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°.
Try itSmall angle approximation
Choose an angle in degrees. Compare sin θ with θ in radians.
Compound angles and factor 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°.
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.