Trigonometry

Trigonometry began as the study of triangles, relating their angles to the lengths of their sides. In a right triangle the sine, cosine and tangent of an acute angle are defined as ratios of the opposite, adjacent and hypotenuse sides. These ratios allow unknown sides or angles to be found from known ones and form the basis of practical methods in surveying, navigation and engineering. The law of sines and the law of cosines extend these relationships to triangles of any shape.

Defining the functions on the unit circle generalises them to all angles and reveals their periodic nature, with sine and cosine repeating after a full turn. This periodicity makes trigonometric functions the natural language for describing oscillations and waves. A large body of identities — the Pythagorean identity, the angle-sum and difference formulas, and the double- and half-angle formulas — allows expressions to be rewritten and trigonometric equations to be solved. Because of periodicity, such equations generally have infinitely many solutions, differing by whole multiples of the period.

Closely related are the hyperbolic functions, defined using the exponential function rather than the circle; they share many algebraic properties with the ordinary trigonometric functions and arise in the study of curves such as the catenary. Euler's formula provides a deep link between these worlds, expressing a complex exponential in terms of cosine and sine and connecting trigonometry to complex analysis. This identity unifies seemingly separate constants and functions in a single relation.

Frequently asked questions

What are sine, cosine and tangent?

In a right triangle they are ratios of pairs of sides relative to an angle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

How do hyperbolic functions differ from trigonometric ones?

Hyperbolic functions are built from the exponential function and relate to a hyperbola rather than a circle, but they obey closely parallel identities.

What does Euler's formula state?

It expresses a complex exponential as the cosine of an angle plus the imaginary unit times its sine, linking trigonometry with complex numbers.