Trigonometry Solver

How to use this tool

Choose a circular, inverse, or hyperbolic function. Select degrees or radians for circular angles or inverse output angles.
Enter an angle, an inverse-function ratio, or a real number for a hyperbolic function.
Read the decimal result and, for recognized special angles, its exact form. Circular functions also have common-angle presets.

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 ✓

Frequently asked questions

Can I calculate in degrees or radians?

Yes. Circular functions use the selected unit for the input angle, and inverse functions (asin, acos, atan) use it for the output angle. Hyperbolic functions take a real number, so degrees and radians do not apply.