Trig Equation Solver

This trig equation solver finds every solution of a basic trigonometric equation of the form sin x = k, cos x = k or tan x = k. Choose the function, enter the value k, and the calculator returns the principal value from your calculator range, a second solution within one full turn, and the general solution that captures all the infinitely many answers. Trig equations have repeating solutions because the functions are periodic, so a complete answer needs the general form with an integer n. For sine and cosine the value k must lie between minus one and one; the tangent accepts any value. The method, a worked example and the assumptions are set out below.

Function

Value k

Second solution (0-360)

General solution

The formula

For sin x = k the solutions are x = arcsin(k) + 360°n and x = 180° − arcsin(k) + 360°n. For cos x = k they are x = ±arccos(k) + 360°n. For tan x = k they are x = arctan(k) + 180°n. Here n is any integer, and for sine and cosine k must be between −1 and 1.

Worked example

Solve sin x = 0.5: the principal value is arcsin(0.5) = 30 degrees, the second solution in one turn is 180 − 30 = 150 degrees, and the general solution is x = 30° + 360°n or 150° + 360°n. Enter sin and 0.5 to confirm.

Frequently asked questions

Why does a trig equation have infinitely many solutions?

Because sine, cosine and tangent are periodic, the same value repeats every 360 or 180 degrees, so each base solution generates a family with an integer n.

What is the general solution for tan x = k?

x = arctan(k) + 180 degrees times n, for any integer n, since the tangent repeats every 180 degrees.

Why does sin x = k sometimes say no solution?

Sine and cosine only take values between minus one and one. If k is outside that range there is no real solution.

Who this calculator is for

This calculator is for students solving basic trigonometric equations and learning the general solution.

What this calculator assumes

You solve sin x = k, cos x = k or tan x = k.
Answers are given in degrees, with n any integer.
Results are rounded for display.

Formula and sources

Related calculators

Trig Calculator: all six trig functions.
Sine Calculator: sine of an angle.
Right Angle Triangle Calculator: solve a right triangle.

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