Trigonometric functions

The trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are fundamental in the study of geometry, physics, engineering, and many other fields. The most commonly used trigonometric functions are sine, cosine, and tangent, often abbreviated as sin, cos, and tan, respectively. These functions are defined for angles, which can be measured in radians or degrees.

Definition

Trigonometric functions are defined using a right-angled triangle or a unit circle (a circle with a radius of one unit). For angles in a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse, the cosine is the ratio of the length of the adjacent side to the hypotenuse, and the tangent is the ratio of the length of the opposite side to the length of the adjacent side.

In the context of the unit circle, these functions are defined for any real angle as follows:

• The sine of an angle is the y-coordinate of the point on the unit circle at that angle,
• The cosine of an angle is the x-coordinate,
• The tangent of an angle is the y-coordinate divided by the x-coordinate (when the x-coordinate is not zero).

Properties and Applications

Trigonometric functions have many important properties, such as periodicity and symmetry. They are periodic in that they repeat their values in regular intervals, making them especially useful in the analysis of waves and oscillations. The sine and cosine functions, for example, have a period of \(2\pi\) radians (or 360 degrees), meaning that their values repeat every \(2\pi\) radians.

These functions are also used to solve triangles, particularly in trigonometry, where the lengths of a triangle's sides and its angles are related. This is crucial in many areas of applied mathematics, physics, and engineering, such as in the design of structures, the analysis of waves, and in navigation.

Extensions

Beyond the basic trigonometric functions, there are several related functions that are useful in various contexts. These include:

• The secant (sec), which is the reciprocal of the cosine,
• The cosecant (csc), the reciprocal of the sine,
• The cotangent (cot), the reciprocal of the tangent.

There are also hyperbolic trigonometric functions which are analogs of the trigonometric functions but for a hyperbola instead of a circle. These include hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh), among others.

Complex Numbers

Trigonometric functions can also be defined for complex numbers, greatly extending their applicability. This is done using Euler's formula, which relates complex exponentials to trigonometric functions, providing a powerful tool in complex analysis and other areas of mathematics.

See Also

• Inverse trigonometric functions
• Trigonometry
• Unit circle
• Euler's formula