Trigonometric Equation Solver

Type sin(x)=1/2, cos(x)>0, or any trig expression โ€” use the Visual editor for fractions and powers. Equation: sin(x)=1/2 · Inequality: sin(x)>1/2 · Simplify: any trig expression.
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How to Solve Trigonometric Equations

A trigonometric equation contains trig functions of an unknown angle. Unlike identities (true for all angles), trig equations are satisfied only by specific angle values. Because trig functions are periodic, most equations have infinitely many solutions — you find solutions in one period, then express the general solution.

Step 1: Isolate

Isolate the trig function on one side. Convert to a single function if possible using identities.

Step 2: Solve in [0, 2π)

Use inverse trig functions and the unit circle to find all solutions in one period.

Step 3: Generalize

Add the period (2nπ for sin/cos, nπ for tan) to express all infinite solutions.

Step 4: Verify

Substitute solutions back into the original equation. Discard any extraneous roots.

Common Solution Methods

MethodWhen to UseExample
Direct InverseSimple form: sin(x) = k, cos(x) = k, tan(x) = ksin(x) = 1/2 → x = π/6, 5π/6
FactoringQuadratic in trig function or product equals zero2cos²x − cos x − 1 = 0
Identity SubstitutionMultiple trig functions — reduce to one functionsin²x + sin x = 0
Double AngleContains 2x terms alongside x termssin(2x) = cos(x)
Squaring Both SidesMixed functions (check for extraneous!)sin x + cos x = 1
Example: Solve sin(x) = 1/2
1. Reference angle: arcsin(1/2) = π/6 = 30°
2. sin is positive in Q1 and Q2
3. Solutions in [0, 2π): x = π/6 and x = 5π/6
4. General: x = π/6 + 2nπ and x = 5π/6 + 2nπ, n ∈ ℤ  ✓

General Solutions & the “No Solution” Cases

Standard general-solution forms:

sin(x) = k  (|k| ≤ 1):   x = arcsin(k) + 2nπ  or  x = π − arcsin(k) + 2nπ
cos(x) = k  (|k| ≤ 1):   x = ±arccos(k) + 2nπ
tan(x) = k  (any real k):   x = arctan(k) + nπ

Sine and cosine are bounded between −1 and 1, so equations like sin(x) = 2, cos(x) = −3, or csc(x) = 0.5 have no solution. Our solver detects these automatically.

Trig Inequalities & Simplification

Inequalities: solve the corresponding equation to find critical points, test intervals, then write the answer in interval notation with periodicity.

Example: sin(x) > 1/2  →  (π/6 + 2nπ,   5π/6 + 2nπ),   n ∈ ℤ

Simplification: rewrite an expression in a more compact equivalent form using Pythagorean, double-angle, and sum-to-product identities. Example: (sin⁴x − cos⁴x)/(sin²x − cos²x) = 1.

Frequently asked

Yes. Click the green 📷 Scan button next to the expression input, upload a photo or PDF of your homework, and our AI extracts every trig equation along with the mode (equation / inequality / simplify) and angle unit.
Enter the equation like sin(x)=1/2 or 2cos²(x)−1=0. For simple forms, solutions compute instantly; for complex equations the AI solver provides step-by-step work with all solutions in [0, 2π) plus the general solution.
Trig functions are periodic, so equations have infinitely many solutions. We add +2nπ for sine and cosine or +nπ for tangent (n any integer) to capture them all.
Yes — switch to Inequality mode and enter inequalities like sin(x)>1/2 or cos(x)≤0. The solver finds critical points, tests intervals, and writes the solution set in interval notation with periodicity.
Enter any trig expression like (sin⁴x−cos⁴x)/(sin²x−cos²x). The solver applies Pythagorean, double-angle, and other identities step-by-step, then verifies the result by checking a specific angle.
The solver detects impossible equations like sin(x)=2 (sine is bounded between −1 and 1) and impossible inequalities like sin(x)>2, and reports “No Solution” with a range explanation.
Simple forms (sin(x)=k, cos(x)=k, tan(x)=k), quadratic forms (2cos²x−1=0), multi-function equations (sin(2x)=cos(x)), product equations (sinx·cosx=1/2), and any combination using standard trig notation.