{16}\) . Be sure you can verify each of these measures.
Example \(\PageIndex{2}\): Finding a Radian Measure
Find the radian measure of one-third of a full rotation.
Solution
For any circle, the arc length along such a rotation would be one-third of the circumference. We know that
\[1 \text{ rotation}=2πr\]
So,
\[\begin{align}s &= \dfrac{1}{3}(2πr) \\[2mm] &=\dfrac{2πr}{3} \end{align}\]
The radian measure would be the arc length divided by the radius.
\[\begin{align} \text{radian measure} & = \dfrac{\frac{2πr}{3}}{r} \\[2mm] &= \dfrac{2πr}{3r} \\[2mm] & = \dfrac{2π}{3} \end{align}\]
\(\PageIndex{2}\)
Find the radian measure of three-fourths of a full rotation.
Solution
\[\dfrac{3π}{2}\]
Converting between Radians and Degrees
Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion.
\[\dfrac{θ}{180}=\dfrac{θ^R}{π}\]
This proportion shows that the measure of angle \(θ\) in degrees divided by 180 equals the measure of angle \(θ\) in radians divided by \(π.\) Or, phrased another way, degrees is to 180 as radians is to \(π\).
\[\dfrac{\text{Degrees}}{180}=\dfrac{Radians}{π} \]
CONVERTING BETWEEN RADIANS AND DEGREES
To convert between degrees and radians, use the proportion
\[\dfrac{θ}{180}=\dfrac{θ^R}{π}\]
Example \(\PageIndex{3}\): Converting Radians to Degrees
Convert each radian measure to degrees.
- \(\frac{π}{6}\)
- 3
Solution
Because we are given radians and we want degrees, we should set up a proportion and solve it.
-
We use the proportion, substituting the given information.
\[\begin{align}\dfrac{θ}{180} &=\dfrac{θ^R}{π} \\[2mm] \dfrac{θ}{180} & =\dfrac{\frac{π}{6}}{π} \\ θ & =\dfrac{180}{6} \\ θ & =30^∘ \end{align}\]
-
We use the proportion, substituting the given information.
\[\begin{align}\dfrac{θ}{180} &= \dfrac{θ^R}{π} \\ \dfrac{θ}{180} &=\dfrac{3}{π} \\ θ &= \dfrac{3(180)}{π} \\ θ & ≈172^∘ \end{align}\]
\(\PageIndex{3}\)
Convert \(−\frac{3π}{4}\) radians to degrees.
Solution
−135°
Example \(\PageIndex{4}\): Converting Degrees to Radians
Convert \(15\) degrees to radians.
Solution
In this example, we start with degrees and want radians, so we again set up a proportion and solve it, but we substitute the given information into a different part of the proportion.
\[\begin{align}\dfrac{θ}{180} &= \dfrac{θ^R}{π} \\ \dfrac{15}{180} &=\dfrac{θ^R}{π} \\ \dfrac{15π}{180} & =θ^R \\ \dfrac{π}{12} & =θ^R \end{align}\]
Analysis
Another way to think about this problem is by remembering that \(30^∘=\frac{π}{6}\). Because \(15^∘=\frac{1}{2}(30^∘)\), ,we can find that \(\frac{1}{2}(\frac{π}{6})\) is \(\frac{π}{12}\).
\(\PageIndex{4}\)
Convert 126° to radians.
Solution
\(\frac{7π}{10}\)
Finding Coterminal Angles
Converting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of 0° to 360°, or 0 to \(2π\). It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution.
It is possible for more than one angle to have the same terminal side. Look at Figure \(\PageIndex{17}\) . The angle of 140° is a positive angle , measured counterclockwise. The angle of –220° is a negative angle , measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are coterminal angles. Every angle greater than 360° or less than 0° is coterminal with an angle between 0° and 360°, and it is often more convenient to find the coterminal angle within the range of 0° to 360° than to work with an angle that is outside that range.
Any angle has infinitely many coterminal angles because each time we add 360° to that angle—or subtract 360° from it—the resulting value has a terminal side in the same location. For example, 100° and 460° are coterminal for this reason, as is −260°. Recognizing that any angle has infinitely many coterminal angles explains the repetitive shape in the graphs of trigonometric functions.
An angle’s reference angle is the measure of the smallest, positive, acute angle \(t\) formed by the terminal side of the angle \(t\) and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See Figure \(\PageIndex{18}\) for examples of reference angles for angles in different quadrants.
COTERMINAL AND REFERENCE ANGLES
- Coterminal angles are two angles in standard position that have the same terminal side.
- An angle’s reference angle is the size of the smallest acute angle, \(t′\), formed by the terminal side of the angle \(t\) and the horizontal axis.
Given an angle greater than 360°, find a coterminal angle between 0° and 360°
- Subtract 360° from the given angle.
- If the result is still greater than 360°, subtract 360° again till the result is between 0° and 360°.
- The resulting angle is coterminal with the original angle.
Example \(\PageIndex{5}\): Finding an Angle Coterminal with an Angle of Measure Greater Than 360°
Find the least positive angle θ that is coterminal with an angle measuring 800°, where \(0°≤θ<360°\).
Solution
An angle with measure 800° is coterminal with an angle with measure 800 − 360 = 440°, but 440° is still greater than 360°, so we subtract 360° again to find another coterminal angle: 440 − 360 = 80°.
The angle \(θ=80°\) is coterminal with 800°. To put it another way, 800° equals 80° plus two full rotations, as shown in Figure \(\PageIndex{19}\) .
\(\PageIndex{5}\)
Find an angle \(α\) that is coterminal with an angle measuring 870°, where \(0°≤α<360°\).
Solution
\(α=150°\)
Given an angle with measure less than 0°, find a coterminal angle having a measure between 0° and 360°.
- Add 360° to the given angle.
- If the result is still less than 0°, add 360° again until the result is between 0° and 360°.
- The resulting angle is coterminal with the original angle.

Solution
Since 45° is half of 90°, we can start at the positive horizontal axis and measure clockwise half of a 90° angle.
Because we can find coterminal angles by adding or subtracting a full rotation of 360°, we can find a positive coterminal angle here by adding 360°:
\[−45°+360°=315°\]
We can then show the angle on a circle, as in Figure \(\PageIndex{20}\) .
\(\PageIndex{6}\)
Find an angle β that is coterminal with an angle measuring −300° such that \(0°≤β<360°\).
Solution
\(β=60°\)
Finding Coterminal Angles Measured in Radians
We can find coterminal angles measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations.
Given an angle greater than \(2\pi\), find a coterminal angle between 0 and \(2\pi\).
- Subtract \(2π\) from the given angle.
- If the result is still greater than \(2π\), subtract \(2π\) again until the result is between \(0\) and \(2π\).
- The resulting angle is coterminal with the original angle.
Example \(\PageIndex{7}\): Finding Coterminal Angles Using Radians
Find an angle \(β\) that is coterminal with \(\frac{19π}{4}\), where \(0≤β<2π.\)
Solution
When working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of \(2π\) radians:
\[\begin{align} \dfrac{19π}{4}−2π & =\dfrac{19π}{4}−\dfrac{8π}{4} \\ &=\dfrac{11π}{4} \end{align}\]
The angle \(\frac{11π}{4}\) is coterminal, but not less than \(2π\), so we subtract another rotation:
\[\begin{align} \dfrac{11π}{4}−2π &= \dfrac{11π}{4}−\dfrac{8π}{4} \\ &= \dfrac{3π}{4} \end{align}\]
The angle \(\frac{3π}{4}\) is coterminal with \(\frac{19π}{4}\), as shown in Figure \(\PageIndex{21}\) .
\(\PageIndex{7}\)
Find an angle of measure \(θ\) that is coterminal with an angle of measure \(−\frac{17π}{6}\) where \(0≤θ<2π.\)
Solution
\(\frac{7π}{6}\)
Determining the Length of an Arc
Recall that the radian measure \(θ\) of an angle was defined as the ratio of the arc length \(s\) of a circular arc to the radius \(r\) of the circle, \(θ=\frac{s}{r}\). From this relationship, we can find arc length along a circle, given an angle.
ARC LENGTH ON A CIRCLE
In a circle of radius r , the length of an arc \(s\) subtended by an angle with measure \(θ\) in radians, shown in Figure \(\PageIndex{22}\) , is
\[s=rθ\]
Given a circle of radius \(r,\) calculate the length \(s\) of the arc subtended by a given angle of measure \(θ\).
- If necessary, convert \(θ\) to radians.
- Multiply the radius \(r\) by the radian measure of \(θ:s=rθ\).
Example \(\PageIndex{8}\): Finding the Length of an Arc
Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.
- In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?
- Use your answer from part (a) to determine the radian measure for Mercury’s movement in one Earth day.
