Sine, Cosine & Tangent Calculator

Trigonometry sits at the heart of geometry, physics, engineering, signal processing, and computer graphics. Sine (sin), cosine (cos) and tangent (tan) are the three primary ratios that relate an angle in a right triangle to the lengths of its sides — and through the unit circle they extend naturally to any angle, positive or negative, however large.

This calculator gives you instant access to all six trigonometric functions (sin, cos, tan, csc, sec, cot), their inverse counterparts (arcsin, arccos, arctan, atan2), a live unit-circle diagram with colour-coded projections, double-angle and half-angle expansions, Pythagorean identity verification, and a wave graph that marks your angle on the full periodic curve. Everything runs inside your browser — nothing is ever uploaded.

How it works

The right-triangle definitions

For a right triangle with angle θ, hypotenuse H, opposite side O and adjacent side A:

• sin θ = O / H — how much of the hypotenuse is “projected” vertically
• cos θ = A / H — horizontal projection
• tan θ = O / A = sin θ / cos θ — the slope of the hypotenuse relative to the base

The three reciprocal functions follow immediately: csc θ = H/O, sec θ = H/A, cot θ = A/O.

The unit circle extension

When the hypotenuse is exactly 1 (the unit circle), the point on the circle at angle θ is simply (cos θ, sin θ). This means the definitions extend to any angle: a negative angle rotates clockwise, an angle greater than 360° wraps around, and the values repeat with period 2π (360°). The tan function repeats every π (180°) because opposite and adjacent both flip sign in the same half-turn.

Pythagorean identity

Because the point (cos θ, sin θ) lies on a circle of radius 1, the distance formula gives sin²θ + cos²θ = 1 for every θ without exception. Dividing both sides by cos²θ yields tan²θ + 1 = sec²θ, and dividing by sin²θ gives 1 + cot²θ = csc²θ.

Inverse functions (arcsine, arccosine, arctangent)

Inverting sin gives arcsin: given a ratio r, arcsin(r) returns the angle whose sine is r. Because sine is many-to-one, the principal value is restricted to [-90°, 90°]. The calculator lists all equivalent angles in [0°, 360°) alongside the principal value. The two-argument form atan2(y, x) determines the correct quadrant from the signs of both components — essential in navigation and vector graphics.

Worked example

You are designing a ramp with a 30° angle of inclination. The ramp is 5 m long.

• Vertical rise = 5 × sin(30°) = 5 × 0.5 = 2.5 m
• Horizontal run = 5 × cos(30°) = 5 × (√3/2) ≈ 4.33 m
• tan(30°) = 2.5 / 4.33 ≈ 0.5774 = 1/√3

Now suppose you only know the rise (2.5 m) and want the angle — use the Inverse tab: arctan(2.5 / 4.33) = arctan(0.5774) ≈ 30°. That is the exact angle you started with.

Formula reference

All computations use IEEE 754 double-precision floating point via JavaScript’s Math.sin, Math.cos, Math.tan, Math.asin, Math.acos, Math.atan and Math.atan2. The underlying C library routine is accurate to within 1 ULP for primary functions. Results are rounded to 10 significant decimal places to suppress floating-point artefacts — for example, sin(180°) displays as 0, not 1.2246e-16.

Double-angle formulas shown: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ, tan 2θ = 2 tan θ / (1 − tan²θ). Half-angle: sin(θ/2) = ±√((1 − cos θ)/2), cos(θ/2) = ±√((1 + cos θ)/2). The sign depends on the quadrant of θ/2.

Every figure is calculated entirely in your browser — no numbers are uploaded or stored.