Trigonometric Functions

The trigonometric functions are periodic wave functions that are used throughout math and physics. The functions can be grouped in three related groups: the main functions: sine, cosine, and tangent. The reciprocal functions: cosecant, secant and cotangent. And the arc functions: arc-sine, arc-cosine and arc-tangent.

Introduction

The three main functions are usually introduced using the geometry of the right-triangle. Each function takes an angle of a right triangle as input and returns the ratio of two of its sides.

This figure illustrates the right triangle.
Figure 1: Right Triangle

Later, these defintions are extended to the domain of all real numbers using the geometry of the circle[1]. This is shown by figure 2 which illustrates the geometry of the circle defined by the point in both the Polar and Cartesian Coordinate System. Each of these definitions takes in an angle defined in radians and returns a ratio that can be visualized using the unit circle

This figure illustrates the trigonometric functions defined using the circle.
Figure 2: Circle Functions

This website refers to these definitions of the trigonometric functions as the circle functions. These definitions are the ones discussed on this page.

Main Functions

The main functions are the most commonly used trigonometric functions. Calculators often have the option to toggle between the radian and degree angle systems, but some programming languages and calculators only provide the functions using radians. This is because radians are the preferred unit for measuring angles in math[2].

Sine

Given an angle in radians the sine function returns the component of the corresponding point on the unit circle. The plot of the sine function is shown below for angles from to (tau) radians. Note, represents a full rotation around the circle in radians.

Plot of Sine Function from 0 to τ (tau) radians.
Figure 3: Sine Function

For example, given the angle of radians equivalent to the sine function [eturns the ratio .

Since the function returns a ratio, the relative size of the right-triangle does not matter, only its measured angle. All possible output of the function can be visualized by constraining the hypotenuse to be of length , which forms the shape of the unit circle. Click and drag either of the blue control points in the interactive below to see the input and output of the function.

Cosine

Given an angle in radians the cosine function returns the ratio of the adjacent side over its hypotenuse. The plot of the cosine function is shown below for the domain to (tau) radians, where represents a full rotation. The numeric values for radians are shown on the top edge of the plot’s grid.

Plot of Cosine from 0 to τ (tau) radians.
Figure 4: Cosine Function

For example, given the angle of radians equivalent to the cosine function returns the ratio .

Since the function returns a ratio, the relative size of the right-triangle does not matter, only its measured angle. All possible output of the function can be visualized by constraining the hypotenuse to be of length , which forms the shape of the unit circle. Click and drag either of the blue control points in the interactive below to see the input and output of the function.

Tangent

Given an angle in radians the tangent function returns the tangent line to the point on the unit circle. Since the component of a point goes to zero at a quarter turn and three-quarters turn, the function diverges at those angles.

Plot of the tangent function from 0 to τ (tau) radians.
Figure 5: Tangent Function

Reciprocal Functions

The functions secant, cosecant, and cotangent are the reciprocal functions of cosine, sine, and tangent. Just like the main functions, these functions take in an angle on the circle as input and return the ratio of two components. Each function can be expressed as the reciprocal of its counter-part.

Note: Note, cosecant and secant are not as common as cotangent and, while still relevant, are included for historical reasons.

Cosecant

The cosecant function returns the reciprocal of the sine function. This is visualized in the graph below where the sine function is shown as a green line and the cosecant function is shown as a solid line. The function diverges to infinity where sin approaches zero.

Plot of the cosecant function from 0 to τ (tau) radians.
Figure 6: Cosecant Function

Secant

The secant function returns the reciprocal of the cosine function. This is visualized in the graph below where the cosine function is shown as a dashed line and the secant function is shown as a solid line.

Plot of the secant function from 0 to τ (tau) radians.
Figure 7: Secant Function

Cotangent

The cotangent function returns the reciprocal of the tangent function. This is visualized in the graph below which draws the tangent function in green and cotangent in black.

Plot of the cotangent function from 0 to τ (tau) radians.
Figure 8: Cotangent Function

Arc Functions

The arc-functions are the inverse of cosine, sine, and tangent. Given the ratio of two sides of the right-triangle the functions return the corresponding angle. The prefix arc is used since the angle output is returned using radians, which corresponds to the arc-length of a circle’s circumference,

Note: For real number input, the arc functions will sometimes return a related angle rather than the true angle in the coordinate plane. There are two strategies for returning all angles. One strategy is to use the atan2 function which takes in two numbers as input. Another strategy is to use extend the arc function definitions to use complex numbers.

ArcSine

The ArcSine function is the inverse of the sine function. Given a number value the ArcSine function returns the corresponding angle or a related angle. The output for real number input is shown in the plot below.

Plot of the ArcSine function.
Figure 9: ArcSine Function

Note, for the domain of real numbers, the function only returns angles within the first and fourth quadrant of the coordinate system. The following equation will return the correct angle when given a value corresponding to the first and fourth quadrants. However, it will onlly return a related angle for the second and third quadrants.

For vector related problems, see the atan2 function below which returns angles in all quadrants. Alternatively, the arcsin function can be defined for complex number input.

ArcCosine

Given a value the arccos function returns the corresponding angle. Note, the function only returns angles within the first and second quadrant of the coordinate system.

Plot of the ArcCosine function.
Figure 10: ArcCosine Function

See the atan2 below for a function that returns angles in all quadrants.

ArcTangent

Given a value the ArcTangent function returns the corresponding angle. Note, the function only returns angles within the first and fourth quadrant of the coordinate system. See the atan2 below for a function that returns angles in all quadrants.

Plot of the ArcTangent function.
Figure 11: ArcTangent Function

ArcTangent 2

The Arc Tangent 2 function is the improved version of the Arc Functions and returns angles in all quadrants of the coordinate system.

Note: Be careful of the order of arguments in calculators and programming languages since they are not consistent. Some list before and vice versa.

references

  1. Extend Trigonometric Functions Using the Circle
    Wumbo (internal)
  2. Radians Versus Degrees
    Wumbo (internal)