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: arcsine, arccosine and arctangent.
The three main functions are usually introduced using the geometry of the righttriangle. Each function takes an angle of a right triangle as input and returns the ratio of two of its sides.
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 website refers to these definitions of the trigonometric functions as the circle functions. These definitions are the ones discussed on this page.
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]}.
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.
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 righttriangle 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.
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.
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 righttriangle 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.
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 threequarters turn, the function diverges at those angles.
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 counterpart.
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.
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.
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.
The arcfunctions are the inverse of cosine, sine, and tangent. Given the ratio of two sides of the righttriangle the functions return the corresponding angle. The prefix arc is used since the angle output is returned using radians, which corresponds to the arclength of a circle’s circumference,
atan2
function which takes in two numbers as input. Another strategy is to use extend the arc function definitions to use complex numbers.
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.
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.
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.
See the atan2 below for a function that returns angles in all quadrants.
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.
The Arc Tangent 2 function is the improved version of the Arc Functions and returns angles in all quadrants of the coordinate system.
A complex number is an extension of the real number line where in addition to the "real" part of the number there is a complex part of the number. The properties of complex numbers are useful in applied physics as they elegantly describe rotation.
The sine function returns the sine of a number provided in radians. In geometric terms, the function returns the vertical component of the point formed by the angle on the unit circle.
The cosine function returns the cosine of an angle provided in radians. In geometric terms, the function returns the ratio of the righttriangle's adjacent side over its hypotenuse.
Given the angle of a right triangle as input returns the ratio of the triangle's opposite side over its adjacent side.
Radians are a unit that measure angle using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle.
The circle functions are the extended definitions of the trigonometric functions. These definitions extend the domain of the functions such as sine, cosine and tangent to all real numbers.
Degrees are a unit of measure for angles. A full rotation is equal to 360 degrees. In the XY Cartesian Coordinate System, degrees are measured starting from the rightmost edge of the circle.
Given the angle of a right triangle as input, returns the ratio of the hypotenuse over the adjacent side. The secant function is the inverse of the cosine function.
Given the angle of a right triangle as input, returns the ratio of the hypotenuse over the opposite side. The cosecant function is the inverse of the sine function.
Given the angle of a right triangle as input, returns the ratio of the adjacent side over the opposite side.

Extend Trigonometric Functions Using the CircleWumbo (internal)

Radians Versus DegreesWumbo (internal)