Compound angles and factor formulae

Ratios and small angles

The six ratios are sin, cos, tan and their reciprocals cosec, sec and cot. Two basic identities are sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ. For a small angle θ in radians, sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − θ²/2. To convert, π radians = 180°.

For small angles (θ in radians) the sine curve is almost the straight line y = θ. The graph shows the difference θ − sin θ, which is tiny near 0.

Try itSmall angle approximation

Choose an angle in degrees. Compare sin θ with θ in radians.

Compound angle formulae.

Double angle formulae.

Factor formulae, from the compound angle formulae.

Trigonometric functions and equations

The sine and cosine curves have amplitude 1 and period 360° (2π). The tangent curve has period 180° and vertical asymptotes. A function y = a sin(bx + c) has amplitude a, period 360°/b and a phase shift. To solve an equation, find the principal value, then use the symmetry of the curve to find all solutions in the range.

Worked exampleSolving an equation

Solve 2 sin x = 1 for 0° ≤ x ≤ 360°.

sin x = 1/2, principal value 30°.
Sine is positive in the second quadrant: 180° − 30°.

Answer: x = 30° or 150°.

Amplitude 2 and period 180°.

Check yourself sin 2A =

Check yourself The period of y = sin 3x is

Check yourself For small θ in radians, sin θ ≈

Key terms

Radian: The angle at the centre when the arc equals the radius.
Identity: An equation true for all values.
Principal value: The first solution given by the calculator.
Amplitude: The maximum value of a wave.

Summary

sin²θ + cos²θ = 1; π rad = 180°.
For small θ: sin θ ≈ θ, tan θ ≈ θ.
Use compound, double-angle and factor formulae.
Find all solutions of an equation using the symmetry of the curve.