Sometimes several solutions come from one value. For sin x = 0.3, the principal value is 17.5°, and the second solution is 180° − 17.5° = 162.5°. If the interval is larger, keep adding 360°.
Some equations are quadratics in disguise. 2cos²x − 5cos x + 2 = 0 factorises as (2cos x − 1)(cos x − 2) = 0. cos x = 2 is impossible, so cos x = 1/2, giving 60° and 300°. Identities like sin²x + cos²x = 1 can turn an equation into one trig function.
Worked example
Solve tan x = −1 for −180° ≤ x < 180°.
- tan⁻¹(−1) = −45°.
- Add 180°: 135°. Both are in the interval.
Answer:x = −45° or x = 135°
Key words
- identity
- true for every angle
- quadratic in cos x
- an equation like 2c² − 5c + 2 = 0 with c = cos x
- impossible value
- sin and cos must lie between −1 and 1
Practise this
New questions every time.
Common Core-referenced: HSF-TF.B.7