Sine Function

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

Summary

The sine function returns the sine of an angle provided in radians. In geometric terms, the function returns the vertical component of the point formed by the angle on the unit circle.

Syntax

sin(θ)

Arguments

Name Description
θ (theta) The radian angle

Return Value

Returns the sine of an angle provided in radians.

Usage

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 corresponding angle (theta) on the unit circle. For example, given the angle of radians (one third of a turn) the function returns the value .

Visually the return value is equal to the y-component of the point on the unit circle corresponding to the angle of radians. This is shown in the figure below which illustrates the unit circle in the XY coordinate plane and highlights the angle in red and the vertical component of the point in blue.

Visualization of the return value of sine on the unit circle.
Figure 1: Sine of One Third Tau

Examples

These examples show the return value of sine for some common angles formed from dividing the unit circle into equal parts. Note, the circle constant (tau) represents a full rotation in radians.

These examples show the return value of sine for some common angles formed from dividing the unit circle into equal parts.

 sin(0*TAU/8) = 0  
 sin(1*TAU/8) = 0.7071... // sqrt(2)/2  
 sin(2*TAU/8) = 1 
 sin(3*TAU/8) = 0.7071... // sqrt(2)/2  

Interactive

The interactive above visualizes the geometric relation sine has on the unit circle. Click and drag either of the two control points to watch the input and output change. The input is highlighted in red and represents the input angle in radians. The output is highlighted in blue.

Explanation

The sine function is part of a group of trigonometric functions that return ratios related to the geometry of the circle. The sine function is one of the three main functions: sine, cosine and tangent which take in an angle as input and return a ratio as output. The graph of the sine function is shown below which demonstrates the possible output of the sine function for the domain of to , where . This plot demonstrates the output of one full rotation around a circle.

Plot of Sine Function from 0 to τ (tau) radians.
Figure 2: Sine Plot

The sine function is periodic, meaning that the output of the function repeats indefinitely. Geometrically, the function’s output can be visualized as the vertical component of a point on the unit circle corresponding to the input angle. This can be seen from the definition of the function shown below in terms of the circle.

Definition

This figure illustrates the geometry used to define the circle definitions of the trigonometric functions.
Figure 3: Circle Functions

Conceptually, the sine of an angle is defined by the ratio of a correspoding point’s coordinate over the radius (distance from the origin). Geometrically, the function returns the coordinate of the point formed by the angle on the unit circle. This can be seen by substituting into the definition above.

This figure illustrates the trigonometry of the unit circle.
Figure 4: Unit Circle Trigonometry

Taylor Series

The sine function can be defined using calculus by a Taylor Series approximation[1].

Links

References

  1. Derive Sine Function (Taylor Series)
    Wumbo (internal)