Trigonometry Calculator

Trigonometry is the branch of mathematics that studies the relationships between the sides and angles of triangles. Its name comes from the Greek words for triangle (trigonon) and measure (metron). From ancient astronomers measuring the heavens to modern engineers designing bridges and software rendering 3D graphics, trigonometry is one of the most practical and widely applied areas of mathematics.

The Six Trigonometric Functions

The six trigonometric functions — sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot) — are defined based on the ratios of sides in a right triangle relative to a given angle. For an angle θ in a right triangle: sin θ equals the opposite side divided by the hypotenuse, cos θ equals the adjacent side divided by the hypotenuse, and tan θ equals the opposite side divided by the adjacent side.

The reciprocal functions follow directly: cosecant is 1/sin θ, secant is 1/cos θ, and cotangent is 1/tan θ. While the primary three functions are used most frequently, the reciprocal functions appear often in calculus, physics, and engineering contexts.

These definitions extend beyond right triangles through the unit circle — a circle of radius 1 centered at the origin. On the unit circle, the angle θ (measured counterclockwise from the positive x-axis) corresponds to the point (cos θ, sin θ). This definition allows trigonometric functions to accept any real number as input, not just angles between 0° and 90°.

Degrees vs. Radians

Angles can be measured in degrees or radians. A full rotation is 360 degrees or 2π radians. The conversion formula is straightforward: multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. Common angles: 30° = π/6 rad, 45° = π/4 rad, 60° = π/3 rad, 90° = π/2 rad, 180° = π rad.

Radians are the natural unit for calculus and higher mathematics because they simplify many formulas. For example, the derivative of sin(x) is cos(x) only when x is in radians. In physics and engineering, radians are almost universally used. Degrees remain common in everyday contexts and elementary geometry.

Fundamental Trigonometric Identities

Trigonometric identities are equations that hold true for all valid values of the variable. The most fundamental is the Pythagorean identity: sin²θ + cos²θ = 1. This follows directly from the Pythagorean theorem applied to the unit circle. Two related identities are 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ.

The angle addition formulas are also essential: sin(A + B) = sin A cos B + cos A sin B, and cos(A + B) = cos A cos B − sin A sin B. From these, the double-angle formulas can be derived: sin(2θ) = 2 sin θ cos θ and cos(2θ) = cos²θ − sin²θ. These identities are indispensable in calculus for evaluating integrals and simplifying expressions.

Inverse Trigonometric Functions

The inverse trigonometric functions — arcsin, arccos, and arctan (written sin⁻¹, cos⁻¹, tan⁻¹) — return the angle that produces a given trigonometric value. For example, arcsin(0.5) = 30° because sin(30°) = 0.5. These functions are essential for finding unknown angles when the side lengths of a triangle are known.

Because trigonometric functions are periodic, their inverses have restricted domains to ensure a single output. The range of arcsin is −90° to 90° (−π/2 to π/2), the range of arccos is 0° to 180° (0 to π), and the range of arctan is −90° to 90° (−π/2 to π/2). The function atan2(y, x), available in most programming languages, is a two-argument variant of arctan that correctly handles all four quadrants.

Real-World Applications

Trigonometry underpins an enormous range of practical applications. In architecture and construction, it is used to calculate roof pitches, structural loads, and the dimensions of angular cuts. In navigation, the sine and cosine rules allow sailors and pilots to calculate headings and distances. Surveyors use trigonometry to measure land boundaries and elevations without physically traversing every point.

In physics, waves — sound, light, water — are naturally described using sine and cosine functions. The simple harmonic motion of a pendulum or a spring follows a sinusoidal pattern. In electrical engineering, alternating current (AC) voltage and current are sinusoidal, making trigonometry fundamental to circuit analysis.

Computer graphics rely heavily on trigonometry for rotation matrices, perspective projections, and animation. Game developers use atan2 to calculate the direction an object faces, and sine curves to produce smooth, natural-looking motion. Signal processing, GPS positioning, medical imaging (CT scans use the Radon transform), and machine learning (Fourier transforms, positional encodings in transformers) all build on trigonometric foundations.