Trigonometry
Trigonometry
Trigonometry in this paper covers radian measure, arc length and sector area, the Pythagorean and double-angle identities, solving trigonometric equations over a stated interval, and expressing a sin θ + b cos θ as a single sine or cosine. Equation solving and the harmonic form are the two highest-frequency question types.
Radians and circular measure
π radians = 180°. Arc length s = rθ and sector area = ½r²θ with θ in radians; segment area is sector minus triangle. Convert to radians before using these formulae, and convert back if the question demands degrees.
Core identities
sin²θ + cos²θ = 1; tan θ = sin θ/cos θ; sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1. The last two forms let you turn a quadratic in cos 2θ into a quadratic in sin θ, which is usually the intended route.
The harmonic form
The expression a sin θ + b cos θ equals R sin(θ + α) with R = √(a² + b²) and tan α = b/a, matching sine to sine and cosine to cosine; for R cos(θ − α) use tan α = b/a with the terms in the order a cos θ + b sin θ.
Worked example
Solve 3 sin θ + 4 cos θ = 2 for 0° ≤ θ ≤ 360°. R = √(9 + 16) = 5 and tan α = 4/3, so α = 53.13°. Write 5 cos(θ − 53.13°) = 2, giving cos(θ − 53.13°) = 0.4. Then θ − 53.13° = ±66.42°, so θ = 119.55° or θ = −13.29°. Adding 360° to the negative value gives 346.71°. Both 119.6° and 346.7° lie in the required interval, so there are two solutions.
Traps that cost marks
- Losing the second solution obtained from the negative branch of the inverse cosine.
- Using degrees in s = rθ or ½r²θ.
- Sign errors in the α of the harmonic form, which shifts every solution.
- Dividing by cos θ and silently discarding the cos θ = 0 case.
- Not checking each candidate against the stated interval.