Trigonometric Functions: Radians, Identities and Graphs
Core ideas
Advanced trigonometry works in radians: arc length l = rθ, sector area A = ½r²θ, and angular measure linked to the unit circle. You must know exact values in both degree and radian form, the ASTC sign pattern by quadrant, and the symmetry identities such as sin(π − θ) = sin θ and cos(π − θ) = −cos θ.
Graphs of y = sin x, y = cos x and y = tan x must be sketched with amplitude, period and phase shift identified: y = a sin(bx + c) + d has amplitude |a|, period 2π/b and phase shift −c/b. For tan, the period is π/b and asymptotes occur where the argument equals π/2 + kπ.
Identities you must handle: sin²θ + cos²θ = 1, tan θ = sin θ/cos θ, and the double-angle forms sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1.
Traps
- Solving sin θ = k gives two base solutions per revolution; state the general solution using + 2kπ or + kπ before restricting to a given interval.
- Squaring both sides of an equation can introduce extraneous roots - always check in the original.
- Calculator mode: radian versus degree is the commonest lost mark.
- When dividing by cos θ or sin θ, confirm it is not zero first.
Worked rule
Solve 2sin²x − sin x − 1 = 0 for 0 ≤ x ≤ 2π. Factor: (2sin x + 1)(sin x − 1) = 0, so sin x = −½ or sin x = 1. The first gives x = 7π/6, 11π/6; the second gives x = π/2. Four solutions, all checked.
Rule of thumb
Find all solutions over one full period, then add or subtract periods until the required interval is covered.