Trigonometry Calculator
Enter an angle in degrees or radians to get sine, cosine, tangent and their reciprocals, with exact values at multiples of 30 and 45 degrees. You can also find an angle from a ratio, or solve a right triangle from any two known values, one of which must be a side.
Last updated November 4, 2026
Result
sin 30° = 0.5, cos = 0.866025, tan = 0.57735
| Measure | Value |
|---|---|
| sin(30°) | 0.5 (exact 1/2) |
| cos(30°) | 0.866025 (exact √3/2) |
| tan(30°) | 0.57735 (exact √3/3) |
| csc(30°) | 2 |
| sec(30°) | 1.154701 (exact 2√3/3) |
| cot(30°) | 1.732051 (exact √3) |
Values are rounded to 6 decimals. Undefined means the function has no value at that angle.
Show the work
Formula: tan = sin / cos; csc = 1 / sin; sec = 1 / cos; cot = cos / sin
- Angle: 30° = 0.523599 rad (degrees x pi / 180).
- sin = 0.5 and cos = 0.866025 for this angle.
- tan = sin / cos = 0.5 / 0.866025 = 0.57735.
- csc = 1 / sin = 2, sec = 1 / cos = 1.154701, cot = 1 / tan = 1.732051.
Result: sin 30° = 0.5, cos = 0.866025, tan = 0.57735
Trigonometry formulas
sin A = a / c · cos A = b / c · tan A = a / b · a² + b² = c² · A + B = 90°
In a right triangle, the sine of an acute angle is the opposite side divided by the hypotenuse, the cosine is the adjacent side divided by the hypotenuse, and the tangent is the opposite side divided by the adjacent side. The mnemonic SOH-CAH-TOA captures this. The other three functions are reciprocals: cosecant is 1 / sin, secant is 1 / cos and cotangent is 1 / tan.
Angles convert between units with degrees = radians × 180 / π. A full turn is 360 degrees or 2π radians, and a right angle is 90 degrees or π/2 radians. The functions repeat every full turn, so sin 390° equals sin 30°, and a negative angle turns the opposite way, so sin(-30°) = -0.5.
See how we calculate for the formulas, sources and review schedule behind every calculator.
Also useful alongside this calculator: Geometry Calculator, Distance Calculator and Perimeter Calculator.
Exact values at special angles
At multiples of 30 and 45 degrees the functions have closed forms. Sine of 30° is exactly 1/2, of 45° is √2/2 and of 60° is √3/2. Cosine runs the same list in reverse. Floating-point arithmetic usually returns tiny errors here, for example 0.49999999999999994 for sin 30°, so this calculator snaps those angles to their exact values. You see sin 180° = 0 and cos 90° = 0 instead of numbers like 1.2e-16.
When a function is undefined
Tangent is sine divided by cosine, and cosine is zero at 90°, 270° and every angle 180° apart from them. At those angles the tangent and secant have no value, and the calculator reports undefined instead of a huge number or Infinity. In the same way cosecant and cotangent are undefined at 0°, 180° and 360°, where the sine is zero.
Inverse functions: finding an angle
Arcsine, arccosine and arctangent answer the opposite question: which angle has this ratio? Arcsine and arccosine accept only values from -1 to 1, because no angle has a sine or cosine outside that range, so anything else is rejected with a message. Arctangent accepts any real number.
Each returns the principal value. Arcsine and arctangent give angles from -90° to 90°, and arccosine gives 0° to 180°. Other angles share the same ratio, for example 150° also has a sine of 0.5, so add multiples of 360° or reflect across the axis when your problem needs them.
Solving a right triangle
A right triangle has five unknowns besides the 90° angle: sides a, b and c, and angles A and B. Two known values fix the rest as long as at least one is a side. Two sides use the Pythagorean theorem and arctangent. A side with an angle uses sine, cosine or tangent to find the other sides. Two angles alone cannot work, because similar triangles of any size share the same angles.
A and B must add up to 90°, so each is strictly between 0° and 90°. When you give a leg and the hypotenuse, the hypotenuse has to be the longer one, otherwise no triangle exists.
How to use the Trigonometry Calculator
- Choose a mode: trig functions of an angle, an inverse function, or the right triangle solver.
- Enter the angle and its unit, the ratio value, or exactly two known triangle values, with at least one side.
- Read the results table, then open Show the work to see each formula applied.
- Use Copy or Share to keep the answer or send the same inputs to someone else.
Worked example
A ladder of length 10 leans against a wall at 30° to the ground. The wall height is the side opposite that angle, so a = 10 × sin 30° = 5, and the foot of the ladder is b = 10 × cos 30° ≈ 8.660254 from the wall. In the solver, enter c = 10 and A = 30 to get the same sides, with B = 60°. As a second check, a 3-4-5 triangle gives c = 5, A ≈ 36.8699° and B ≈ 53.1301°. The table lists common values.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.5 | 0.866025 | 0.57735 |
| 45° | 0.707107 | 0.707107 | 1 |
| 60° | 0.866025 | 0.5 | 1.732051 |
| 90° | 1 | 0 | undefined |
Frequently asked questions
Why is tan 90 degrees undefined?
Tangent is sine divided by cosine, and cos 90° is exactly 0. Division by zero has no value, so the calculator reports undefined.
Should I use degrees or radians?
Use whichever your problem is written in. Degrees are common in geometry and navigation, radians in calculus and physics. 180° equals π radians, about 3.14159.
Why does arcsine reject a value like 2?
Sine never goes above 1 or below -1 for a real angle, so no angle has a sine of 2. The inverse functions only accept values from -1 to 1.
Can I solve a triangle when I only know its angles?
No. Triangles of every size share the same angles, so at least one side length is needed to fix the size of the triangle.
Does this work for triangles without a right angle?
The solver covers right triangles only. The functions still apply to any angle, but oblique triangles need the law of sines or the law of cosines.