Trigonometry

Introduction

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.

History

The earliest known work on trigonometry was recorded in Egypt and Babylon. Ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. However, it was the Greeks who are seen as the original pioneers of trigonometry.

Ancient Greek mathematicians working on geometrical problems.

Basic concepts

Trigonometry is based on certain ratios, called trigonometric functions, to be explained in the next section. In a right triangle, the trigonometric ratios of sine, cosine, and tangent (often abbreviated as sin, cos, and tan respectively) are used to relate the angles of the triangle to the lengths of its sides.

Trigonometric functions

The primary trigonometric functions are the sine and cosine of an angle. These are defined for a right triangle as follows:

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

The other trigonometric functions are defined in terms of sine and cosine and can be calculated using the Pythagorean theorem. These include:

The tangent of an angle is the ratio of the sine to the cosine.
The cotangent is the reciprocal of the tangent.
The secant is the reciprocal of the cosine.
The cosecant is the reciprocal of the sine.

Applications of Trigonometry

Trigonometry has a wide range of applications in various fields such as physics, engineering, computer science, architecture, and more. It is used to calculate distances and angles in navigation, physics, and engineering. In computer science, it is used in graphics, creating sound waves, and other numerical computations.

An architect using trigonometry in his work.

Trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the equation are defined. These identities are useful in simplifying complex trigonometric expressions and in solving trigonometric equations.

Trigonometric equations

Trigonometric equations are mathematical equations that involve trigonometric functions. These equations often have infinitely many solutions, and one way to solve them is by using the unit circle.

Conclusion

Trigonometry is a powerful mathematical tool that has a wide range of applications in various fields. Its concepts and formulas are fundamental to many areas of mathematics and science.