Trig Equation Solver

This trig equation solver works out every solution to a basic trigonometric equation of the form sin x = k, cos x = k or tan x = k, the kind of question that comes up constantly in NCEA and university maths. You choose the function from the dropdown and enter the value of k, and the calculator returns three results: the principal value, the single answer your calculator's inverse function gives directly; the second solution within one full turn from 0 to 360 degrees, because sine and cosine cross the same value twice per cycle; and the general solution, the formula with an integer n that captures every one of the infinitely many answers the equation has. Trigonometric functions repeat every 360 degrees, or 180 for tangent, so a single principal value is never the whole story, and exams usually want the general form. For sine and cosine, k must sit between minus one and one; enter a value outside that range and the calculator tells you plainly there is no solution rather than guessing. Tangent has no such limit and accepts any real value of k. Try the default of sin x = 0.5 to see the principal value of 30 degrees, the second solution of 150 degrees, and the general solution appear together, then swap in your own numbers to check homework or explore a new problem.

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Function

Value k

x = 30° (principal)

Second solution (0-360)150°

General solutionx = 30° + 360°n or 150° + 360°n

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.