Sine Function

Sine Function

Summary

The sine function returns the sine of a number provided in radians. In geometric terms, given the angle of a right-triangle as input, the function returns the ratio of the triangle's opposite side over its hypotenuse.

Syntax

sin(θ)

Arguments

Name Description
θ (theta) The angle provided in radians.

Return Value

Returns the sine of a number.

Usage Notes

The sine function returns the sine of a number provided in radians. For example, given the angle of radians (equivalent to ) the function returns the ratio .

Degrees Examples

Most calculators allow the user to specify the radians angle system or the degrees angle system. These examples use the degrees angle system to demonstrate the input and output of the sine function.

Sine of 30 Degrees
Special Right Triangle 30 Degrees

The sine of is given by the length of the opposite side oer the length of the hypotenuse of the corresponding right triangle. In this case, we can use the properties of the 30-60-90 Right Triangle, where the opposite length is and the hypotenuse length is to find the ratio.

Steps
  1. Start by setting up the definition of the sine funtion.

  2. Substitute the angle, opposite and hypotenuse values into the expression.

  3. The sine of is equal to .

Sine of 45 Degrees
Special Right Triangle 45 Degrees

The sine of is given by the length of the opposite side over the length of the hypotenuse of the corresponding right triangle. In this case, we can use the properties of the 45-45-90 Right Triangle, where the opposite length is and the hypotenuse length is to find the ratio.

Steps
  1. Start by setting up the definition of the sine funtion. Then substitute the angle, opposite and hypotenuse values into the expression.

  2. The sine of is equal to the ratio . Some mathematicians prefer not to have a radical in the denominator and so they multiply by in the form of .

  3. The sine of is equal to the ratio .

  4. The sin of is approximately equal to .

Sine of 60 Degrees
Special Right Triangle 60 Degrees

The sine of is given by the length of the opposite side over the length of the hypotenuse of the corresponding right triangle. In this case, we can use the properties of the 30-60-90 Right Triangle, where the opposite length is and the hypotenuse length is to find the ratio.

Steps
  1. Start by setting up the definition of the sine funtion. Then substitute the angle, opposite and hypotenuse values into the expression.

  2. The sine of is equal to the ratio .

  3. The sine of is approximately equal to .

Radians Examples

The radians angle system measures angles using the radius of a circle. A full rotation in radians is equal to (tau) radians. For reference, a full rotation in degrees is . These examples use the radians angle system to demonstrate the input and output of the sine function.

Sine of Tau Over 12
Special Right Triangle Radians

The sine of radians is equal to the ratio . This is given by the special triangle with a hypotenuse of length and an opposite side of length .

Steps
  1. Start by setting up the definition of the sine funtion.

  2. Substitute the angle, opposite and hypotenuse values into the expression.

  3. The sine of radians is equal to the ratio .

Sine of Tau Over 8
Special Right Triangle Radians

The sine of radians is equal to the ratio . This is given by the special triangle with a hypotenuse of length and an opposite side of length .

Steps
  1. Start by setting up the definition of the sine funtion. Then substitute the angle, opposite and hypotenuse values into the expression.

  2. The sine of radians is equal to the ratio . Some mathematicians prefer not to have a radical in the denominator and so they multiply by in the form of .

  3. The sine of is equal to the ratio .

  4. The sine of is approximately equal to .

Sine of Tau Over 6
Special Right Triangle Radians

The sine of radians is equal to the ratio . This is given by the special triangle with a hypotenuse of length and an opposite side of length .

Steps
  1. Start by setting up the definition of the sine funtion.

  2. Substitute the angle, opposite and hypotenuse values into the expression.

  3. The sine of radians is approximately equal to the ratio .

Explanation

The graph of the sine function is shown below which demonstrates all possible output of the sine function using the range of to , where .

Sine Graph

The sine function is periodic, meaning that the output of the function repeats indefinitely. Geometrically, the function can be visualized with a right-triangle of hypotenuse length placed on the unit circle. The function returns the length of the opposite side, which since the hypotenuse is of length represents the ratio for all triagnles with the given angle. After a full rotation is completed the output of the function starts to repeat.

The interactive above demonstrates this geometry 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.

Angle | Concept

An angle is defined as the amount of rotation between two rays. Angles are measured using degrees and radians. A full rotation in degrees is 360°. A full rotation in radians is approximately 6.283 radians or τ (tau) radians.

Radians | Concept

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.

Right Triangle | Concept

A right triangle is a triangle where one of the three angles is a perpendicular angle. There are three sides of the right triangle: the adjacent, opposite, and hypotenuse sides.

Law of Sines | Concept

The law of sines is an equation that relates the three sides of a triangle with the three angles of a triangle using the sine function.

Unit Circle | Concept

The unit circle is a unifying idea in mathematics that connects many useful concepts together. This article goes over the basic properties of the circle using interactive examples and explains how they connect to the trigonometric functions and pythagorean theorem.

Trigonometric Identities | Concept

The trigonometric identites are a set of equations derived from the properties of the right triangle and the circle.