Solution
-
Let’s begin by finding the circumference of Mercury’s orbit.
\[\begin{align} C &=2πr \\ &=2π(\text{36 million miles}) \\ &≈226 \text{ million miles} \end{align}\]
Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:
\[(0.0114)226 \text{ million miles} = 2.58 \text{ million miles}\]
-
Now, we convert to radians:
\[\begin{align} \text{radian} & = \dfrac{\text{arc length}}{\text{radius}} \\ &= \dfrac{2.58 \text{ million miles}}{36 \text{ million miles}} \\ & =0.0717 \end{align} \]
\(\PageIndex{8}\)
Find the arc length along a circle of radius 10 units subtended by an angle of 215°.
Solution
\[\dfrac{215π}{18}=37.525 \text{ units} \]
Finding the Area of a Sector of a Circle
In addition to arc length, we can also use angles to find the area of a sector of a circle . A sector is a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie. Recall that the area of a circle with radius \(r\) can be found using the formula \(A=πr^2\). If the two radii form an angle of \(θ\), measured in radians, then \(\frac{θ}{2π}\) is the ratio of the angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of the circle. Thus, the area of a sector is the fraction \(\frac{θ}{2π}\) multiplied by the entire area. (Always remember that this formula only applies if \(θ\) is in radians.)
\[\begin{align} \text{Area of sector} & =(\dfrac{θ}{2π})πr^2 \\ &=\dfrac{θπr^2}{2π} \\ & =\dfrac{1}{2}θr^2 \end{align}\]
AREA OF A SECTOR
The area of a sector of a circle with radius \(r\) subtended by an angle \(θ\), measured in radians, is
\[A=\dfrac{1}{2}θr^2\]
Given a circle of radius \(r,\) find the area of a sector defined by a given angle \(θ.\)
- If necessary, convert \(θ\) to radians.
- Multiply half the radian measure of \(θ\) by the square of the radius \(r: A=\frac{1}{2}θr^2.\)
Example \(\PageIndex{9}\): Finding the Area of a Sector
An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in Figure \(\PageIndex{24}\) . What is the area of the sector of grass the sprinkler waters?
Solution
First, we need to convert the angle measure into radians. Because 30 degrees is one of our special angles, we already know the equivalent radian measure, but we can also convert:
\[\begin{align} \text{30 degrees} &=30⋅\frac{π}{180} \\ &=\frac{π}{6} \text{ radians} \end{align}\]
The area of the sector is then
\[\begin{align} \text{Area} &= \dfrac{1}{2}(\dfrac{π}{6})(20)^2 \\ & ≈104.72 \end{align}\]
So the area is about \(\mathrm{104.72 \; ft^2}\).
\(\PageIndex{9}\)
In central pivot irrigation, a large irrigation pipe on wheels rotates around a center point. A farmer has a central pivot system with a radius of 400 meters. If water restrictions only allow her to water 150 thousand square meters a day, what angle should she set the system to cover? Write the answer in radian measure to two decimal places.
Solution
1.88
Use Linear and Angular Speed to Describe Motion on a Circular Path
In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed. Linear speed is speed along a straight path and can be determined by the distance it moves along (its displacement ) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or 10π inches, every second. So the linear speed of the point is \(10π\) in./s. The equation for linear speed is as follows where \(v\) is linear speed, \(s\) is displacement, and \(t\) is time.
\[v=\dfrac{s}{t}\]
Angular speed results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed is angular rotation per unit time. So, for instance, if a gear makes a full rotation every 4 seconds, we can calculate its angular speed as \(\frac{360 \text{ degrees}}{4 \text{ seconds}}= \) 90 degrees per second. Angular speed can be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where \(ω\) (read as omega) is angular speed, \(θ\) is the angle traversed, and \(t\) is time.
\[ω=\dfrac{θ}{t}\]
Combining the definition of angular speed with the arc length equation, \(s=rθ\), we can find a relationship between angular and linear speeds. The angular speed equation can be solved for \(θ\), giving \(θ=ωt.\) Substituting this into the arc length equation gives:
\[\begin{align}s &=rθ \\ &=rωt \end{align}\]
Substituting this into the linear speed equation gives:
\[\begin{align} v & = \dfrac{s}{t} &=\dfrac{rωt}{t} &=rω \end{align}\]
ANGULAR AND LINEAR SPEED
As a point moves along a circle of radius \(r,\) its angular speed , \(ω\), is the angular rotation \(θ\) per unit time, \(t\).
\[ω=\dfrac{θ}{t}\]
The linear speed . \(v\), of the point can be found as the distance traveled, arc length \(s\), per unit time, \(t.\)
\[v=\frac{s}{t}\]
When the angular speed is measured in radians per unit time, linear speed and angular speed are related by the equation
\[v=rω\]
This equation states that the angular speed in radians, \(ω\), representing the amount of rotation occurring in a unit of time, can be multiplied by the radius \(r\) to calculate the total arc length traveled in a unit of time, which is the definition of linear speed.
Given the amount of angle rotation and the time elapsed, calculate the angular speed
- If necessary, convert the angle measure to radians.
- Divide the angle in radians by the number of time units elapsed: \(ω=\frac{θ}{t}.\)
- The resulting speed will be in radians per time unit.
Example \(\PageIndex{10}\): Finding Angular Speed
A water wheel, shown in Figure \(\PageIndex{25}\) , completes 1 rotation every 5 seconds. Find the angular speed in radians per second.
Solution
The wheel completes 1 rotation, or passes through an angle of \(2π\) radians in 5 seconds, so the angular speed would be \(ω=\frac{2π}{5}≈1.257\) radians per second.
\(\PageIndex{10}\)
An old vinyl record is played on a turntable rotating clockwise at a rate of 45 rotations per minute. Find the angular speed in radians per second.
Solution
\(−\frac{3π}{2}\) rad/s
Given the radius of a circle, an angle of rotation, and a length of elapsed time, determine the linear speed
- Convert the total rotation to radians if necessary.
- Divide the total rotation in radians by the elapsed time to find the angular speed: apply \(ω=\frac{θ}{t}\).
- Multiply the angular speed by the length of the radius to find the linear speed, expressed in terms of the length unit used for the radius and the time unit used for the elapsed time: apply \(v=rω\).
Example \(\PageIndex{11}\): Finding a Linear Speed
A bicycle has wheels 28 inches in diameter. A tachometer determines the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is traveling down the road.
Solution
Here, we have an angular speed and need to find the corresponding linear speed, since the linear speed of the outside of the tires is the speed at which the bicycle travels down the road.
We begin by converting from rotations per minute to radians per minute. It can be helpful to utilize the units to make this conversion:
\[\mathrm{180 \dfrac{\cancel{rotations}}{minute}⋅\dfrac{2π \; radians}{\cancel{rotation}}=360π\dfrac{radians}{minute}}\]
Using the formula from above along with the radius of the wheels, we can find the linear speed:
\[\begin{align} v & =(14 \text{ inches})(360π \dfrac{\text{radians}}{\text{minute}}) \\ &=5040π \dfrac{\text{inches}}{\text{minute}} \end{align}\]
Remember that radians are a unitless measure, so it is not necessary to include them.unitless measure, so it is not necessary to include them.
Finally, we may wish to convert this linear speed into a more familiar measurement, like miles per hour.
\[\mathrm{5040π\dfrac{\cancel{inches}}{\cancel{minute}}⋅\dfrac{1 \; \cancel{ feet}}{12 \; \cancel{ inches}}⋅\dfrac{1 \; \cancel{ mile}}{5280 \cancel{ feet}}⋅\dfrac{60 \cancel{ minutes}}{1 \; hour}≈14.99 \; miles \; per \; hour \; (mph)}\]
\(\PageIndex{11}\)
A satellite is rotating around Earth at 0.25 radians per hour at an altitude of 242 km above Earth. If the radius of Earth is 6378 kilometers, find the linear speed of the satellite in kilometers per hour.
Solution
1655 kilometers per hour
Media
Access these online resources for additional instruction and practice with angles, arc length, and areas of sectors.
Key Equations
| arc length | \(s=rθ\) |
| area of a sector | \(A=\frac{1}{2}θr^2\) |
| angular speed | \(ω=\frac{θ}{t}\) |
| linear speed | \(v=\frac{s}{t}\) |
| linear speed related to angular speed | \(v=rω\) |
Key Concepts
- An angle is formed from the union of two rays, by keeping the initial side fixed and rotating the terminal side. The amount of rotation determines the measure of the angle.
- An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x -axis. A positive angle is measured counterclockwise from the initial side and a negative angle is measured clockwise.
- To draw an angle in standard position, draw the initial side along the positive x -axis and then place the terminal side according to the fraction of a full rotation the angle represents. See Example .
- In addition to degrees, the measure of an angle can be described in radians. See Example .
- To convert between degrees and radians, use the proportion \(\frac{θ}{180}=\frac{θ^R}{π}\). See Example and Example .
