Trigonometry: identities, equations, inequalities, inverse functions
Identities first, equations second
The backbone is sin²x + cos²x = 1, tan x = sin x/cos x, the double-angle formulas (sin 2x = 2 sin x cos x; cos 2x = cos²x − sin²x = 1 − 2sin²x = 2cos²x − 1) and the reduction formulas that convert any angle to a first-quadrant reference angle with the correct sign. Know the exact values at 0, π/6, π/4, π/3, π/2 cold.
Equations: reduce to one function of one angle, solve the basic equation, then write the general solution with its period. sin x = a gives x = (−1)ⁿ arcsin a + πn; cos x = a gives x = ±arccos a + 2πn; tan x = a gives x = arctan a + πn. Keep only solutions inside the stated interval and discard any that make tan or a denominator undefined.
Inequalities: solve on the unit circle or over one period, then extend by periodicity; answers are usually unions of intervals.
Inverse functions: arcsin has range [−π/2, π/2], arccos [0, π], arctan (−π/2, π/2). Hence arcsin(sin 5π/6) = π/6, not 5π/6 - a favourite trap.
Worked example. 2sin²x − sin x − 1 = 0 factors as (2 sin x + 1)(sin x − 1) = 0, so sin x = 1 or sin x = −1/2, giving x = π/2 + 2πn or x = (−1)^(k+1)·π/6 + πk.
Never divide an equation by sin x or cos x: you silently delete the solutions where that factor vanishes.