Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between angles and the lengths of the sides of triangles. The name comes from Greek words meaning "measurement of triangles". The foundation of the field rests on functions called trigonometric functions: sine (sin), cosine (cos), and tangent (tan), which are ratios between the sides of a right triangle (a triangle in which one angle is 90 degrees).

Trigonometry was originally used for land surveying, navigation, and astronomy, because distances and areas could be calculated by measuring only a single angle. Today trigonometric functions have extended far beyond triangles alone: they have become periodic functions used to describe waves, oscillating motion, and electrical signals.

Definition of functions

In a right triangle, when looking at an angle θ that is not the right angle: sine θ is the ratio of the side opposite θ to the hypotenuse (the longest side, opposite the right angle); cosine θ is the ratio of the side adjacent to θ to the hypotenuse; tangent θ is the ratio of sine to cosine. There are three additional functions: cosecant, secant, and cotangent. To define angles greater than 90 degrees or negative angles, the unit circle is used, which extends these functions to all real numbers.

Angle measurement

Angles are measured in two standard systems: degrees, where a full rotation is 360 degrees, and radians, where a full rotation is 2π radians. Radians are more convenient for higher mathematics (calculus), because the derivative formulas for sine and cosine are simpler when radians are used.

Laws and identities

For triangles without right angles, the law of sines and law of cosines are used, which relate the sides and angles of any triangle. There are also many identities (equations that are always true), the most well-known being the statement that sin²θ plus cos²θ always equals 1. Identities for angle addition and double angles simplify calculation and the solving of equations.

Applications

Engineering, land surveying, navigation of ships and aircraft, and architectural design use trigonometry to calculate distances and directions that cannot be measured directly. Physics and electrical engineering rely on how these functions describe waves; Fourier analysis decomposes any signal into a combination of sines and cosines. Computer graphics and satellite positioning systems also depend on these methods.