- Two angles that have the same terminal side are called coterminal angles.
- We can find coterminal angles by adding or subtracting 360° or \(2π\). See Example and Example .
- Coterminal angles can be found using radians just as they are for degrees. See Example .
- The length of a circular arc is a fraction of the circumference of the entire circle. See Example .
- The area of sector is a fraction of the area of the entire circle. See Example .
- An object moving in a circular path has both linear and angular speed.
- The angular speed of an object traveling in a circular path is the measure of the angle through which it turns in a unit of time. See Example .
- The linear speed of an object traveling along a circular path is the distance it travels in a unit of time. See Example .
Glossary
- angle
- the union of two rays having a common endpoint
- angular speed
- the angle through which a rotating object travels in a unit of time
- arc length
- the length of the curve formed by an arc
- area of a sector
- area of a portion of a circle bordered by two radii and the intercepted arc; the fraction \(\frac{θ}{2π}\) multiplied by the area of the entire circle
- coterminal angles
- description of positive and negative angles in standard position sharing the same terminal side
- degree
- a unit of measure describing the size of an angle as one-360th of a full revolution of a circle
- initial side
- the side of an angle from which rotation begins
- linear speed
- the distance along a straight path a rotating object travels in a unit of time; determined by the arc length
- measure of an angle
- the amount of rotation from the initial side to the terminal side
- negative angle
- description of an angle measured clockwise from the positive x -axis
- positive angle
- description of an angle measured counterclockwise from the positive x -axis
- quadrantal angle
- an angle whose terminal side lies on an axis
- radian measure
- the ratio of the arc length formed by an angle divided by the radius of the circle
- radian
- the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle
- ray
- one point on a line and all points extending in one direction from that point; one side of an angle
- reference angle
- the measure of the acute angle formed by the terminal side of the angle and the horizontal axis
- standard position
- the position of an angle having the vertex at the origin and the initial side along the positive x -axis
- terminal side
- the side of an angle at which rotation ends
- vertex
- the common endpoint of two rays that form an angle


11.9: XI Glossary
Clubbing: A gradual enlargement of the fingertips in patients with respiratory conditions that cause chronic hypoxia.
Cyanosis: A bluish discoloration of the skin and mucous membranes caused by lack of oxygenation to the tissues.
Dyspnea: A subjective feeling of not being able to get enough air; also called shortness of breath.
FiO2: Fraction of inspired oxygen (i.e., the concentration of oxygen inhaled). Room air contains 21% oxygen levels, and oxygenation devices can increase the inhaled concentration of oxygen up to 100%. However, FiO2 levels should be decreased as soon as feasible to do so to prevent lung injury.
Hypercapnia: Elevated carbon dioxide levels in the blood, indicated by PaCO2 level greater than 45 in an ABG test.
Hypoxemia: Decreased dissolved oxygen in the arterial blood, indicated by a PaO2 level less than 80 mmHg in an ABG test.
Hypoxia: A reduced level of tissue oxygenation.
Obstructive Sleep Apnea (OSA): Characterized by repeated occurrences of complete or partial obstructions of the upper airway during sleep, resulting in apneic episodes.
PaO2: The partial pressure of dissolved oxygen in the blood measured by arterial blood gas samples.
SpO2: An estimated oxygenation level based on the saturation level of hemoglobin measured by a pulse oximeter.
Ventilation: The mechanical movement of air into and out of the lungs.


15.3: Examples of Sensory Terms Easily Defined By Their Word Components
Here are examples of common medical terms related to the sensory systems that can be easily defined by breaking the terms into their word components.
Audiologist
- Break down the medical term into word components: Audi / o / logist
- Label the word components: Audi = WR; o = CV; logist = S
- Define the word components: Audi = hearing; logist = physician who specializes in diagnosing and treating
- Create a final definition of the medical term: Specialist who diagnoses and treats hearing
Ophthalmoscopy
- Break down the medical term into word components: Ophthalm / o / scopy
- Label the word components: Ophthalm = WR; o = CV; scopy = S
- Define the word components: Ophthalm = eye; scopy = process of viewing
- Create a final definition of the medical term: Process of viewing the eye
Tympanoplasty
- Break down the medical term into word components: Tympan / o / plasty
- Label the word components: Tympan = WR; o = CV; plasty = S
- Define the word components: Tympan = tympanic membrane; plasty = surgical repair
- Create a final definition of the medical term: Surgical repair of the tympanic membrane
Interactive Learning Activity: Practice defining and pronouncing sensory system medical terms by breaking them into word parts.
Query \(\PageIndex{1}\)
You can also print these flashcard activities as a Chapter 15 Student Companion Worksheet and check your answers with this Answer Key .


17.8: References
Centers for Disease Control and Prevention. (2022). About adult BMI . www.cdc.gov/healthyweight/as...erpretedAdults
Centers for Disease Control and Prevention. (2022). US adults drink 17 billion binge drinks a year . www.cdc.gov/alcohol/features...-drinking.html
Centers for Disease Control and Prevention. (2023). Healthy weight, nutrition and physical activity. https://www.cdc.gov/healthyweight/index.html
Harriger, J. A., Evans, J. A., Thompson, J. K., & Tylka, T. L. (2022). The dangers of the rabbit hole: Reflections on social media as a portal into a distorted world of edited bodies and eating disorder risk and the role of algorithms. Body Image , 41 , 292–297. https://doi.org/10.1016/j.bodyim.2022.03.007
Hood, W. A. (2020). Nutritional status assessment in adults laboratory medicine. Medscape . emedicine.medscape.com/article/2141861-labs
U.S. Department of Agriculture & U.S. Department of Health and Human Services. (2020). Dietary guidelines for Americans 2020–2025 (9th ed.). https://www.dietaryguidelines.gov/si..._2020-2025.pdf
U.S. Department of Agriculture & U.S. Department of Health and Human Services. (2023, April). Nutrition . www.nutrition.gov/
U.S. Department of Agriculture & U.S. Department of Health and Human Services. (2023). MyPlate . https://www.myplate.gov/
U.S. Department of Agriculture, Agricultural Research Service, Beltsville Human Nutrition Research Center. (2024). FoodData central . [Internet]. https://fdc.nal.usda.gov/


6.2: Overview of Major Religions
6.2.1 Types of Religions
We often break religions into one of two basic types: ethnic and universalizing. Ethnic religions are associated within one group of people. They make little to no effort at proselytizing (converting others), although that possibility may exist. The largest ethnic religion is Hinduism. Judaism is another well-known ethnic religion. Through migration, both of these religions have become dispersed around much of the world, but they are closely tied to their own ethnic groups.
Universalizing religions seek to convert others. For some religions, it is a requirement for practitioners to spend part of their lives in missionary work attempting to convert others.
Another way of dividing religions is into the categories of polytheistic (many gods) and monotheistic (one god). Although the difference between monotheistic religions and polytheistic religions seems unbridgeable, there are religions that have managed to combine elements of polytheism and monotheism into the same religion. For example, in Voudon (Voodoo), entities that had previously represented African gods are recast as Catholic saints, who themselves are semidivine in the Catholic cosmology. Combining two religions to create a new religion is known as syncretism.
6.2.2 A Brief Description of the Major Religions
Christianity
Christianity is a monotheistic religion centered on the life and teachings of Jesus of Nazareth. It dates to some time in the first century AD since the Western world uses the Christian calendar. Christianity began as an offshoot of Judaism, and includes the Hebrew Bible (known to Christians as the Old Testament) as well as the New Testament as its canonized scriptures. It has three main branches: Catholic, Orthodox, and Protestant ( Figure 6.4 ). Catholic and Orthodox Christianity split roughly one thousand years ago, while the Protestant/Catholic Schism began in the sixteenth century. The split between the Orthodox and Catholic hierarchies centered around whose authority in the church was final. The split between Protestantism and Catholicism mostly centered on practices conducted by the Catholic Church that the future Protestants did not believe were suitable for a religious organization.
The three branches of Christianity have their own spatiality, with a great deal of overlap between them. Orthodox Christianity is mostly seen in Russia, Eastern Europe, and Southern Europe with notable examples in Africa (Ethiopia) and in places where large numbers of people from these places have migrated (the United States, Canada). Catholic Christianity is seen in a wider range of places. It largely formed around the historic Roman Empire, then spread to the north and west of Europe. Catholicism did not stop there, however. The age of colonial expansion transplanted Catholicism to such widespread places as the Philippines, much of the Americas and Caribbean, and large parts of Africa. Protestantism is the most recently developed Christian branch, but it has also diffused widely. The initial Protestant countries were in northern Europe, but again due to colonialism, Protestant Christianity was exported to places like the United States, South Africa, Ghana, and New Zealand. The current expansion of Christianity, particularly in Asia, is largely due to the growth of Protestantism.
