## The Lesson

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:

## 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.

# 1

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

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.

## Lesson Slides

The slider below gives more information about the sine function. Open the slider in a new tab

## Interactive Widget

Here is an interactive widget to help you learn about the sine function on a right triangle.

## Trigonometry and Right Angles

The sine function is a function in trigonometry (called a trigonometric function). The word trigonometry comes from the Greek words 'trigonon' ("triangle") and 'metron' ("measure"). Trigonometry is the branch of mathematics that studies the relationships between the sides and the angles of right triangles. When an angle is defined in a right triangle, the three sides can be defined.

• The side next to the angle is called the adjacent.
• The side opposite the angle is called the opposite.
• The longest side is called the hypotenuse.

## Other Trigonometric Functions

The sine function is only one of the trigonometric functions:

## The Sine Function and the Unit Circle

The sine function can be related to the unit circle. Consider a point on the unit circle (x, y). It can be joined to the center by a radius of length 1. A right triangle can be formed under the radius.

The hypotenuse is the radius of 1, the adjacent side has length x and the opposite side length y.
opposite = sin θ × hypotenuse y = sin θ × 1 y = sin θ