solving trig equations calculator

Interactive Solver

Use this calculator to solve equations of the form f(ax + b) = c, where f is sin, cos, or tan.

Equation: sin(1x + 0) = 0.5

Trig Function

Angle Mode

a (coefficient of x)

b (constant inside function)

c (right side value)

Interval Start (x)

Interval End (x)

Enter values and click Solve Equation to see general and interval solutions.

How this solving trig equations calculator works

Trigonometric equations usually have infinitely many solutions because sine, cosine, and tangent are periodic. This calculator handles that by giving you:

General solutions using an integer parameter n.
Specific solutions inside your chosen interval (for example, 0° to 360°).

The input equation format is: sin(ax + b) = c, cos(ax + b) = c, or tan(ax + b) = c.

What equations you can solve

1) Sine equations

For sin(ax + b) = c, real solutions exist only when -1 ≤ c ≤ 1. The calculator computes both sine branches and then isolates x.

2) Cosine equations

For cos(ax + b) = c, real solutions also require -1 ≤ c ≤ 1. The tool uses the cosine symmetry branches and converts them to x-values.

3) Tangent equations

For tan(ax + b) = c, any real c is allowed. Because tangent repeats every 180° (or π radians), the solution family has one main branch plus the tangent period.

Step-by-step usage

Select sin, cos, or tan.
Choose angle mode: Degrees or Radians.
Enter values for a, b, and c.
Set the interval where you want explicit solutions listed.
Click Solve Equation.

If a = 0, the equation becomes constant (no x inside). The solver correctly reports either no solution or infinitely many solutions.

Example problems you can test

Example A

sin(x) = 1/2 in degrees, interval 0 to 360.

Expected interval answers: 30°, 150°.

Example B

cos(2x - 30) = 0.5 in degrees, interval 0 to 360.

The calculator handles the inside linear expression and gives all valid x-values in the interval.

Example C

tan(3x + 0.2) = -1 in radians, interval 0 to 2π.

You will get a general expression with period adjustment and all numeric roots in your selected range.

Common mistakes when solving trig equations

Forgetting the second branch for sine and cosine.
Ignoring periodicity (adding only one answer instead of infinitely many).
Mixing radians and degrees.
Using invalid right-side values for sine/cosine (outside [-1, 1]).
Dropping parentheses when isolating x from ax + b.

Quick trig equation tips

Always isolate the trig expression first. Then find reference angles with inverse trig, apply all periodic branches, and finally solve for x. If you only need answers in one cycle, set your interval accordingly (0 to 360 for degrees or 0 to 2π for radians).