What this calculator does
The three primary trigonometric functions describe the ratios between the sides of a right triangle, but they are more usefully understood through the unit circle. There, cosine is the horizontal coordinate of a point and sine is the vertical one, which is why both oscillate between −1 and 1 and why they extend naturally beyond the 0 to 90° range a triangle allows.
Tangent is their ratio, and it behaves differently: it grows without bound as the angle approaches 90°, where cosine reaches zero and the division becomes undefined. This calculator reports all three along with their reciprocals and converts between degrees and radians.
The formula
Radians measure angle by arc length on a unit circle, so a full turn is 2π rather than 360. Programming languages and calculus both work in radians, which is why the conversion appears so often. The reciprocal functions, secant, cosecant and cotangent, are simply one divided by cosine, sine and tangent respectively.
| Term | Meaning |
|---|---|
| sin | Opposite ÷ hypotenuse; the vertical coordinate on the unit circle. |
| cos | Adjacent ÷ hypotenuse; the horizontal coordinate. |
| tan | sin ÷ cos; the gradient of the radius line. |
| Radian | The angle subtending an arc equal to the radius. 180° = π radians. |
The inputs explained
| Field | What to enter |
|---|---|
| Angle | The angle. Values outside 0 to 360 are handled fine, since the functions repeat. |
| Units | Whether the angle is in degrees or radians. |
When to use it
Finding a height from an angle
Measuring the angle of elevation to the top of a structure and your distance from it gives the height as distance × tan(angle).
Working with gradients and slopes
A roof pitch or road gradient is the tangent of the angle. A 1-in-10 slope is an angle whose tangent is 0.1, about 5.7°.
Converting for programming
Nearly every programming language expects radians. The conversion line gives the value to pass in when your source data is in degrees.
Understanding periodic behaviour
Waves, alternating current, tides and orbits are all described by sine and cosine. The functions repeat every 360°, which is what makes them the natural language for anything cyclical.
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.
The standard angles
The angles whose exact values are worth knowing by heart.
| Angle | sin | cos | tan | In the other unit |
|---|---|---|---|---|
| 0° | 0 | 1.000 | 0 | 0 rad |
| 30° | 0.5000 | 0.8660 | 0.5774 | 0.5236 rad |
| 45° | 0.7071 | 0.7071 | 1.0000 | 0.7854 rad |
| 60° | 0.8660 | 0.5000 | 1.732 | 1.047 rad |
| 90° | 1.000 | 6.1232e-17 | 1.6331e+16 | 1.571 rad |
| 180° | 1.2246e-16 | -1.000 | -1.2246e-16 | 3.142 rad |
Values in radians
The same key angles expressed in radians, which is how mathematics and programming handle them.
| Angle | sin | cos | In the other unit |
|---|---|---|---|
| 0 rad | 0 | 1.000 | 0° |
| 0.5236 rad | 0.5000 | 0.8660 | 30.000° |
| 0.7854 rad | 0.7071 | 0.7071 | 45.000° |
| 1.0472 rad | 0.8660 | 0.5000 | 60.000° |
| 1.5708 rad | 1.0000 | -0.000004 | 90.000° |
| 3.1416 rad | -0.000007 | -1.0000 | 180.000° |
Questions
What is a radian?
The angle at which the arc length equals the radius. A full circle is 2π radians, so 180° equals π radians and one radian is about 57.3°. It is the natural unit for calculus, because the derivative rules only take their simple form in radians.
How do I convert degrees to radians?
Multiply by π and divide by 180. To go the other way, multiply by 180 and divide by π. The calculator shows the conversion alongside the trigonometric values.
Why is tan(90°) undefined?
Because tangent is sine divided by cosine, and cosine is zero at 90°. Division by zero has no value. Approaching 90° from below, the tangent grows without limit.
What are secant, cosecant and cotangent?
The reciprocals: secant is 1/cos, cosecant is 1/sin, cotangent is 1/tan. They appear mainly in calculus and in some engineering conventions, and are rarely needed otherwise.
Can I use angles over 360°?
Yes. The functions repeat every 360°, or 2π radians, so 450° gives the same values as 90°. Negative angles work too, measuring clockwise.
To solve a whole triangle, see the triangle solver. For angle unit conversion on its own, use the angle converter.