Trig equation solver calculator
Trigonometric equations crop up everywhere a quantity rises and falls in a wave: alternating current, tides, sound, oscillations and orbits. This calculator solves the most common form, sin x = k, for the unknown angle x. Given a value of k between minus one and one, it returns both solutions that lie in one full turn, from 0 up to 360 degrees, and states the general solution family that captures every angle, in any turn, satisfying the equation. The principal value comes from the inverse sine, and the second solution in the range is 180 degrees minus that principal value, because the sine function takes each value twice per cycle. The general solution then adds whole multiples of 360 degrees to each. When k lies outside minus one to one there is no real solution, since the sine of a real angle never leaves that band, and the calculator says so. Enter your own value of k to solve a wave problem, find phase angles or check a trigonometry exercise. Every figure here is computed deterministically from the inverse sine and the symmetry of the sine curve, shown in full below with a worked example that reconciles exactly to the calculator.
For sin x = k the two solutions in 0 to 360 degrees are x = arcsin(k) and x = 180 minus arcsin(k). With k = 0.5 the solutions are 30.00 degrees and 150.00 degrees, and the general family is 30 + 360n and 150 + 360n.
Solving sin x = k
principal: x1 = arcsin(k) (in degrees)
second in range: x2 = 180 - x1
general solution: x = x1 + 360n or x = x2 + 360n
n is any integer
requires -1 <= k <= 1, else no real solution
The inverse sine gives the principal angle. Because the sine curve is symmetric about 90 degrees, the second solution in a single turn is 180 degrees minus the first. Adding any whole number of 360-degree turns to either gives the complete general solution.
Worked example
Solve sin x = 0.5 for x between 0 and 360 degrees.
- Principal value: x1 = arcsin(0.5) = 30 degrees
- Second value in range: x2 = 180 - 30 = 150 degrees
- Check: sin 30 = 0.5 and sin 150 = 0.5
- General solution: x = 30 + 360n or x = 150 + 360n
- So within one turn the solutions are 30.00 and 150.00 degrees
The solutions in range are 30.00 and 150.00 degrees. These are the calculator's default inputs, so the results above match the widget exactly.
Trig equation solver calculator: frequently asked questions
How many solutions does sin x = k have?
Over all real angles it has infinitely many, because the sine function repeats every 360 degrees. Within a single turn from 0 to 360 degrees there are generally two solutions, except at the peaks where k equals plus or minus one, which give a single solution, and outside the range minus one to one, which give none.
Why is the second solution 180 minus the first?
The sine curve is symmetric about 90 degrees, so sin of an angle equals sin of 180 degrees minus that angle. That means whenever arcsin(k) is a solution, 180 degrees minus it is also a solution with the same sine value, giving the second answer in the range.
What is the general solution?
The general solution lists every angle, in any turn, that satisfies the equation. For sin x = k it is x equals arcsin(k) plus 360 times any integer, or 180 degrees minus arcsin(k) plus 360 times any integer. The whole-turn multiples capture the periodic repetition of the sine function.
What if k is outside minus one to one?
The sine of a real angle always lies between minus one and one, so if k is greater than one or less than minus one there is no real solution. The calculator reports that case rather than returning a spurious answer.
Are the answers in degrees or radians?
This calculator works in degrees throughout, both for the principal value and the general solution. To convert a result to radians, multiply by pi and divide by 180. The arithmetic is deterministic and the worked example reconciles exactly to the calculator.
Official sources
- Mathematical functions and reference definitions: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.