Trig Functions Calculator
Calculate sin, cos, tan, csc, sec, and cot for any angle in degrees or radians. Inverse trig functions and a unit circle reference table included.
About Trig Functions Calculator
Calculate all six trigonometric function values (sin, cos, tan, csc, sec, cot) for any angle in degrees or radians. Includes inverse trig functions, identity verification, and a reference table of exact values at standard angles.
The Six Trigonometric Functions
Given a right triangle with angle θ, opposite side, adjacent side, and hypotenuse:
| Function | Ratio | Reciprocal |
|---|---|---|
| sin(θ) | opposite / hypotenuse | csc(θ) = 1/sin(θ) |
| cos(θ) | adjacent / hypotenuse | sec(θ) = 1/cos(θ) |
| tan(θ) | opposite / adjacent = sin(θ)/cos(θ) | cot(θ) = 1/tan(θ) = cos(θ)/sin(θ) |
The mnemonic SOH-CAH-TOA helps remember: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Exact Values at Standard Angles
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 120° | 2π/3 | √3/2 | -1/2 | -√3 |
| 135° | 3π/4 | √2/2 | -√2/2 | -1 |
| 150° | 5π/6 | 1/2 | -√3/2 | -1/√3 |
| 180° | π | 0 | -1 | 0 |
| 270° | 3π/2 | -1 | 0 | undefined |
| 360° | 2π | 0 | 1 | 0 |
Key Trigonometric Identities
The calculator verifies these identities for every input:
| Identity | Formula |
|---|---|
| Pythagorean | sin²(θ) + cos²(θ) = 1 |
| Pythagorean (tan) | 1 + tan²(θ) = sec²(θ) |
| Pythagorean (cot) | 1 + cot²(θ) = csc²(θ) |
| Quotient | tan(θ) = sin(θ) / cos(θ) |
| Reciprocal | sin(θ) × csc(θ) = 1 |
| Even/odd | cos(-θ) = cos(θ), sin(-θ) = -sin(θ) |
Inverse Trig Functions
Inverse trig functions find the angle when you know the ratio:
| Function | Input Range | Output Range | Example |
|---|---|---|---|
| arcsin (sin⁻¹) | [-1, 1] | [-90°, 90°] | arcsin(0.5) = 30° |
| arccos (cos⁻¹) | [-1, 1] | [0°, 180°] | arccos(0.5) = 60° |
| arctan (tan⁻¹) | all real numbers | (-90°, 90°) | arctan(1) = 45° |
Each inverse function has a restricted output range (principal value) to ensure a unique answer. For example, sin(30°) = sin(150°) = 0.5, but arcsin(0.5) returns only 30° (the principal value).
Sign Rules by Quadrant
| Quadrant | Angle Range | sin | cos | tan |
|---|---|---|---|---|
| I | 0° - 90° | + | + | + |
| II | 90° - 180° | + | - | - |
| III | 180° - 270° | - | - | + |
| IV | 270° - 360° | - | + | - |
The mnemonic "All Students Take Calculus" tells which functions are positive: All (Q1), Sin (Q2), Tan (Q3), Cos (Q4).
Degrees vs Radians
Conversion: radians = degrees × π/180. Degrees = radians × 180/π.
| Degrees | Radians (exact) | Radians (decimal) |
|---|---|---|
| 30° | π/6 | 0.5236 |
| 45° | π/4 | 0.7854 |
| 60° | π/3 | 1.0472 |
| 90° | π/2 | 1.5708 |
| 180° | π | 3.1416 |
| 360° | 2π | 6.2832 |
For a complete visual reference of trig values on the unit circle, see the unit circle tool. For solving triangles with known sides and angles, the law of sines and law of cosines calculators apply these functions directly.
All calculations run in your browser. No data is sent to any server.
Frequently Asked Questions
What are the six trig functions?
Sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Sin, cos, and tan are the primary functions. Csc, sec, and cot are their reciprocals.
What is the difference between degrees and radians?
Degrees measure angles out of 360 for a full circle. Radians measure angles based on the radius, with 2 pi radians in a full circle. To convert, multiply degrees by pi/180 to get radians.
What are inverse trig functions?
Inverse trig functions find the angle when you know the ratio. Arcsin(0.5) = 30 degrees because sin(30) = 0.5. Arcsin and arccos only accept values from -1 to 1, while arctan accepts any number.
When is tangent undefined?
Tangent is undefined when cosine equals zero, which happens at 90, 270, 450 degrees and so on. At these angles, the ratio sin/cos involves dividing by zero.
What is the Pythagorean identity?
The identity sin squared plus cos squared equals 1 holds for any angle. The calculator verifies this as an identity check. It comes from the Pythagorean theorem applied to the unit circle.
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