Trigonometry Calculator

Trigonometry is the branch of mathematics relating the angles and side lengths of triangles. The six trigonometric functions (sine, cosine, tangent, and their three reciprocals) are the foundation for everything from engineering and physics to signal processing and computer graphics.

Primary trigonometric functions (right triangle definition): sin(θ) = Opposite ÷ Hypotenuse cos(θ) = Adjacent ÷ Hypotenuse tan(θ) = Opposite ÷ Adjacent = sin(θ) ÷ cos(θ)

Reciprocal functions: csc(θ) = 1 ÷ sin(θ) (cosecant) sec(θ) = 1 ÷ cos(θ) (secant) cot(θ) = 1 ÷ tan(θ) (cotangent)

Inverse functions (find angle from ratio): θ = arcsin(x): also written sin⁻¹(x) θ = arccos(x): also written cos⁻¹(x) θ = arctan(y/x): also written tan⁻¹(x)

Key identities:

• Pythagorean: sin²(θ) + cos²(θ) = 1
• Angle conversion: Radians = Degrees × π ÷ 180
• Double angle: sin(2θ) = 2sin(θ)cos(θ)

Exact values at common angles:

Worked example: finding a triangle’s missing side: A ladder leans against a wall. The ladder is 5 meters long and makes a 65° angle with the ground.

• Height up the wall = 5 × sin(65°) = 5 × 0.9063 = 4.53 meters
• Distance from wall at base = 5 × cos(65°) = 5 × 0.4226 = 2.11 meters
• Check: 4.53² + 2.11² = 20.52 + 4.45 = 24.97 ≈ 5² = 25 ✓

Real-world applications: Navigation (finding bearing from distance and angle), construction (calculating rafter lengths), physics (resolving force vectors), astronomy (measuring stellar distances via parallax).