Trigonometry Calculator
Instantly compute sin, cos, tan, and more — find an angle with inverse trig — or solve a right triangle from whatever sides or angle you know. Supports degrees and radians, with step-by-step working and a diagram that actually shows where sin and cos come from.
Background
Trigonometry connects angles to side ratios. In a right triangle: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent. The same ratios extend to any angle using the unit circle, where sin(θ) and cos(θ) are simply the y- and x-coordinates of a point moving around a circle of radius 1.
How to use this calculator
- Pick a mode: trig values, inverse trig, or the right triangle solver.
- Choose degrees or radians — switching later automatically converts any angle you've typed.
- Click Calculate for the result, an optional diagram, and step-by-step working.
How this calculator works
- Trig values: converts your angle to radians, computes sin/cos/tan (or their reciprocals cot/sec/csc), and checks whether it's one of the 17 special unit-circle angles with a clean exact form.
- Inverse trig: finds the principal-value angle for arcsin/arccos/arctan, since a single ratio actually matches infinitely many angles.
- Right triangle: uses SOH-CAH-TOA and a² + b² = c². If you enter all three sides, it checks they're mutually consistent before using them, instead of silently overriding whichever one it recomputes first.
Formulas & Equations Used
SOH: sin(θ) = opposite / hypotenuse
CAH: cos(θ) = adjacent / hypotenuse
TOA: tan(θ) = opposite / adjacent
Reciprocals: cot(θ)=1/tan(θ), sec(θ)=1/cos(θ), csc(θ)=1/sin(θ)
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 sin⁻¹(0.5)
- Principal angle: 30° (or π/6 radians)
Example 3 — Right triangle from two sides
Given opposite a=3 and adjacent b=4, solve the triangle.
- Hypotenuse: c = √(a²+b²) = √(9+16) = 5
- Angle: θ = arctan(a/b) = arctan(0.75)
- Answer: c=5, θ ≈ 36.87°, φ ≈ 53.13°
Example 4 — Catching an inconsistent triangle
Someone enters a=3, b=4, and c=100 by mistake.
- Check: a² + b² = 9 + 16 = 25, but c² = 10000 — these don't match.
- Expected c given a and b would be √25 = 5, not 100.
- The calculator flags this instead of silently guessing which value you meant.
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 most precalculus/calculus work.
Q: What does "sin⁻¹" mean?
It means inverse sine, also called arcsin. It does not mean 1/sin(x) — that's csc(x) instead.
Q: What inputs solve a right triangle?
Either two sides, or one side plus one acute angle. If you enter all three sides, they need to be mutually consistent — the calculator checks this rather than silently picking one.
Q: Why is tan undefined sometimes?
tan(θ) = sin(θ)/cos(θ). When cos(θ) = 0 — at 90° and 270° — that's division by zero, so tan is undefined there.
Q: I switched units and my angle changed — is that a bug?
No — that's intentional. Switching between degrees and radians converts whatever angle you'd typed so it keeps meaning the same angle, rather than silently reading the same number under a different unit.
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