1 Geometrical definitions

TrigonometryMathworldPlanetmath arose in ancient times out of attempts to measure the lengths of various lines associated with circles. For example, consider this diagram showing the upper right quadrantMathworldPlanetmath with a unit circle:


The point B, external to the circle with center O, determines a unique secant BO and two tangent linesMathworldPlanetmath of equal length, one of which is BA. It thus also uniquely determines an angle α=AOB. The trigonometric functionsDlmfMathworldPlanetmath sec and tan simply measure the ratio of the lengths of the secant and tangent lines to the radius of the circle.

Thus, in the diagram, tanα=AB=AB:OA. But triangles OAB and ODC are obviously similarMathworldPlanetmathPlanetmath, so tanα is also equal to CD:OD, and we recover the usual current-day definition of tan. Similarly, from the diagram, secα=OB=OB:OA and again, using similarity, we also have secα=OC:OD, which is again the usual modern definition.

The cosecant, csc, is the ratio of the secant to the other leg of the right triangle. Using the standard definitions today for sec and csc, we see that this is correct:


The sine arose in attempts to measure the length of a chord traversing a given angle on a circle. Consider the following diagram:


Here, the arc CE is the circular angle α, which is also CFE, COD, and DOE (recall that a circumferential angle is the same measure as the arc subtended, while a central angle is twice the arc subtended). The length of the chord CE is in fact 2sinα. Presumably because central angles are easier to work with, the common definition was that of the half-chord, and one can see by considering COD that in fact CD is sinα using the modern definitions.

The term “sine” has an interesting history. The Hindus gave the name jiva to the half-chord CD; the Arabs used (or created) the word jiba for this concept. When Robert of Chester, an early translator of al-Khowarizmi’s Algebra, translated this word, he mistook it for a similar Arabic word, jaib, which means “bay” or “inlet”. As a result, he used the Latin term “sinus”, which also means bay or inlet.

To understand how the trigonometric functions are defined today, consider the right triangle ABC below.

Noting that 0<α<90, we define the trigonometric functions sine, cosine, tangent, secant, cosecant and cotangent for angle α respectively as:


We will discuss later how to extend these definitions to a broader set of values for α.

Several identities follow directly from the definitions:

  • tan(α)=sin(α)/cos(α).

  • sec(α)=1/cos(α),csc(α)=1/sin(α),cot(α)=1/tan(α).

  • sin(α)=cos(90-α),tan(α)=cot(90-α),sec(α)=csc(90-α).

The last property follows from the fact that ACB=90-α.

The Pythagorean theoremMathworldPlanetmathPlanetmath states that CA2=AB2+BC2 and thus


It is customary to write (sin(α))n, (cos(α))n, etc. as sinn(α), cosn(α), etc. respectively, so the previous identity is usually written as


and it is known as the Pythagorean identity. Notice that the first three identities let us to express any expression involving trigonometric functions using only sines and cosines, whereas the Pythagorean identity lets us reduce it further, using only sines. This technique is sometimes used when proving trigonometric identities.

Two other identities that can be obtained from the Pythagorean theorem or from the Pythagorean identity are


2 Extending the domain

There are several approaches for extending the domain of the trigonometric functions so they are not restricted to angles between 0 and 90. We could use the angle sum identities below in order to calculate the value of the trigonometric functions outside the given range (for instance, sin(120) could be found as sin(60+60). Some other more formal approaches use power series expansion to define the functionsMathworldPlanetmath for any real value (or even complex!), but we will use a more geometrical approach, using the unit circle.

Consider a unit circle centered at origin of the plane (that is, a circle of radius 1 and center (0,0)), and draw a ray from the center making an angle of α with the horizontal axis, measured counter-clockwise from the positive x-axis. (This ray is the terminal ray of the angle.) Let P be the point of intersection of this line with the circle, and T the intersection with the line x=1. Finally, let us denote by (x,y) the coordinatesMathworldPlanetmathPlanetmath of P and (1,t) the coordinates of T.

Since the circle has radius 1, the distance from origin to P is also 1 and thus


in other words, the coordinates of P are precisely (cosα,sinα).

Now, the two right triangles made with the projections of P and T to the x-axis are similar, so we have


that is, the point t has ordinate equal to tanα.

This formulation makes it much easier to generalize the definitions to arbitrary angles: cosα,sinα are the coordinates of the intersection of the circle with the line through the origin making angle α with the positive x-axis:

You may be wondering now why we put -tanα instead of tanα if the analogy was to be continued. The reason is that we only need sinα and cosα to define all the other trigonometric functions, and given that


we use that identity to extend the domain of tanα. In the particular case of the drawing above, both cosα and sinα are negative, so their quotient must be positive, and that’s why we added a minus sign to tanα.

The geometrical approach lets us easily verify the following relations:

sin(180-α)=sinα, sin(180+α)=-sinα, sin(-α)=-sinα
cos(180-α)=-cosα, cos(180+α)=-cosα, cos(-α)=cosα
tan(180-α)=-tanα, tan(180+α)=tanα, tan(-α)=-tanα

3 Graphs

The graph of sin appears below. Note that the x-axis is scaled not in degrees, but rather in radians.

The graph of cos is below. Note the similarity between the two graphs: the graphs of cos is just like the graph of sin, but shifted by 90 (π/2 radians). This precisely reflects the identity sinα=cos(α-90).

Also notice that you can read off directly from the graphs the facts that


4 Trigonometrical identities

There are literally thousands of trigonometric identities ( Some of the most common (and most useful) are:

Sum and difference of angles

sin(x+y) =sinxcosy+cosxsiny,  sin(x-y) =sinxcosy-cosxsiny,
cos(x+y) =cosxcosy-sinxsiny,  cos(x-y) =cosxcosy+sinxsiny,
tan(x+y) =tanx+tany1-tanxtany,  tan(x-y) =tanx-tany1+tanxtany.

Half and double angles

The double angle formulas are derived directly from the sum of angles formulas above. The half-angle formulas can then be derived from the double angle formulas by substituting x/2 for x and simplifying, using the identity sin2+cos2=1.

sin(x/2)=±1-cosx2, sin(2x) =2sinxcosx,
cos(x/2)=±1+cosx2, cos(2x) =cos2x-sin2x=2cos2x-1=1-2sin2x.
Title trigonometry
Canonical name Trigonometry
Date of creation 2013-03-22 15:05:00
Last modified on 2013-03-22 15:05:00
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 31
Author rm50 (10146)
Entry type Topic
Classification msc 43-00
Classification msc 51-00
Classification msc 42-00
Classification msc 33B10
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Defines sine
Defines cosine
Defines tangent
Defines secant
Defines cosecant
Defines cotangent