Trigonometry Calculator – Sin, Cos, Tan & Inverse Functions
Calculate sine, cosine, tangent, and inverse trig functions. Solve right triangles and convert between degrees and radians. Free online trig calculator.
The Six Trigonometric Functions
Trigonometry is built on six fundamental functions relating angles to ratios of sides in a right triangle. For an angle θ in a right triangle with opposite side O, adjacent side A, and hypotenuse H:
| Function | Abbreviation | Formula | Reciprocal |
|---|---|---|---|
| Sine | sin θ | O/H | cosecant (csc) |
| Cosine | cos θ | A/H | secant (sec) |
| Tangent | tan θ | O/A | cotangent (cot) |
| Cosecant | csc θ | H/O | sine |
| Secant | sec θ | H/A | cosine |
| Cotangent | cot θ | A/O | tangent |
The memory aid SOH-CAH-TOA helps: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Common Angle Values
Certain angles appear frequently and are worth memorizing:
| 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 |
| 180° | π | 0 | −1 | 0 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 360° | 2π | 0 | 1 | 0 |
Degrees vs. Radians
Angles can be measured in degrees or radians. Degrees divide a full circle into 360 equal parts. Radians measure angle as the ratio of arc length to radius — a full circle equals 2π radians.
Conversion formulas:
- Degrees to radians: radians = degrees × π / 180
- Radians to degrees: degrees = radians × 180 / π
Radians are the natural unit in calculus and physics. The derivative of sin(x) is cos(x) only when x is in radians. In programming, most math libraries use radians by default.
Pythagorean Theorem and Trig Identities
The most important identity in trigonometry: sin²θ + cos²θ = 1. This follows directly from the Pythagorean theorem (O² + A² = H²) divided by H².
Other key identities:
- Double angle: sin(2θ) = 2sin(θ)cos(θ); cos(2θ) = cos²θ − sin²θ
- Sum of angles: sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
- Tangent identity: tan(θ) = sin(θ)/cos(θ)
- Reciprocal: csc(θ) = 1/sin(θ); sec(θ) = 1/cos(θ); cot(θ) = 1/tan(θ)
💡 Did you know?
- The word "trigonometry" comes from Greek: trigonon (triangle) + metron (measure).
- Hipparchus of Nicaea (190–120 BC) is often called the "father of trigonometry" — he compiled the first known trigonometric table.
- GPS navigation, computer graphics, sound waves, and AC electrical circuits all rely on trigonometric functions.
Frequently Asked Questions
What is the difference between sin, cos, and tan?
In a right triangle: sine is the ratio of the opposite side to the hypotenuse; cosine is adjacent to hypotenuse; tangent is opposite to adjacent. Remember SOH-CAH-TOA. Each function produces a value between −1 and 1 (tan can be any value), representing the relationship between an angle and side lengths.
How do I use inverse trig functions (arcsin, arccos, arctan)?
Inverse functions find the angle given a ratio. If sin(θ) = 0.5, then θ = arcsin(0.5) = 30°. Use arcsin when you know opposite/hypotenuse; arccos when you know adjacent/hypotenuse; arctan when you know opposite/adjacent. Calculator buttons: sin⁻¹, cos⁻¹, tan⁻¹ (same as arcsin, arccos, arctan).
Why does tan(90°) not exist?
Tangent = sin/cos. At 90°, cos(90°) = 0, and division by zero is undefined. On a graph, tangent approaches ±infinity as the angle approaches 90°. This is called a vertical asymptote. Similarly, tan(270°) is undefined.
What are trig functions used for in real life?
Trigonometry is used in: navigation (GPS, aviation, sailing), architecture and construction (roof angles, staircase calculations), physics (waves, oscillations, AC circuits), computer graphics (3D rotation, game engines), astronomy (measuring distances to stars), and engineering (structural analysis, signal processing).