Each Christian branch has developed a distinct appearance in the landscape. Orthodox churches are meant to invoke a sense of the divine. Buildings are elaborate, both inside and outside. Catholic churches also tend to be elaborate, in asimilar vein to Orthodox churches, but with a different architectural tradition. This is understandable due to the fact that these two branches of Christianity arose in different places with different ideas of architectural grandeur and beauty. Protestant churches as a collective are less elaborate that their close relatives. This is a reflection of the early history of Protestant churches, which were often specific rejections of the elaborate ceremony and ostentatious display of the Catholic Church.
Islam
Islam is a monotheistic religion originating with the teachings of Muhammad (570-632), an Arab religious and political figure. The word Islam means “submission”, or the total surrender of oneself to God. An adherent of Islam is known as a Muslim, meaning “one who submits (to God).” Both Islam and Christianity inherited the idea of the chain of prophecy from Judaism. This means that figures such as Moses (Judaism) and Jesus (Christianity) are considered prophets in Islam. Muslims believe that Muhammad is the very last in that chain of prophecy. Islam has two main branches, and many smaller ones. Of the two main branches -Sunni and Shi’a, Sunni is much larger, comprising roughly 80% of all Muslims ( Figure 6.5 ). The split between the two largest branches of Islam centered around the question of succession, that is to say, who would be the rightful leader of the Muslim world. Currently, there is no single voice for the global Muslim community. Other forms of Islam include Sufi (mystical) Islam and Ahmadi Islam. India is the number three Muslim country, but there are five times as many Hindus in India as there are Muslims.
The Muslim world is somewhat more contiguous than the Christian world. This is mostly due to the fact that the Muslim expansion did not occur in two phases in the same way that Christianity did. As can be seen in the following map, Sunni and Shi’a countries are somewhat spatially separated. Only the countries of Iran, Iraq, Azerbaijan, and Bahrain are majority Shi’a. There are sizable minority Muslim sects in the world. Many of these groups, such as the Ahmadiyya, are subject to discrimination by other Muslim populations and/or governments. The world’s most theocratic governments are Muslim, particularly those of Iran and Saudi Arabia. This is notable in that these two countries are also regional rivals and the two most powerful states in the Muslim world.
Buddhism
Buddhism is an offshoot of Hinduism that dates to the fifth century BCE. It was founded by Siddhartha Gautama near the modern border between Nepal and India. The three largest branches of Buddhism are Theravada, Mahayana and Vajrayana ( Figure 6.6 ). The main differences between the branches are their approaches to canonized doctrine .
Hinduism
Hinduism is a religious tradition that originated in the Indian subcontinent ( Figure 6.7 ). Its origins can be traced to the ancient Vedic civilization (1500 BCE), a product of the invasion of Indo-European peoples. A conglomerate of diverse beliefs and traditions that assembled organically over a period of centuries, Hinduism has no single founder. Due to its concurrent growth with Indian civilization, Hinduism has historically been tightly bound to the caste system, although the modern Indian state has worked to ameliorate the more damaging effects of this relationship.
Chinese Religions
Not strictly located in China, Chinese religions are closely tied to Daoism (a nature religion), Confucianism (a philosophy of living), and ancestor worship. Chinese religious structures are associated with people of Chinese descent within and external to China. Because of the diversity of religious practices and beliefs, this category is best thought of as a complex of beliefs, rather than a defined set of beliefs and practices.
Sikhism
Sikhism is a 15th Century amalgamation of Islam and Hinduism. It is in many ways emblematic of syncretic religions. Syncretic religions are created by the combination of two or more religions, with the addition of doctrinal elements to create cohesion between the disparate pieces. Founded by Nanak Dev Ji (1469 – 1539) Sikhism reconciles Hinduism and Islam by recasting Hindu gods as aspects of a single god, in a manner similar to the Catholic Trinity. Although heavily associated with the Punjab region of the Indian subcontinent, Sikhism has spread widely through relocation diffusion ( Figure 6.8 ). It has about 26 million adherents.
Judaism
Judaism is a monotheistic religion originating in the Bronze Age in the eastern Mediterranean ( Figure 6.9 ). Although it has no single founder, it holds the Torah as its holy book. In the modern context of Judaism, there are three major forms—Orthodox, Conservative and Reform—each with their own set of interpretations of correct practice. Judaism, as the initial Abrahamic religion, influenced other religions (particularly Christianity and Islam).
Animism, Jainism, Bahai, Shinto and Others
This catch-all category combines together religions that are all quite different. They are here due to their similar ties to places or ethnicities, not because they share any doctrinal or historical connections. Before continuing on a discussion of the following religions, it is important to make a point clear. It is possible to practice more than one religion. Many people in the world practice two or more religions with no sense of contradiction. In many parts of the world, pre-Christian or pre-Islamic beliefs persist alongside the newer religions.
Animism is a broad category, found in a variety of environments ( Figure 6.10 ). The underlying theme is the idea that almost anything in the environmentpeople, mountains, rivers, rain, etc. is alive and worthy of recognition as such. Animism is frequently practiced with other ideologies or philosophies.
Baha’i was founded by Mirza Husayn Ali Nuri (1817-1892) in 1863. Baha’i was an offshoot of another religion, Babism, that in itself was a derivative of Islam. Although traditional Muslims believe that Muhammad was the last of the prophets (the seal of the prophets) many religions have been founded on the idea that there could be other, later people who also spoke for god. Baha’is believe that new messengers would be sent to humanity to remind people of their universal relationship to god and one another. The late date and historic context of this religion informed a religion that explicitly rejected racism and nationalism. One of the notable characteristics is that although Baha’is are not one of the larger religions on Earth, they have a temple on every permanently inhabited continent.
Jainism is another ancient religion that arose in India. It is best known for its concept of ahimsa , or nonviolence.
Shinto , the ethnic religion of Japan is often practiced in conjunction with Buddhism. It is polytheistic and dates back centuries. The most important consideration of Shinto is that the rituals are so ingrained in Japanese national identity that the religion can either be considered vibrant and relevant, or moribund and ritualistic, depending on the perspective of the viewer.
Syncretic Religions
Syncretic religions are formed by the combination of two or more existing religions to produce a new religion ( Figure 6.11 ). Some of the larger syncretic religions have already been mentioned, such as Baha’i or Sikhism.
Cao Dai is a religion founded in twentieth century Vietnam. It has strains of Taoism and Buddhism and represents an attempt to reconcile many diverse religious traditions into a single religion.
Voodoo arose in French Caribbean colonies as a combination of Catholicism and the beliefs of another set of West African peoples, the Ewe and the Fon. Practitioners speak to God using intercessors called loa that function as saints do in both the Catholic and Sufi worlds.
Candomble is a syncretism formed from many West African religious traditions and Catholicism. It has existed in Brazil for centuries. It believes in a creator god (Oludumare) and a series of demi-gods (Orishas).
New Religions
Much like any other human phenomena, new religions are formed continuously. They are usually adaptations or combinations of existing religions. It is impossible to list the most recent arrivals. This category includes such religions as Scientology, the Unification Church, Seicho no Ie and Wicca.
What about the Nonreligious?
Sometimes the nonreligious are considered a religion unto themselves. This is generally not true. The nonreligious category is amorphous. There are no documents of beliefs that all nonreligious people must abide by. There is no overarching nonreligious creed. It is another catch-all category that contains a large, diverse population with divergent beliefs and practices. Within these categories, however, there are notable manifestations. First, there are those places which are officially atheistic or non-religious. This label is problematic. It provides only the perspective of the government of these places. In many places that officially have no religions, practitioners simply do not advertise their religious affiliations. In other places, religious attendance has declined to a point that many people have no connection to a particular religious tradition. The label agnostic refers to the idea that the existence of a higher power is unknowable. It is important to point out that religions do not necessarily require the existence of a god-like force. Daoism relies on nature as its driving force.


37.5: How Hormones Work - Plasma Membrane Hormone Receptors
- Describe the events that occur when a hormone binds to a plasma hormone receptor
Plasma Membrane Hormone Receptors
Amino acid-derived hormones and polypeptide hormones are not lipid-derived (lipid-soluble or fat-soluble); therefore, they cannot diffuse through the plasma membrane of cells. Lipid-insoluble hormones bind to receptors on the outer surface of the plasma membrane, via plasma membrane hormone receptors. Unlike steroid hormones, lipid-insoluble hormones do not directly affect the target cell because they cannot enter the cell and act directly on DNA. Binding of these hormones to a cell surface receptor results in activation of a signaling pathway; this triggers intracellular activity to carry out the specific effects associated with the hormone. In this way, nothing passes through the cell membrane; the hormone that binds at the surface remains at the surface of the cell while the intracellular product remains inside the cell. The hormone that initiates the signaling pathway is called a first messenger, which activates a second messenger in the cytoplasm.
