Trigonometric Identities & Equations

Solving Trigonometric Equations

Solving trigonometric equations algebraically over a given interval.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Sum & Difference Identities

How Many Solutions Does a Trig Equation Have?

An equation like x + 5 = 12 has exactly one solution. Before reading on, think about sin(x) = 1/2 on the interval [0, 2π) — since sine is periodic and repeats the same values, could this equation reasonably have more than one solution in just one revolution?

It does — sine reaches 1/2 at two different points within one revolution, since the unit circle's symmetry means two different angles (one in Quadrant I, one in Quadrant II) share the same sine value. And since sine repeats every 2π, there are actually infinitely many solutions across all real numbers, one pair for every additional full revolution.

Worked Example — Solving over a Restricted Interval

Solve sin(x) = 1/2 for x in [0, 2π). The reference angle is π/6 (since sin(π/6) = 1/2). Sine is positive in Quadrants I and II: x = π/6 and x = π − π/6 = 5π/6.

Worked Example — Writing the General Solution

Write the general solution to sin(x) = 1/2 over all real numbers. Add every full revolution, 2πn, to each solution found on [0, 2π): x = π/6 + 2πn or x = 5π/6 + 2πn, for any integer n.

Worked Example — Solving an Equation Requiring an Identity First

Solve 2sin²(x) − sin(x) − 1 = 0 over [0, 2π). Let u = sin(x): 2u² − u − 1 = 0. Factor: (2u + 1)(u − 1) = 0, giving u = −1/2 or u = 1. For sin(x) = 1: x = π/2. For sin(x) = −1/2: reference angle π/6, sine negative in Quadrants III and IV: x = π + π/6 = 7π/6, and x = 2π − π/6 = 11π/6. The full solution set is x = π/2, 7π/6, 11π/6.

Tip

For any sine or cosine equation with a solution in Quadrant I, there's almost always a second solution reflected into an adjacent quadrant by the unit circle's own symmetry — check for it before finalizing an answer over a full interval.

Common Mistakes

Reporting only one solution over an interval like [0, 2π), missing the second solution the unit circle's symmetry produces.
Check both quadrants where the trig function has the needed sign — sine is positive in Quadrants I and II, so a positive sine value almost always has two solutions in one revolution, not just one.
Forgetting to add + 2πn when a general (all real number) solution is requested, instead of a solution restricted to one interval.
A trigonometric function repeats every full period — always add the period's multiple (2πn for sine/cosine, πn for tangent) when a general solution over all real numbers is requested.

Key Takeaways

Trig equations often have multiple solutions within a single period, due to the unit circle's symmetry.
A general solution (over all real numbers) adds the function's period, multiplied by any integer, to each solution found in one period.
An equation with a squared trig function can be solved by substitution, turning it into an ordinary quadratic first.

Summary

Solving trig equations combines algebra with the unit circle's built-in symmetry and periodicity. The next unit turns to a different kind of quantity entirely — vectors, which carry both size and direction.

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