Trigonometry Calculator
Trigonometry connects angles to the ratios of a right triangle’s sides. This calculator returns all six trigonometric functions for any angle, tells you which quadrant it lies in and finds the reference angle. Switch to inverse mode to turn a ratio such as 0.5 back into an angle.
How to use the trigonometry calculator
- Choose whether you are finding functions of an angle or an angle from a ratio.
- Enter the angle and select degrees or radians. For the inverse, enter the ratio (between −1 and 1 for arcsin and arccos).
- Read the results. “Undefined” means the function has no value at that angle, such as tan 90°.
Formula
- sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent
- csc = 1/sin, sec = 1/cos, cot = 1/tan
- Radians = degrees × π/180
- Reference angle: the acute angle between the terminal side and the x-axis
Worked examples
A special angle
sin 30° = 0.5, and 30° is in Quadrant I. Its reference angle is 30°.
An angle in Quadrant II
150° has a reference angle of 30°, so its sine is +0.5 and its cosine is −0.866.
Undefined tangent
tan 90° is undefined because cos 90° = 0 and you cannot divide by zero.
The unit circle and exact values
Place an angle at the centre of a circle of radius 1. The point where it meets the circle is (cos θ, sin θ). That single picture gives you every trig value.
Exact values to memorise
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Signs by quadrant
“All Students Take Calculus”: in Quadrant I all are positive, II only sine, III only tangent, IV only cosine. Combine the sign with the reference angle to get any value: sin 150° = +sin 30° = 0.5, cos 150° = −cos 30° ≈ −0.866.
Why the inverse is tricky
Many angles share the same sine, such as 30° and 150°. Inverse functions return one principal value: arcsin gives −90° to 90°, arccos gives 0° to 180°, arctan gives −90° to 90°. If your triangle angle is obtuse, adjust accordingly.
To practise, try the trigonometry quiz.
Practice problems
Try these yourself first, then check your answer. The answers come from the calculator above.
- Find sin, cos and tan of 45°.
Show answer
sin: 0.7071067812 · cos: 0.7071067812 · tan: 1
- What is the reference angle of 210°?
Show answer
sin: -0.5 · cos: -0.8660254038 · tan: 0.5773502692
Common mistakes to avoid
- Using the wrong angle mode. sin(30) is 0.5 in degrees but about −0.988 in radians.
- Giving arcsin or arccos a value outside −1 to 1, which has no real answer.
- Forgetting that inverse functions return a principal value, so other angles may share the same ratio.
Frequently asked questions
What are sine, cosine and tangent?
Ratios of the sides of a right triangle: opposite/hypotenuse, adjacent/hypotenuse and opposite/adjacent.
What is a reference angle?
The acute angle formed with the x-axis. It lets you find any angle’s trig values from a first-quadrant angle.
How do I convert degrees to radians?
Multiply by π/180. See degrees vs radians.
Why is tan 90° undefined?
Because tan = sin ÷ cos and cos 90° = 0.
Reviewed September 26, 2026 by the SolveCalcPro editorial team. Found a mistake? Tell us; see our editorial policy.