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.
Calculators often have the option to toggle between the radian and degree angle systems, but some programming languages and calculators only provide implementations of the functions using radians. This is because radians are the preferred unit for measuring angles in math^{[1]} and by using radians the functions ineherit useful properties.
The trigonometric functions are usually introduced using the geometry of the righttriangle as shown on this page. Later, the definitions are extended using the geometry of the circle as shown below. This website refers to these as the “circle definitions” of the trigonometric functions.
Each of the functions takes in a radian angle as input and returns a ratio as output. Geometrically, the defintions correspond to a circle with a radius and a point that forms the angle (theta). The functions can be visualized by the geometry of the unit circle. This can be seen by substituting into the circle definitions above.
The main functions are the most commonly used trigonometric functions.
Given an angle in radians, the sine function returns the sine of the angle. For example, given the angle of radians the sine function returns the value .
This example can be visualized by drawing a circle and point corresponding to the input angle. Since the function returns a ratio, the size of the circle doesn’t matter. This is shown below
All possible output of the function can be visualized on the unit circle. Geometrically, given an angle as input the sine function returns the vertical component of the point corresponding to the angle on the unit circle. This is shown in the interactive below. Click and drag either blue control point.
Given an angle in radians, the cosine function returns the cosine of the angle. For example, given the angle the cosine function returns the value .
This example can be visualized by drawing a circle and point corresponding to the input angle. The size of the circle doesn’t matter since the function returns a ratio. This circle is shown beow.
All possible output of the function can be visualized on the unit circle. Geometrically, given an angle as input the cosine function returns the horizontal component of the point corresponding to the angle on the unit circle. This is shown in the interactive below. Click and drag either blue control point.
Given an angle in radians, the tangent function returns the tangent of the angle. Geometrically, the output of the function can be visualized by the line drawn tangent to the point on the unit circle formed from the angle. Shown below is the plot of the tangent function.
Below is an interactive which visualizes the output of the tangent function on the unit circle. Click and drag either blue control point.
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 a trigonometric ratio. Each function can be expressed as the reciprocal of its counterpart.
The cosecant function returns the reciprocal of the sine function. The function can also be written in terms of sine.
The plot below shows the output of the cosecant function in black and the sine function in green.
The secant function returns the reciprocal of the cosine function. The function can also be written in terms of cosine.
The plot below shows the output of the secant function in black and the cosine function in green.
The cotangent function returns the reciprocal of the tangent function. The function can also be written in terms of tangent.
The function is visualized in the plot below which draws the tangent function in green and cotangent in black.
The output of cotangent can be visualized on the unit circle as the other line tangent to the point on the unit circle corresponding to the angle.
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.
The ArcSine function is the inverse of the sine function. Given a number value the ArcSine function returns the corresponding angle. The plot of the function is given below.
Note, the function only returns angles within the first and fourth quadrant of the coordinate system. So the following expression is not true for all angles.
The ArcCosine function is the inverse of the cosine function. Given a value the ArcCosine function returns the corresponding angle. The plot of the function is given below.
Note, the function only returns angles within the first and second quadrant of the coordinate system. So the following expression is not true for all angles.
See the atan2 below for a function that returns angles in all quadrants.
The ArcTangent function is the inverse of tangent. Given a value the ArcTangent function returns the corresponding angle. The plot of the function is given below.
Note, the function only returns angles within the first and fourth quadrant of the coordinate system. So the following expression is not true for all angles.
However, the improved version shown below does.
The Arc Tangent 2 function is the improved version of the Arc Functions and returns angles in all quadrants of the coordinate system. Given the coordinates of a point, the function returns the angle corresponding to the point.
ATAN2
function in calculators and programming languages since their order is not consistent. Some list before and vice versa.
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.
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.
There are six trigonometric functions that relate to the geometry of the righttriangle sine, cosine, tangent, cosecant, secant, and cotangent. The functions take the angle of a right triangle as input and return a ratio of two of its sides.
The unit circle is a circle of radius one placed at the origin of the coordinate system. This article discusses how the unit circle represents the output of the trigonometric functions for all real numbers.
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.
Given a number representing the ratio of a right triangle's opposite side over its hypotenuse returns the corresponding angle.
Given a number representing the ratio of a right triangle's adjacent side over its hypotenuse returns the corresponding angle.
Given a number representing the ratio of a right triangle's opposite side over its adjecent side returns the corresponding angle.
Given the x and y components of a point, returns the angle associated with the point.

Radians Versus DegreesWumbo (internal)