What this calculator does
In trigonometry, theta (θ) is the standard placeholder for an unknown angle, and the most common reason to search for a theta calculator is to work backwards: a sine, cosine or tangent ratio is already known, from a triangle, a wave or a physics problem, and the angle that produced it still needs finding.
This calculator does exactly that. Enter whichever ratio is known, sin(θ), cos(θ) or tan(θ), and it returns theta directly, in both degrees and radians, along with the complementary angle and a check of the ratio it started from.
The formula
Finding theta from a ratio uses the inverse trigonometric functions: θ = sin⁻¹(x), θ = cos⁻¹(x) or θ = tan⁻¹(x), depending on which ratio is given. Sine and cosine ratios are only ever defined between −1 and 1, since neither can exceed the hypotenuse in a right triangle, while tangent is unrestricted, since it can grow without bound as an angle approaches 90°.
| Term | Meaning |
|---|---|
| θ (theta) | The unknown angle being solved for. |
| sin⁻¹, cos⁻¹, tan⁻¹ | The inverse trigonometric functions: given a ratio, they return the angle that produces it. |
| Radian | The other standard unit for angle, where a full turn is 2π rather than 360°. |
The inputs explained
| Field | What to enter |
|---|---|
| Known ratio | Choose which ratio is already known: sine, cosine or tangent of the unknown angle. |
| Ratio value | The numeric value of that ratio. Sine and cosine values must fall between −1 and 1; tangent can be any number. |
When to use it
How to find theta from a right triangle’s sides
A right triangle with two known sides gives a ratio, such as opposite over hypotenuse for sine, and this calculator turns that ratio straight into the angle theta, in degrees.
How to calculate theta in a physics problem
A physics question giving a component of a force, velocity or displacement as a ratio of the whole, such as a slope’s rise over its length for tangent, is solved directly here for the angle theta involved.
Checking a theta value against its known ratio
A textbook or working already gives an angle, and this calculator confirms it by running the reverse check: entering the ratio and seeing that the returned theta matches what was expected.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Finding theta from a range of sine values
The angle theta recovered from a range of sine ratio values, from 0 up to 1.
Finding theta from tangent, including values above 1
The angle theta recovered from a range of tangent ratio values, including values greater than 1, which sine and cosine cannot take.
Questions
What does theta mean in maths?
Theta (θ) is a Greek letter used as the standard symbol for an unknown or general angle, in the same way x commonly stands for an unknown number in algebra.
How do you find theta from a sine, cosine or tangent value?
Apply the matching inverse trigonometric function: θ = sin⁻¹(x) if sine is known, θ = cos⁻¹(x) if cosine is known, or θ = tan⁻¹(x) if tangent is known. This calculator applies whichever one matches the ratio entered.
Why can sine and cosine only take values between −1 and 1?
Both are defined as a triangle side divided by the hypotenuse, and no side of a right triangle can be longer than its hypotenuse, so the ratio can never exceed 1 in size, in either direction.
How is this different from a standard trigonometry calculator?
A standard trig calculator like the sin, cos and tan function calculator starts from a known angle and works out its ratios. This one runs the other direction, starting from a known ratio and solving for the angle theta itself.
To go the other way, from a known angle to its sine, cosine and tangent, see the trigonometric functions calculator. To solve a whole triangle from two sides and an included angle, see the solve a triangle calculator.