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:

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.

Slider

The slider below gives more information about the sine function.

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See Also

What is an angle? What is a right triangle? What is the opposite? What is the hypotenuse? What is the adjacent? What is the cosine function? What is the tangent function? What is the y-coordinate? Learn more about the sine function on a right triangle ( interactive widget)