Sine: Definition & Examples

Instructor: Jennifer Beddoe
Sine is one of the three main trigonometric ratios. It's based on the measurements of a right triangle and helps you find angle measures and distances, among other things. This lesson defines the sine function and gives examples of when it is used.

Trigonometric Functions

There are six main trigonometric functions:

  1. Sine (sin)
  2. Cosine (cos)
  3. Tangent (tan)
  4. Secant (sec)
  5. Cosecant (csc)
  6. Cotangent (cot)

These functions are used to relate the angles of a triangle with the sides of that triangle. Trigonometric functions are important when studying triangles and modeling periodic phenomena, such as waves, sound and light.

The Sine Function

The sine function is defined as the ratio of the side of the triangle opposite the angle divided by the hypotenuse.

Right triangle

One way to remember this ratio along with the ratios for the other most common trigonometric functions is with the mnemonic SOH-CAH-TOA:

SOH = Sine is Opposite over Hypotenuse

CAH = Cosine is Adjacent over Hypotenuse

TOA = Tangent is Opposite over Adjacent

This ratio can be used to solve problems involving distance or height, or if you need to know an angle measure.

Example:

Imagine a ship that is tethered to an anchor on the ocean floor.

boat problem

The anchor's chain is 30 m long and creates a line that is the hypotenuse of a right triangle. The angle between that hypotenuse and the ocean floor is 39 degrees. What is the depth of the anchor?

Since it isn't practical to dive down and measure how deep the anchor is, we can use a trigonometric ratio to figure it out.

We know the angle the cable makes with the ocean floor and the length of the cable (hypotenuse). To find the length of the side opposite the angle, d, we use the sine function.

sin 39 = d/30

0.63 = d/30

d = 18.9 m

The Sine Function as a Periodic Function

Trigonometric functions are called periodic functions because they repeat over a given period.

Look at the graph of the sine function:

Sine function graph, image by Geek3
Sine graph

You can see that the graph repeats itself at a distance of 2π. Therefore, we can say that the sine function has a period of 2π. Usually when looking at the sine function in this way, you don't use degree measure, but radians. Radian is the standard unit of angle measurement used in mathematics. A full circle is 2π radians, which is equal to 360°.

Sine and Its Relation to a Unit Circle.

In trigonometry, a unit circle is a circle centered on the origin (0,0) of a coordinate plane with a radius of 1.

Unit circle, image by Brews ohare
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Any point on the circle corresponds to the sine and cosine made by the angle created by a line from the center of the circle to that point and the x-axis. If the point is (x, y), the relationship looks like this:

sinθ = y

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