Trigonometric equations: quadratic and systems
Solve trigonometric equations by finding every angle from 0° to 360°: tan x = −1, quadratic equations in sin x or cos x (one needs an identity first), and a system of two equations in x and y.
- UK: Year 12
- Germany: grade 10
- Iran: grade 12
- Afghanistan: grade 11
- Palestine: grade 10
- Ukraine: grade 10
- Russia: grade 10
Guide
For Afghan Grade 11 (chapter 2: trigonometric equations, quadratic trigonometric equations, systems of two unknowns), Iran Grade 12 and A-level. Place a point on the line for every solution between 0° and 360°; the line is marked every 45° and snaps to 15°.
Find the reference angle from the value, then use the quadrants: sin is positive in the first and second, cos in the first and fourth, tan in the first and third. For tan the solutions repeat every 180°; for sin and cos every 360°.
A quadratic in sin x or cos x factorises like any quadratic: 2sin2x − sin x − 1 = (2sinx + 1)(sin x − 1). If both sin and cos appear, use sin2x = 1 − cos2x so that only one of them is left.
In a system, use the simpler equation to remove one unknown (here y = 90° − x), solve for the other, and then find the first. Check every answer in both equations.