# What Is the Sine Function?

## What Is the Sine Function?

The sine function relates a given angle to the opposite side and hypotenuse of a right triangle.

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.

### Dictionary Definition

The Merriam-Webster dictionary defines the sine function as "the trigonometric function that for an acute angle is the ratio between the leg opposite the angle when it is considered part of a right triangle and the hypotenuse."

The sine of an angle is given by the formula below:

In this formula, **sin** denotes the sine function, **θ** is an angle of a right triangle, the opposite is the length of the side opposite the angle and the hypotenuse is the length of longest side. The image below shows what we mean:

## Interactive Widget

Use this **interactive widget** to create a right-angled triangle and then use the sine function to calculate the hidden element. Start by selecting which element you want to hide (using the green buttons) and then clicking in the shaded area.

Angle: 0° | |

Opposite: ? | |

Hypotenuse: 0 |

**Oops, it's broken!Turn your phone on its side to use this widget.**

## A Real Example of the Sine Function

It is easier to understand the sine function with an example.

### Question

Find sin 30° using the right triangle shown below.

### Step-by-Step:

# 1

Start with the formula:

sin θ = opposite / hypotenuse

**Don't forget:** / means ÷

# 2

Substitute the angle θ, the length of the opposite and the length of the hypotenuse into the formula. In our example, θ = 30°, the opposite is 2 cm and the hypotenuse is 4 cm.

sin (30°) = 2 / 4

sin (30°) = 2 ÷ 4

sin (30°) = 0.5

### Answer:

sin 30° = 0.5.

## The Graph of the Sine Function

The sine function can be plotted on a graph.

Find the angle along the horizontal axis, then go up until you reach the sine graph. Go across and read the value of sin θ from the vertical axis.

We can see from the graph above that sin 30° = 0.5.

## The Sine Function and the Unit Circle

The sine function can be related to a unit circle, which is a circle with a radius of 1 that is centered at the origin in the Cartesian coordinate system.

For a point at any angle θ, sin θ is given by the y-coordinate of the point.