Trigonometry Calculator
Instantly compute sin, cos, tan, and more — or solve a right triangle (missing sides & angles) from whatever you know. Supports degrees and radians, with optional step-by-step and a mini diagram.
Background
Trigonometry connects angles to side ratios. In a right triangle: sin(θ)=opposite/hypotenuse, cos(θ)=adjacent/hypotenuse, tan(θ)=opposite/adjacent. This calculator can (1) evaluate trig functions for an angle, (2) find an angle from a trig value (inverse trig), and (3) solve a right triangle using SOH-CAH-TOA + the Pythagorean theorem.
How to use this calculator
- Pick a mode: Trig values, Inverse trig, or Right triangle solver.
- Choose degrees or radians (you can switch anytime).
- Click Calculate for results + optional steps and a mini diagram.
How this calculator works
- Trig values: compute sin/cos/tan/etc. from θ (in deg or rad).
- Inverse trig: compute θ from a trig value using arcsin/arccos/arctan (principal value).
- Right triangle: use SOH-CAH-TOA and a² + b² = c² to solve missing parts.
Formula & Equation Used
SOH: sin(θ) = opposite / hypotenuse
CAH: cos(θ) = adjacent / hypotenuse
TOA: tan(θ) = opposite / adjacent
Pythagorean theorem: a² + b² = c²
Example Problems & Step-by-Step Solutions
Example 1 — Trig value
Find sin(30°).
- 30° is a special unit-circle angle.
- sin(30°)=1/2
- Answer: 0.5
Example 2 — Inverse trig
Find θ if sin(θ)=0.5 (principal value).
- θ = arcsin(0.5) (also written as sin⁻¹(0.5))
- Principal angle: 30° (or π/6 radians)
Example 3 — Right triangle
Given opposite a=3 and adjacent b=4, solve the triangle.
- Hypotenuse: c = √(a²+b²)=√(9+16)=5
- Angle: θ = arctan(a/b)=arctan(3/4)
- Answer: c=5, θ ≈ 36.87°, φ ≈ 53.13°
Frequently Asked Questions
Q: Degrees or radians — which should I use?
Use degrees for most right-triangle word problems. Use radians for unit-circle angles and many precalculus/calc problems.
Q: What does “sin⁻¹” mean?
It means inverse sine, also called arcsin. It does not mean 1/sin(x).
Q: What inputs solve a right triangle?
You need either two sides or one side + one acute angle (plus the right angle).
Q: Why is tan undefined sometimes?
tan(θ)=sin(θ)/cos(θ). If cos(θ)=0, tan is undefined (division by zero).
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