One very important second messenger is cyclic adenosine monophosphate (cAMP). When a hormone binds to its membrane receptor, a G protein that is associated with the receptor is activated. G proteins are proteins separate from receptors that are found in the cell membrane. When a hormone is not bound to the receptor, the G protein is inactive and is bound to guanosine diphosphate, or GDP. When a hormone binds to the receptor, the G protein is activated by binding guanosine triphosphate, or GTP, in place of GDP. After binding, GTP is hydrolyzed by the G protein into GDP and becomes inactive.
The activated G protein in turn activates a membrane-bound enzyme called adenylyl cyclase. Adenylyl cyclase catalyzes the conversion of ATP to cAMP. cAMP, in turn, activates a group of proteins called protein kinases, which transfer a phosphate group from ATP to a substrate molecule in a process called phosphorylation. The phosphorylation of a substrate molecule changes its structural orientation, thereby activating it. These activated molecules can then mediate changes in cellular processes.
The effect of a hormone is amplified as the signaling pathway progresses. The binding of a hormone at a single receptor causes the activation of many G-proteins, which activates adenylyl cyclase. Each molecule of adenylyl cyclase then triggers the formation of many molecules of cAMP. Further amplification occurs as protein kinases, once activated by cAMP, can catalyze many reactions. In this way, a small amount of hormone can trigger the formation of a large amount of cellular product. To stop hormone activity, cAMP is deactivated by the cytoplasmic enzyme phosphodiesterase, or PDE. PDE is always present in the cell, breaking down cAMP to control hormone activity; thus, preventing overproduction of cellular products.
The specific response of a cell to a lipid-insoluble hormone depends on the type of receptors that are present on the cell membrane and the substrate molecules present in the cell cytoplasm. Cellular responses to hormone binding of a receptor include altering membrane permeability and metabolic pathways, stimulating synthesis of proteins and enzymes, and activating hormone release.
Contributions and Attributions
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Key Points
- When a lipid (fat) insoluble hormone binds to a plasma membrane hormone receptor, this triggers specific actions inside the cell that alter the cell’s activities, such as gene expression.
- Because the first event in this sequence is the binding of the hormone to the plasma membrane receptor, the hormone is called the “first messenger”, while the molecule that is activated within the cell and carries out intracellular change is called the ” second messenger “.
- In many cases, a hormone binding to a plasma membrane receptor activates a special kind of protein called a G protein, which in turn activates an enzyme that generates cAMP, a second messenger.
- cAMP activates another group of proteins called protein kinases, which can change the structure of other molecules by adding a phosphate group to them; these activated molecules can then affect changes within the cell.
Key Terms
- second messenger : any substance used to transmit a signal within a cell, especially one which triggers a cascade of events by activating cellular components
- cyclic adenosine monophosphate : cAMP, a second messenger derived from ATP that is involved in the activation of protein kinases and regulates the effects of adrenaline
- G protein : any of a class of proteins, found in cell membranes, that pass signals between hormone receptors and effector enzymes


1.7: The Magic Feather
THE MAGIC FEATHER.
N the depths of the forest in the land of the Dacotahs stood a wigwam many leagues distant from any other. The old man who had been known to live in it was supposed to have died; but he kept himself in hiding for the sake of his little grandson, whose mother had brought him there to escape the giants.
The Dacotahs had once been a brave and mighty people. They were swift runners and proud of their fleetness. It had been told among the nations for many generations that a great chief should spring from this tribe, and that he should conquer all his enemies, even the giants who had made themselves strong by eating the flesh of those they took in battle and drinking their blood. This great chief should wear a white feather and should be known by its name.
The giants believed the story and sought to prevent it coming true. So they said to the Dacotahs: “Let us run a race. If you win you shall have our sons and our daughters to do with them as you please, and if we win we will take yours.”
Some of the wise Indians shook their heads and said: “Suppose the giants win; they will kill our children and will serve them as dainty food upon their tables.” But the young men answered: “Kaw: who can outrun the Dacotahs? We shall return from the race with the young giants bound hand and foot, to fetch and carry for us all our days.” So they agreed to the wager and ran with the giants.
Now, it was not to be supposed that the giants would act fairly. They dug pitfalls on the prairie, covering them with leaves and grass, which caused the runners to stumble, and lose the race.
The Dacotahs, therefore, had to bring out their children and give them to the giants. When they were counted one child was missing. The giants roared with anger and made the whole tribe search for him, but he could not be found. Then the giants killed the father instead and ate his flesh, grumbling and muttering vengeance with every mouthful.
This was the child whose home was in the forest. When he was still a very little fellow his grandfather made him a tiny bow and some smooth, light arrows, and taught him how to use them.
The first time he ventured from the lodge he brought home a rabbit, the second time a squirrel, and he shot a fine, large deer long before he was strong enough to drag it home.
 You do not yet wear it, but you are worthy of it.”
He looked about, but at first saw no one. At last he caught sight of the head of a little old man among the trees. On going up to it he discovered that the body from the heart downwards was wood and fast in the earth. He thought some hunter must have leaped upon a rotten stump and, it giving way, had caught and held him fast; but he soon recognized the roots of an old oak that he well knew. Its top had been blighted by a stroke of lightning, and the lower branches were so dark that no birds built their nests on them, and few even lighted upon them.
The boy knew nothing of the world except what his grandfather had taught him. He had once found some lodge poles on the edge of the forest and a heap of ashes like those about their own wigwam, by which he guessed that there were other people living. He had never been told why he was living with an old man so far away from others, or of his father, but the time had come for him to know these things.
The head which had called him, said as he came near: “Go home, White Feather, and lie down to sleep. You will dream, and on waking will find a pipe, a pouch of smoking mixture, and a long white feather beside you. Put the feather on your head, and as you smoke you will see the cloud which rises from your pipe pass out of the doorway as a flock of pigeons.” The voice then told him who he was, and also that the giants had never given up looking for him. He was to wait for them no longer, but to go boldly to their lodge and offer to race with them. “Here,” said the voice, “is an enchanted vine which you are to throw over the head of every one who runs with you.”
White Feather, as he was thenceforth called, picked up the vine, went quickly home and did as he had been told. He heard the voice, awoke and found the pouch of tobacco, the pipe, and the white feather. Placing the feather on his head, he filled the pipe and sat down to smoke.
His grandfather, who was at work not far from the wigwam, was astonished to see flocks of pigeons flying over his head, and still more surprised to find that they came from his own doorway. When he went in and saw the boy wearing the white feather, he knew what it all meant and became very sad, for he loved the boy so much that he could not bear the thought of losing him.
The next morning White Feather went in search of the giants. He passed through the forest, out upon the prairie and through other woods across another prairie, until at last he saw a tall lodge pole in the middle of the forest. He went boldly up to it, thinking to surprise the giants, but his coming was not unexpected, for the little spirits which carry the news had heard the voice speaking to him and had hastened to tell those whom it most concerned.
The giants were six brothers who lived in a lodge that was ill-kept and dirty. When they saw the boy coming they made fun of him among themselves; but when he entered the lodge they pretended that they were glad to see him and flattered him, telling him that his fame as a brave had already reached them.
White Feather knew well what they wanted. He proposed the race; and though this was just what they had intended doing, they laughed at his offer. At last they said that if he would have it so, he should try first with the smallest and weakest of their number.
They were to run towards the east until they came to a certain tree which had been stripped of its bark, and then back to the starting point, where a war-club made of iron was driven into the ground. Whoever reached this first was to beat the other’s brains out with it.
White Feather and the youngest giant ran nimbly on, and the giants, who were watching, were rejoiced to see their brother gain slowly but surely, and at last shoot ahead of White Feather. When his enemy was almost at the goal, the boy, who was only a few feet behind, threw the enchanted vine over the giant’s head, which caused him to fall back helpless. No one suspected anything more than an accident, for the vine could not be seen except by him who carried it.
After White Feather had cut off the giant’s head, the brothers thought to get the better of him, and begged him to leave the head with them, for they thought that by magic they might bring it back to life, but he claimed his right to take it home to his grandfather.
The next morning he returned to run with the second giant, whom he defeated in the same manner; the third morning the third, and so on until all but one were killed.
As he went towards the giant’s lodge on the sixth morning he heard the voice of the old man of the oak tree who had first appeared to him. It came to warn him. It told him that the sixth giant was afraid to race with him, and would therefore try to deceive him and work enchantment on him. As he went through the wood he would meet a beautiful woman, the most beautiful in the world. To avoid danger he must wish himself an elk and he would be changed into that animal. Even then he must keep out of her way, for she meant to do him harm.
White Feather had not gone far from the tree when he met her. He had never seen a woman before, and this one was so beautiful that he wished himself an elk at once for he was sure she would bewitch him. He could not tear himself away from the spot, however, but kept browsing near her, raising his eyes now and then to look at her.
