Trigonometric Functions

The trigonometric functions relate the geometry of the right triangle together. There are three main functions: sine, cosine, and tangent. Each function takes the angle of a right triangle as input and returns the ratio of two of its sides.

Right Triangle

Cosine Function

Given an angle the cosine function returns the ratio of the adjacent side over its hypotenuse. The function can be visualized with a right-triangle of hypotenuse one, since this triangle is the general case for all similar triangles.

Sine Function

Given an angle the sine function returns the ratio of the opposite side over its hypotenuse. Same as cosine, the function can be visualized with a right-triangle of hypotenuse one, since this triangle is the general case for all similar triangles.

Tangent Function

Given an angle the tangent function returns the ratio of the opposite side over its adjacent side.

Graph of the tangent function

Unit Circle

The unit circle can be used to visualize the geometric properties that each function represents. Since the radius of the circle is one, the hypotenuse of the right triangle can be subsituted and the equations simplified.

Cartesian Coordinate System

The right triangle is often placed at the center of the Cartesian Coordinate system, and instead of using the adjacent, opposite, and hypotenuse sides of the triangle, the variables x, y, and r are used instead. This is convenient for applications of the trigonometric functions in physics and other areas.

Right Triangle
Right Triangle in Cartesian Coordinate System

The variable r is used to represent the radius of the circle which is also the hypotenuse of the right triangle. In the case of the Unit Circle, r will always have a value of 1, which is why the functions are sometimes simplified as shown on the right.

Arc Functions

The arc functions take in a ratio of sides as input and return the corresponding angle. These functions are sometimes called the inverse functions, but to hopefully reduce confusion with the reciprocal functions (below) they are referred to as the arc functions on this website.

Arc Cosine Function

Graph of the arc cosine function Graph of the cosine function

Arc Sine Function

Graph of the arc sine function Graph of the sine function

Arc Tangent Function

Graph of the arc tangent function Graph of the tangent function

Reciprocal Functions

The reciprocal functions are the reciprocals of the three main functions: cosine, sine, and tangent. The functions can be visualized geometrically by the larger triangle formed on the unit circle.

Trigonometric Functions

Cosecant Function

The cosecant function returns the reciprocal of the sine function. This is visualized in the graph below where the sine function is highlighted in green.

Graph of the cosecant function

Secant Function

The secant function returns the reciprocal of the cosine function. This is visualized in the graph below where the cosine function is highlighted in green.

Graph of the secant function

Cotangent Function

The cotangent function returns the reciprocal of the tangent function. This is visualized in the graph below.

Cotangent Function