Trigonometry Calculator
Calculate all 6 trigonometric functions and all 6 inverse trig functions. Enter an angle in degrees or radians, or enter a value to find its inverse.
Trigonometric Ratios Reference Table
Trigonometric values for common angles are listed below in exact and decimal form. These appear frequently in geometry, physics, and engineering problems.
| Angle (°) | Radians | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | undef. | 1 | undef. |
| 30° | π/6 | 0.5 | 0.8660 | 0.5774 | 2 | 1.1547 | 1.7321 |
| 45° | π/4 | 0.7071 | 0.7071 | 1 | 1.4142 | 1.4142 | 1 |
| 60° | π/3 | 0.8660 | 0.5 | 1.7321 | 1.1547 | 2 | 0.5774 |
| 90° | π/2 | 1 | 0 | undef. | 1 | undef. | 0 |
| 120° | 2π/3 | 0.8660 | -0.5 | -1.7321 | 1.1547 | -2 | -0.5774 |
| 135° | 3π/4 | 0.7071 | -0.7071 | -1 | 1.4142 | -1.4142 | -1 |
| 150° | 5π/6 | 0.5 | -0.8660 | -0.5774 | 2 | -1.1547 | -1.7321 |
| 180° | π | 0 | -1 | 0 | undef. | -1 | undef. |
| 270° | 3π/2 | -1 | 0 | undef. | -1 | undef. | 0 |
| 360° | 2π | 0 | 1 | 0 | undef. | 1 | undef. |
Understanding Trigonometry Functions
Trigonometry functions describe the relationships between angles and the sides of right triangles. These relationships extend beyond right triangles to all angles on the unit circle, making them foundational to mathematics, physics, and engineering.
SOH CAH TOA
SOH CAH TOA is the classic memory device for the three primary trig functions in a right triangle.
The Reciprocal Functions
The three reciprocal functions are derived directly from the primary ones. Cosecant (csc) is 1 / sin. Secant (sec) is 1 / cos. Cotangent (cot) is 1 / tan, or equivalently cos / sin. These appear frequently in integration and physics problems.
Degrees vs Radians
Degrees divide a full circle into 360 equal parts. Radians measure angles by arc length on a unit circle. A full circle is 2π radians, which equals 360 degrees. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.
Inverse Trigonometry Functions
Inverse trigonometry functions reverse the normal trig operation. Instead of inputting an angle to get a ratio, you input a ratio to get the angle. They are essential for solving triangles when sides are known but angles are not.
Domain and Range Restrictions
Because trig functions are periodic, their inverses need restricted ranges to be proper functions. Arcsin returns angles from -90° to 90°. Arccos returns angles from 0° to 180°. Arctan returns angles from -90° to 90°. These principal value ranges ensure each input maps to exactly one output.
Real World Applications
Arctan is used in navigation to find bearing angles from coordinate differences. Arcsin and arccos are used in optics (Snell’s law), robotics (joint angle calculation), and structural engineering (roof pitch and slope angle determination).