She went to him, laid her hand upon his neck and stroked his sides.  “For,” said she, “I heard of you in a distant land, and, though many sought me, I came hither to be your wife.”
As White Feather looked at her he saw tears shining in her eyes, and almost before he knew it he wished himself a man again. In a moment he was restored to his natural shape, and the woman flung her arms about his neck and kissed him.
By and by she coaxed him to lie down on the ground and put his head on her lap.  Then she took an ax and broke his back. This done, she changed herself into the giant, turned White Feather into a dog, and bade him follow to the lodge.
The giant took the white feather and placed it on his own head, for he knew there was magic in it; and he wished to make the tribes honor him as the great warrior they had long expected.
In a little village but a woman’s journey from the home of the giants lived a chief named Red Wing. He had two daughters, White Weasel and Crystal Stone, each noted for her beauty and haughtiness, though Crystal Stone was kind to every one but her lovers, who came from far and near, and were a constant source of jealousy to White Weasel, the elder. The eldest of the giants was White Weasel’s suitor, but she was afraid of him, so both the sisters remained unmarried.
When the news of White Feather’s race with the giants came to the village, each of the maidens determined that she would win the young brave for a husband. White Weasel wanted some one who would be a great chief and make all the tribes afraid of him. Crystal Stone loved him beforehand, for she knew he must be good as well as brave, else the white feather would not have been given to him. Each kept the wish to herself and went into the woods to fast, that it might come true.
When they heard that White Feather was on his way through the forest, White Weasel set her lodge in order and dressed herself gaily, hoping thereby to attract his attention. Her sister made no such preparation, for she thought so brave and wise a chief would have too good sense to take notice of a woman’s finery.
When the giant passed through the forest, White Weasel went out and invited him into her lodge. He entered and she did not guess that it was the giant of whom she had been in such fear.
 Now, although the dog was enchanted and could not change his condition, he still had more than human sense and knew all the thoughts of his mistress. He grew to love her more and more every day and looked about for some way to show it.
One day when the giant was hunting on the prairie, the dog went out to hunt also; but he ran down to the bank of the river. He stepped cautiously into the water and drew out a large stone, which was turned into a beaver as soon as it touched the ground. He took it home to his mistress, who showed it to her sister and offered to share it with her. White Weasel refused it, but told her husband he had better follow the dog and discover where such fine beavers could be had.
The giant went, and hiding behind a tree, saw the dog draw out a stone, which turned into a beaver. After the animal had gone home he went down to the water and drew out a stone, which likewise turned into a beaver. He tied it to his belt and took it home, throwing it down at the door of the lodge.
When he had been at home a little while, he told his wife to go and bring in his belt. She did so, but there was no beaver tied to it, only a large, smooth stone such as he had drawn out of the water.
The dog, knowing that he had been watched, would not go for more beavers; but the next day went through the woods until he came to a charred tree. He broke off a small branch, which turned into a bear as soon as he took hold of it to carry it home.
Then White Weasel became very angry and scoffed at her husband, asking him if this was the way he had done the wonderful things that had made his fame.
The next day, after the giant had gone out, she went to the village to tell her father, Red Wing, how badly her husband treated her in not bringing home food. She also told him that her sister, who had taken the dog into her wigwam, always had plenty to eat, and that Crystal Stone pitied the wife of the wearer of the white feather, who often had to go hungry.
Red Wing listened to her story and knew at once that there must be magic at work somewhere. He sent a company of young men and women to the lodge of Crystal Stone to see if White Weasel’s story were true, and if so to bring his younger daughter and the dog to his wigwam.
Meanwhile the dog had asked his mistress to give him a bath such as the Indians take. They went down to the river, where he pointed out a spot on which she was to build him a lodge. She made it of grass and sticks, and after heating some large stones laid them on the floor, leaving only just enough room for the dog to crawl in and lie down. Then she poured water on the stones, which caused a thick steam that almost choked him. He lay in it for a long time, after which, raising himself, he rushed out and jumped into a pool of water formed by the river. He came out a tall, handsome man, but without the power of speech.
The messengers from Red Wing were greatly astonished at finding a man instead of the dog that they had expected to see, but had no trouble in persuading him and Crystal Stone to go with them.
Red Wing was as much astonished as his messengers had been, and called all the wise men of the tribe to witness what should take place, and to give counsel concerning his daughters.
The whole tribe and many strangers soon assembled. The giant came also and brought with him the magic pipe that had been given to White Feather in his dream. He smoked it and passed it to the Indians to smoke, but nothing came of it. Then White Feather motioned to them that he wished to take it. He also asked for the white feather, which he placed on his head; when, at
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“Lo! Clouds of blue and white pigeons rushed from the smoke.”
the first whiff from the pipe, lo! clouds of blue and white pigeons rushed from the smoke.
The men sprang to their feet, astonished to see such magic. White Feather’s speech returned, and in answer to the questions put to him, he told his story to the chief.
Red Wing and the council listened and smoked for a time in silence. Then the oldest and wisest brave ordered the giant to appear before White Feather, who should transform him into a dog. White Feather accomplished this by knocking upon him the ashes from the magic pipe. It was next decreed that the boys of the tribe should take the war-clubs of their fathers and, driving the animal into the forest, beat him to death.
White Feather wished to reward his friends, so he invited them to a buffalo hunt, to take place in four days’ time, and he bade them prepare many arrows. To make ready for them, he cut a buffalo robe into strips, which he sowed upon the prairie.
On the day appointed the warriors found that these shreds of skin had grown into a large herd of buffaloes. They killed as many as they pleased, for White Feather tipped each arrow with magic, so that none missed their aim.
A grand feast followed in honor of White Feather’s triumph over the giants and of his marriage with Crystal Stone.


21.6.4: Graph a Line given its Equation
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Learning Outcomes
- Identify the slope and y-intercept from the equation of a line.
- Plot the y-intercept of a line given its equation.
- Plot a second point on a line given the y-intercept and the slope.
- Graph a line given its equation in slope y-intercept form.
Often we are given an equation of a line and we want to visualize it. For this reason, it is important to be able to graph a line given its equation. We will look at lines that are in slope intercept form: \(y=a + bx\) where \(a\) is the y-intercept of the line and \(b\) is the slope of the line. The y-intercept is the value of \(y\) where the line crosses the y-axis. The slope is the rise over run. If we write the slope as a fraction, then the numerator tells us how far to move up (or down if it is negative) and the denominator tells us how far to the right we need to go. the main application to statistics is in regression analysis which is the study of how to use a line to make a prediction about one variable based on the value of the other variable.
Example \(\PageIndex{1}\)
Graph the line given by the equation:
\[y=1+\frac{3}{2}x \nonumber\]
Solution
We follow the three step process:
Step 1: Plot the y-intercept
The y-intercept is the number that is not associated with the \(x\). For this example, it is 1. The x-coordinate of the y-intercept is always 0. So the coordinates of the y-intercept are (0,1). Thus start at the origin and move up 1:
Step 2: Plot the Slope.
The slope of a line is the coefficient of the \(x\) term. Here it is \(\frac{3}{2}\). What this means is that we rise 3 and run to the right 2. Rising 3 from an original y-coordinate of 1 gives a new y-coordinate of 4. Running 2 to the right from an initial x-coordinate of 0gives a new x-coordinate of 2. Thus we next plot the point (2,4).
Step 3: Connect the Dots
The last thing we need to do is connect the dots with a line:
Example \(\PageIndex{2}\)
A study was done to look at the relationship between the weight of a car, \(x\), in tons and its gas mileage in mpg, \(y\). The equation of the regression line was found to be:
\[y=110-70x\]
Graph this line.
Solution
The fist step is to note that the y-intercept is 110, hence the graph goes through the point (0,110). The next step is to see that the slope is -70. We can always put a number over 1 in order to make it a fraction. The slope of \(-\frac{70}{1}\) tells us that \(y\) goes down by 70 if \(x\) goes up by 1. We use this to find the second point. The y-coordinate is: \(110\:-\:70\:=\:40\). The x-coordinate is 1. Thus, a second point is (1,40). We can now plot the two points and connect the dots with a line.
Exercise
The regression line that relates the ounces of beer consumed just before a test, \(x\), and the score on the test, \(y\), is given by
\(y=93-1.2x\)
Graph this line.


4.3: Classifying Chemical Reactions
- Define three common types of chemical reactions (precipitation, acid-base, and combustion)
- Classify chemical reactions as one of these three types given appropriate descriptions or chemical equations
- Identify common acids and bases
- Predict the solubility of common inorganic compounds by using solubility rules
Humans interact with one another in various and complex ways, and we classify these interactions according to common patterns of behavior. When two humans exchange information, we say they are communicating. When they exchange blows with their fists or feet, we say they are fighting. Faced with a wide range of varied interactions between chemical substances, scientists have likewise found it convenient (or even necessary) to classify chemical interactions by identifying common patterns of reactivity. This module will provide an introduction to three of the most prevalent types of chemical reactions: precipitation, acid-base, and oxidation-reduction.
