3.8 Trigonometric Equations and Inequalities

Solving trig equations

To solve a trigonometric equation, isolate the trig function, then use inverse trig functions or known values to find solutions. Because trig functions are periodic, equations usually have infinitely many solutions. Add multiples of the period (e.g., 2π for sine/cosine, π for tangent) to any base solution to capture all answers, or restrict to a given interval.

Multiple solutions per period

Sine and cosine often have two solutions within one period due to symmetry (e.g., reference angles in different quadrants). Always check the unit circle or graph to find all solutions in the required interval, not just the one your calculator gives first.

Trig inequalities

Solving trigonometric inequalities means finding intervals where a function is above or below a value. Graph the function (or use the unit circle) to identify where it crosses the boundary value, then test regions between those points to determine where the inequality holds true over the given domain.

Key terms

Why do trig equations have infinitely many solutions?

Because trig functions are periodic; the pattern of solutions repeats every period (2π for sin/cos, π for tan).

General solution for sin(x) = k

Find the base angle(s) in [0, 2π), then add 2nπ (n = integer) to each to capture all solutions.

General solution for tan(x) = k

Find one base angle, then add nπ (n = integer), since tangent has period π.

How to solve a trig inequality

Graph or use the unit circle to find where the function equals the boundary value, then test intervals between those points.

Number of solutions to sin(x) = c (0 < c < 1) on [0, 2π)

Two solutions: one in quadrant I and one in quadrant II (symmetric about π/2).

Try an AP-style question

Solve sin x = 1/2 on [0, 2π).

1. π/6 and 5π/6
2. π/6 only
3. π/3 and 2π/3
4. π/6 and 7π/6