IB Mathematics (HL)/Trigonometry

Topic 3: Circular functions and trigonometry discusses functions based on the circle, how trigonometric functions are derived, and various trigonometric identities and rules.
3.1 The circle and radian measure.
Section content: "The circle: radian measure of angles; length of an arc; area of a sector"
3.2 Trigonometry as circular functions.
Section content: "Definition of cos θ, sin θ and tan θ in terms of the unit circle; exact values of sin, cos and tan of 0, π/6, π/4, π/3, π/2 and their multiples; definition of the reciprocal trigonometric ratios sec θ, csc θ and cot θ; Pythagorean identities"
Consider a unit circle (circle of radius 1) on a coordinate plane with centre at origin O(0, 0) and a point P(x, y) on its circumference. The angle θ is the angle made by OP from the positive x-axis and the angle α is the angle made by OP from the x-axis (either positive or negative). OP = 1 unit.
Tip: These relationships are not found in the formula booklet. However, you can use ASTC (all, sine, tangent, cosine) to make a phrase to make this relationship with the quadrants easier to remember; for example: "Add sugar to coffee".
A straight line passing through the origin O(0, 0) has the equation formula_18.
Exact values for acute angles are summarised in the following table. Paper 1 uses this knowledge and you may also be asked to use this in Paper 2.
Tip: The exact values of sine and cosine ratios are the reverse of each other, so you only have to remember one set of values. Also tan θ is just (sin θ)/(cos θ).
The exact values for angles greater than π/2 radians can be derived by using the quadrant relationships above.
The reciprocal trigonometric functions are:
The first two of these are included in the formula booklet.
The Pythagorean identities are:
All three of these are included in the formula booklet.
3.3 Compound and double angle identities.
Section content: "Compound angle identities; double angle identities"
The compound angle identities expresses trigonometric ratios of the sum of two angles in terms of ratios of separate angles. These are:
The proofs of these have been explicitly excluded from the syllabus.
The double angle identities can be derived from the compound angle identities. 
All identities in this section are included in the formula booklet.
3.4 Composite trigonometric functions.
Section content: "Composite functions of the form f(x) = a sin(b(x + c)) + d"&lt;br&gt;
Rewind: 2.3 - Transformation of graphs: translations; stretches; reflections in the axes
Consider a composite trigonometric function f where formula_28
Tip: The phase shift c is sometimes confused with bc due to the common perception that b = 1. Examination questions may trick you by expanding the expression in the trigonometric function. For example formula_29, c is not 8 but instead 8 divided by 2, which is 4.
3.5 Inverse trigonometric functions.
Section content: "The inverse functions arcsin x, arccos x, arctan x; their domains and ranges; their graphs"&lt;br&gt;
Rewind: 2.1 - Inverse function, including domain restriction; 2.3 - The graph of the inverse function as a reflection in y = x
3.6 Solutions to trigonometric equations.
Section content: "Algebraic and graphical methods of solving trigonometric equations in a finite interval, including the use of trigonometric identities and factorisation"&lt;br&gt;
Rewind: 2.6 - Solving polynomial equations both graphically and algebraically; use of technology to solve a variety of equations, including those where there is no appropriate analytic approach
Trigonometric equations can be solved through algebraic and graphical means. To solve such an equation algebraically:
To solve graphically, plot the left hand side of the equation as one graph and right hand side of the equation as another graph on the same pair of axes. Make sure that your GDC is on the correct angle mode (degrees or radians) - assume radians unless stated by the exam paper. The intersection points will be the solutions to the equation.
3.7 Sine and cosine rules.
Section content: "The cosine rule; the sine rule including the ambiguous case; area of a triangle as 1/2 ab sin C"
The sine and cosine rules relate the sides and angles in any triangle.
Consider a triangle ABC where a = BC, b = AC and c = AB.
The area of any triangle, given the lengths of any two sides a and b and the included angle C, is: formula_42
These rules are given in the formula booklet.