Precipitation Reactions and Solubility Rules
A precipitation reaction is one in which dissolved substances react to form one (or more) solid products. Many reactions of this type involve the exchange of ions between ionic compounds in aqueous solution and are sometimes referred to as double displacement , double replacement , or metathesis reactions. These reactions are common in nature and are responsible for the formation of coral reefs in ocean waters and kidney stones in animals. They are used widely in industry for production of a number of commodity and specialty chemicals. Precipitation reactions also play a central role in many chemical analysis techniques, including spot tests used to identify metal ions and gravimetric methods for determining the composition of matter (see the last module of this chapter).
The extent to which a substance may be dissolved in water, or any solvent, is quantitatively expressed as its solubility , defined as the maximum concentration of a substance that can be achieved under specified conditions. Substances with relatively large solubilities are said to be soluble . A substance will precipitate when solution conditions are such that its concentration exceeds its solubility. Substances with relatively low solubilities are said to be insoluble , and these are the substances that readily precipitate from solution. More information on these important concepts is provided in the text chapter on solutions. For purposes of predicting the identities of solids formed by precipitation reactions, one may simply refer to patterns of solubility that have been observed for many ionic compounds (Table \(\PageIndex{1}\)).
| Always Soluble compounds contain | |
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| Usually Soluble compounds contain | Except if they also contain |
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| Usually Insoluble compounds contain | Exceptions include |
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A vivid example of precipitation is observed when solutions of potassium iodide and lead nitrate are mixed, resulting in the formation of solid lead iodide:
\[\ce{2KI}(aq)+\ce{Pb(NO3)2}(aq)\rightarrow \ce{PbI2}(s)+\ce{2KNO3}(aq)\]
This observation is consistent with the solubility guidelines: The only insoluble compound among all those involved is lead iodide, one of the exceptions to the general solubility of iodide salts.
The net ionic equation representing this reaction is:
\[\ce{Pb^2+}(aq)+\ce{2I-}(aq)\rightarrow \ce{PbI2}(s)\]
Lead iodide is a bright yellow solid that was formerly used as an artist’s pigment known as iodine yellow (Figure \(\PageIndex{1}\)). The properties of pure PbI 2 crystals make them useful for fabrication of X-ray and gamma ray detectors.
The solubility guidelines in Table \(\PageIndex{1}\) may be used to predict whether a precipitation reaction will occur when solutions of soluble ionic compounds are mixed together. One merely needs to identify all the ions present in the solution and then consider if possible cation/anion pairing could result in an insoluble compound. For example, mixing solutions of silver nitrate and sodium fluoride will yield a solution containing Ag + , \(\ce{NO3-}\), Na + , and F − ions. Aside from the two ionic compounds originally present in the solutions, AgNO 3 and NaF, two additional ionic compounds may be derived from this collection of ions: NaNO 3 and AgF. The solubility guidelines indicate all nitrate salts are soluble but that AgF is one of the exceptions to the general solubility of fluoride salts. A precipitation reaction, therefore, is predicted to occur, as described by the following equations:
\[\ce{NaF}(aq)+\ce{AgNO3}(aq)\rightarrow \ce{AgF}(s)+\ce{NaNO3}(aq)\hspace{20px}\ce{(molecular)}\]
Predict the result of mixing reasonably concentrated solutions of the following ionic compounds. If precipitation is expected, write a balanced net ionic equation for the reaction.
- potassium sulfate and barium nitrate
- lithium chloride and silver acetate
- lead nitrate and ammonium carbonate
Solution
(a) The two possible products for this combination are KNO 3 and BaSO 4 . The solubility guidelines indicate BaSO 4 is insoluble, and so a precipitation reaction is expected. The net ionic equation for this reaction, derived in the manner detailed in the previous module, is
(b) The two possible products for this combination are LiC 2 H 3 O 2 and AgCl. The solubility guidelines indicate AgCl is insoluble, and so a precipitation reaction is expected. The net ionic equation for this reaction, derived in the manner detailed in the previous module, is
(c) The two possible products for this combination are PbCO 3 and NH 4 NO 3 . The solubility guidelines indicate PbCO 3 is insoluble, and so a precipitation reaction is expected. The net ionic equation for this reaction, derived in the manner detailed in the previous module, is
\[\ce{Pb^2+}(aq)+\ce{CO3^2-}(aq)\rightarrow \ce{PbCO3}(s) \nonumber\]
Which solution could be used to precipitate the barium ion, Ba 2 + , in a water sample: sodium chloride, sodium hydroxide, or sodium sulfate? What is the formula for the expected precipitate?
- Answer
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sodium sulfate, BaSO 4
Acid-Base Reactions
An acid-base reaction is one in which a hydrogen ion, H + , is transferred from one chemical species to another. Such reactions are of central importance to numerous natural and technological processes, ranging from the chemical transformations that take place within cells and the lakes and oceans, to the industrial-scale production of fertilizers, pharmaceuticals, and other substances essential to society.
For purposes of this brief introduction, we will consider only the more common types of acid-base reactions that take place in aqueous solutions. In this context, an acid is a substance that will dissolve in water to yield hydronium ions, H 3 O + . As an example, consider the equation shown here:
The process represented by this equation confirms that hydrogen chloride is an acid. When dissolved in water, H 3 O + ions are produced by a chemical reaction in which H + ions are transferred from HCl molecules to H 2 O molecules (Figure \(\PageIndex{2}\)).
The nature of HCl is such that its reaction with water as just described is essentially 100% efficient: Virtually every HCl molecule that dissolves in water will undergo this reaction. Acids that completely react in this fashion are called strong acids , and HCl is one among just a handful of common acid compounds that are classified as strong (Table \(\PageIndex{1}\)). A far greater number of compounds behave as weak acids and only partially react with water, leaving a large majority of dissolved molecules in their original form and generating a relatively small amount of hydronium ions. Weak acids are commonly encountered in nature, being the substances partly responsible for the tangy taste of citrus fruits, the stinging sensation of insect bites, and the unpleasant smells associated with body odor. A familiar example of a weak acid is acetic acid, the main ingredient in food vinegars:
When dissolved in water under typical conditions, only about 1% of acetic acid molecules are present in the ionized form, \(\ce{CH3CO2-}\) (Figure \(\PageIndex{3}\)). (The use of a double-arrow in the equation above denotes the partial reaction aspect of this process, a concept addressed fully in the chapters on chemical equilibrium.)
| Compound Formula | Name in Aqueous Solution |
|---|---|
| HBr | hydrobromic acid |
| HCl | hydrochloric acid |
| HNO 3 | nitric acid |
| H 2 SO 4 | sulfuric acid |
A base is a substance that will dissolve in water to yield hydroxide ions, OH − . The most common bases are ionic compounds composed of alkali or alkaline earth metal cations (groups 1 and 2) combined with the hydroxide ion—for example, NaOH and Ca(OH) 2 . When these compounds dissolve in water, hydroxide ions are released directly into the solution. For example, KOH and Ba(OH) 2 dissolve in water and dissociate completely to produce cations (K + and Ba 2 + , respectively) and hydroxide ions, OH − . These bases, along with other hydroxides that completely dissociate in water, are considered strong bases .
Consider as an example the dissolution of lye (sodium hydroxide) in water:
This equation confirms that sodium hydroxide is a base. When dissolved in water, NaOH dissociates to yield Na + and OH − ions. This is also true for any other ionic compound containing hydroxide ions. Since the dissociation process is essentially complete when ionic compounds dissolve in water under typical conditions, NaOH and other ionic hydroxides are all classified as strong bases.
Unlike ionic hydroxides, some compounds produce hydroxide ions when dissolved by chemically reacting with water molecules. In all cases, these compounds react only partially and so are classified as weak bases . These types of compounds are also abundant in nature and important commodities in various technologies. For example, global production of the weak base ammonia is typically well over 100 million metric tons annually, being widely used as an agricultural fertilizer, a raw material for chemical synthesis of other compounds, and an active ingredient in household cleaners (Figure \(\PageIndex{4}\)). When dissolved in water, ammonia reacts partially to yield hydroxide ions, as shown here:
This is, by definition, an acid-base reaction, in this case involving the transfer of H + ions from water molecules to ammonia molecules. Under typical conditions, only about 1% of the dissolved ammonia is present as \(\ce{NH4+}\) ions.
