Trigonometric Functions
Core ideas
Work in radians unless degrees are stated. Know exact values for multiples of π/6, π/4, π/3 and the symmetries of the unit circle. Be able to sketch y = a sin(b(x − c)) + d: amplitude |a|, period 2π/b (π/b for tan), phase shift c, vertical shift d. Solve trigonometric equations over a given interval using the general solution and the symmetry of the circle. Know the Pythagorean identity sin²θ + cos²θ = 1 and the double-angle results sin2θ = 2sinθcosθ and cos2θ = cos²θ − sin²θ. Familiarity with reciprocal functions sec, cosec and cot helps with identities.
Traps
- Losing solutions when dividing by sinθ or cosθ; factorise instead.
- Forgetting the second solution in an interval (e.g. sinθ = 0.5 gives θ = π/6 and 5π/6).
- Using degree mode or mixing degrees and radians in calculus.
- Wrong period when the coefficient of θ is inside the bracket.
- Sign errors in quadrants when using inverse trig.
Worked example
Solve 2sin²θ − sinθ − 1 = 0 for 0 ≤ θ ≤ 2π. Factorise: (2sinθ + 1)(sinθ − 1) = 0. So sinθ = −1/2 or sinθ = 1. For sinθ = −1/2, θ = 7π/6, 11π/6. For sinθ = 1, θ = π/2. Solutions: θ = π/2, 7π/6, 11π/6. Rule of thumb: sketch the unit circle to confirm every solution in the interval before writing the answer.