Trigonometry
Trigonometry
Trigonometry in this paper is mostly equation solving and identity work in radians or degrees as specified, together with the sine and cosine rules and the arc length and sector area formulas. The examiner looks for a complete solution set within the stated interval and for identities applied in one direction only.
Core rules
- Radians: arc length s = rθ and sector area A = ½r²θ, with θ in radians.
- Identities: sin²θ + cos²θ = 1; tanθ = sinθ/cosθ; sec²θ = 1 + tan²θ; cosec²θ = 1 + cot²θ.
- Double angle: sin2θ = 2sinθcosθ; cos2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1.
- The form a sinθ + b cosθ = R sin(θ + α), with R = √(a² + b²) and tanα = b/a, solves equations and locates maxima.
- Sine rule a/sinA = b/sinB; cosine rule a² = b² + c² − 2bc cosA.
Worked example
Solve 2 sin2θ = 1 for 0° ≤ θ ≤ 360°. sin2θ = 0.5, so 2θ = 30° or 150° in the first cycle; because 2θ ranges from 0° to 720°, add 360°: 2θ = 30°, 150°, 390°, 510°. Dividing by 2 gives θ = 15°, 75°, 195°, 255°. Four solutions - doubling the interval is what produces the extra pair.
Traps
- Forgetting to extend the interval for 2θ, losing half the solutions.
- Using degrees inside s = rθ.
- Squaring an equation and introducing extraneous roots that must be checked.
- Choosing the ambiguous case of the sine rule without considering the obtuse possibility.