The chemical reactions described in which acids and bases dissolved in water produce hydronium and hydroxide ions, respectively, are, by definition, acid-base reactions. In these reactions, water serves as both a solvent and a reactant. A neutralization reaction is a specific type of acid-base reaction in which the reactants are an acid and a base, the products are often a salt and water, and neither reactant is the water itself:
To illustrate a neutralization reaction, consider what happens when a typical antacid such as milk of magnesia (an aqueous suspension of solid Mg(OH) 2 ) is ingested to ease symptoms associated with excess stomach acid (HCl):
Note that in addition to water, this reaction produces a salt, magnesium chloride.
Write balanced chemical equations for the acid-base reactions described here:
- the weak acid hydrogen hypochlorite reacts with water
- a solution of barium hydroxide is neutralized with a solution of nitric acid
Solution
(a) The two reactants are provided, HOCl and H 2 O. Since the substance is reported to be an acid, its reaction with water will involve the transfer of H + from HOCl to H 2 O to generate hydronium ions, H 3 O + and hypochlorite ions, OCl − .
\[\ce{HOCl}(aq)+\ce{H2O}(l)\rightleftharpoons \ce{OCl-}(aq)+\ce{H3O+}(aq) \nonumber \]
A double-arrow is appropriate in this equation because it indicates the HOCl is a weak acid that has not reacted completely.
(b) The two reactants are provided, Ba(OH) 2 and HNO 3 . Since this is a neutralization reaction, the two products will be water and a salt composed of the cation of the ionic hydroxide (Ba 2 + ) and the anion generated when the acid transfers its hydrogen ion \(\ce{(NO3- )}\).
\[\ce{Ba(OH)2}(aq)+\ce{2HNO3}(aq)\rightarrow \ce{Ba(NO3)2}(aq)+\ce{2H2O}(l) \nonumber \]
Write the net ionic equation representing the neutralization of any strong acid with an ionic hydroxide. (Hint: Consider the ions produced when a strong acid is dissolved in water.)
- Answer
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\[\ce{H3O+}(aq)+\ce{OH-}(aq)\rightarrow \ce{2H2O}(l) \nonumber\]
Explore the microscopic view of strong and weak acids and bases.
Combustion Reactions
Combustion reactions are another common class of reactions. These reactions provide the energy to drive your car, heat your home, and cook your food. Anytime something burns, it is a combustion reaction. These reactions are critical to our economy but they are also responsible for emitting 35 gigatons of CO 2 into the atmosphere in 2020. In a combustion reaction a fuel, like methane, reacts with oxygen to produce carbon dioxide and water. Almost all combustion reactions require additional oxygen, which comes from the atmosphere, with the reactants. In a combustion reaction all the carbon in the reactants is converted into carbon dioxide and all the hydrogen in the reactants is converted into water.
Write balanced chemical equations for the combustion reaction described here:
- the combustion of methane, CH 4
- the combustion of ethane, C 2 H 6
Solution
(a) In addition to methane, CH 4 , the combustion reaction also requires oxygen from the atmosphere. The products of the combustion reaction will be H 2 O and CO 2 .
\[\ce{CH4}(g)+\ce{2O2}(g)\rightarrow \ce{CO2}(g)+\ce{2H2O}(l) \nonumber \]
(b) In addition to ethane, C 2 H 6 , the combustion reaction also requires oxygen from the atmosphere. The products of the combustion reaction will be H 2 O and CO 2 .
\[\ce{2C2H6}(g)+\ce{7O2}(g)\rightarrow \ce{4CO2}(g)+\ce{6H2O}(l) \nonumber \]
Notice that balancing the reaction requires doubling the number of ethane molecules so that the coefficient for oxygen is a whole number.
Write the balanced equation for the combustion of methanol, CH 3 OH
- Answer
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\[\ce{2CH3OH}(l)+\ce{3O2}(g)\rightarrow \ce{2CO2}(g)+\ce{4H2O}(l) \nonumber \]
Summary
Chemical reactions are classified according to similar patterns of behavior. A large number of important reactions are included in three categories: precipitation, acid-base, and oxidation-reduction (redox). Precipitation reactions involve the formation of one or more insoluble products. Acid-base reactions involve the transfer of hydrogen ions between reactants. Redox reactions involve a change in oxidation number for one or more reactant elements. Writing balanced equations for some redox reactions that occur in aqueous solutions is simplified by using a systematic approach called the half-reaction method.
Footnotes
Glossary
- acid
- substance that produces H 3 O + when dissolved in water
- acid-base reaction
- reaction involving the transfer of a hydrogen ion between reactant species
- base
- substance that produces OH − when dissolved in water
- combustion reaction
- vigorous redox reaction producing significant amounts of energy in the form of heat and, sometimes, light
- insoluble
- of relatively low solubility; dissolving only to a slight extent
- neutralization reaction
- reaction between an acid and a base to produce salt and water
- oxidation
- process in which an element’s oxidation number is increased by loss of electrons
- oxidation-reduction reaction
- (also, redox reaction) reaction involving a change in oxidation number for one or more reactant elements
- precipitate
- insoluble product that forms from reaction of soluble reactants
- precipitation reaction
- reaction that produces one or more insoluble products; when reactants are ionic compounds, sometimes called double-displacement or metathesis
- salt
- ionic compound that can be formed by the reaction of an acid with a base that contains a cation and an anion other than hydroxide or oxide
- single-displacement reaction
- (also, replacement) redox reaction involving the oxidation of an elemental substance by an ionic species
- soluble
- of relatively high solubility; dissolving to a relatively large extent
- solubility
- the extent to which a substance may be dissolved in water, or any solvent
- strong acid
- acid that reacts completely when dissolved in water to yield hydronium ions
- strong base
- base that reacts completely when dissolved in water to yield hydroxide ions
- weak acid
- acid that reacts only to a slight extent when dissolved in water to yield hydronium ions
- weak base
- base that reacts only to a slight extent when dissolved in water to yield hydroxide ions
Contributors and Attributions
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Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


7.11: Proficiency Exam
Proficiency Exam
For the following problems, construct a coordinate system and graph the inequality.
\(−6x+4>−14\)
- Answer
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\(x<3\)
\(−8<x+6≤−4\)
- Answer
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\(−14<x≤−10\)
Plot the ordered pairs \((3, 1),(−2, 4),(0, 5),(−2, −2)\).
- Answer
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As accurately as possible, label the coordinates of the points that have been plotted on the graph.
- Answer
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\((0,1),(3,3),(−3,0),(2,−3)\)
What is the geometric structure of the graph of all the solutions to the equation \(2y+3x=−4\)?
- Answer
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straight line
In what form is the linear equation in two variables \(ax+by=c\)?
- Answer
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general form
In what form is the linear equation in two variables \(y=mx+b\)?
- Answer
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slope-intercept
If an ordered pair is a solution to a linear equation in two variables, where does it lie geometrically?
- Answer
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It lies on the line.
Consider the graph of \(y=\dfrac{2}{7}x+16\). If we were to place our pencil at any point on the line and then move it horizontally \(7\) units to the right, how many units and in what direction would we have to move our pencil to get back on the line?
- Answer
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\(2\) units up
For the following problems, find the slope, if it exists, of the line containing the following points.
(−6, −1) and (0, 8)
- Answer
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\(\dfrac{3}{2}\)
(−2, −8) and (−2, 10)
- Answer
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no slope; vertical line at \(x=−2\)
Determine the slope and \(y\)−intercept of the line \(3y+2x+1=0\).
- Answer
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slope = \(-\dfrac{2}{3}\), \(y\)-intercept is \((0, -\dfrac{1}{3})\)
As we look at a graph left to right, do lines with a positive slope rise or decline?
- Answer
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rise
For the following problems, find the equation of the line using the information provided. Write the equation in slope-intercept form
Slope = \(4\), \(y\)-intercept = \(−3\).
- Answer
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\(y=4x−3\)
Slope = \(-\dfrac{3}{2}\), \(y\)-intercept = \(\dfrac{4}{3}\).
- Answer
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\(y = -\dfrac{3}{2}x + \dfrac{4}{3}\)
Slope = \(\dfrac{2}{3}\), passes through \((-1, 2)\)
- Answer
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\(y = \dfrac{2}{3}x + \dfrac{8}{3}\)
Slope = \(7\), passes through \((0, 0)\)
- Answer
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\(y=7x\)
Passes through the points \((5, 2)\) and \((2, 1)\).
- Answer
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\(y = \dfrac{1}{3}x + \dfrac{1}{3}\)
For the following problems, graph the equation of inequality.
\(y = \dfrac{1}{3}x - 2\)
- Answer
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\(y = \dfrac{1}{3}x - 2\)
\(5y−2x+15=0\)
- Answer
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\(4(x+y)=8\)
- Answer
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\(4(x+y)=8\)
\(\dfrac{3}{2}y + 2 = 0\)
- Answer
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\(x=−2\)
- Answer
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\(x=−2\)
\(2x+3y>6\)
- Answer
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Reading only from the graph, determine the equation of the line.
- Answer
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\(y = -\dfrac{1}{3}x + 3\)