Math Calculators
Trigonometry Solver
Calculate sin, cos, tan, and their inverses in degrees or radians. Includes sinh, cosh, tanh and a quick reference table for common angles.
sin(30°) = 0.49999999999999994
Common Angles
SOH-CAH-TOA
For a right triangle with angle θ, hypotenuse (H), opposite (O), and adjacent (A) sides:
sin(θ) = O / H (Sine = Opposite over Hypotenuse)
cos(θ) = A / H (Cosine = Adjacent over Hypotenuse)
tan(θ) = O / A (Tangent = Opposite over Adjacent)Unit circle values
| Angle | sin | cos | tan |
|---|---|---|---|
| 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 |
Inverse trig functions
Use inverse functions to find angles from ratios: arcsin, arccos, arctan (also written sin⁻¹, cos⁻¹, tan⁻¹). For example, if sin(θ) = 0.5, then θ = arcsin(0.5) = 30°.
Reciprocal trig functions
The six trigonometric functions include three primary functions and three reciprocals:
| Function | Definition | Reciprocal of | Mnemonic |
|---|---|---|---|
| sin(θ) | opposite / hypotenuse | - | SOH |
| cos(θ) | adjacent / hypotenuse | - | CAH |
| tan(θ) | opposite / adjacent | - | TOA |
| csc(θ) | hypotenuse / opposite | sin(θ) | "cosecant = 1/sine" |
| sec(θ) | hypotenuse / adjacent | cos(θ) | "secant = 1/cosine" |
| cot(θ) | adjacent / opposite | tan(θ) | "cotangent = 1/tangent" |
Reciprocal functions appear frequently in calculus derivatives: the derivative of tan(x) is sec²(x), and the derivative of sin(x) involves csc(x) in many integral forms.
Radians vs. degrees
Degrees are intuitive (a full circle = 360°), but radians are the natural unit for mathematics and physics. One radian is the angle subtended by an arc equal in length to the radius:
- Full circle: 360° = 2π rad ≈ 6.2832 rad
- Half circle: 180° = π rad ≈ 3.1416 rad
- Right angle: 90° = π/2 rad ≈ 1.5708 rad
- Convert degrees to radians: multiply by π/180
- Convert radians to degrees: multiply by 180/π
Calculus formulas (derivatives, integrals, Taylor series) for trig functions only work simply in radians. Programming languages typically use radians by default.
Worked example - finding a ramp angle
A loading ramp rises 2 m over a horizontal run of 5 m. What angle does it make with the ground?
- Identify: opposite = 2 m, adjacent = 5 m -> use arctan.
- θ = arctan(2/5) = arctan(0.4) ≈ 21.8°
- The hypotenuse (ramp length) = √(2² + 5²) = √29 ≈ 5.39 m
- Verify: sin(21.8°) ≈ 0.371, and 0.371 × 5.39 ≈ 2.0 m ✓