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Trigonometric Functions of Real Numbers: Definition & Examples

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Trigonometric functions are relationships between the angles of a right triangle and the lengths of its sides. They are an important subject in trigonometry and have applications in fields like navigation and architecture. Learn how to calculate the three primary trigonometric functions: sine, cosine, and tangent.

What Are Trigonometric Functions?

Trigonometric functions are mathematical relationships between the angles and sides of a right triangle. The three primary trigonometric functions are sine, cosine, and tangent.

Why might you need to know these relationships? Imagine that there is a tree in your backyard and you are worried that it might one day fall on your house. You know that the base of the tree is 50 m from your house, but you don't know if the tree is taller than 50 m or not. Trigonometric functions can help you!

tree and trig functions

All you have to do is stand in your backyard, measure the distance between yourself and the tree, and use a protractor to gauge the angle of your line of sight to the top of the tree. Using only these two measures, you can calculate the height of the tree and know whether your house is safe.

Calculating the Trigonometric Functions

Let's look at each one of the trigonometric functions and see how it is calculated.

To calculate the sine of an angle in a right triangle, you always divide the length of the side opposite the angle by the length of the hypotenuse of the angle. Sine is usually abbreviated as sin in mathematical statements and on calculators.

sine of an angle

Cosine, typically abbreviated as cos, is calculated by dividing the length of the side adjacent to the angle by the length of the hypotenuse of the triangle.

cosine of an angle

Finally, to calculate the tangent of an angle, divide the length of the side opposite the angle by the side adjacent to the angle. Tangent, abbreviated tan, is the only one of the three trigonometric functions that does not involve the hypotenuse of the triangle at all.

tangent of an angle

To remember how to calculate the three trigonometric functions, think about the acronym SOH CAH TOA.

how to remember the trig functions

Example Problems

1. Let's first try to find the sine, cosine, and tangent of angle B in the right triangle shown below:

Example 1

To find the sine of angle B, look at the hypotenuse and the side opposite the angle:

sin B = opposite/hypotenuse = 6/10

You could simplify this answer and write it as either a fraction or a decimal number:

sin B = 3/5 or sin B = 0.6

To find the cosine of angle A, look at the side adjacent, or next to, angle A:

cos B = adjacent/hypotenuse = 8/10

cos B = 4/5 or cos B = 0.8

To find the tangent of angle B, divide the length of the opposite side by the length of the adjacent side:

tan B = opposite/adjacent = 6/8

tan B = 3/4 or tan B = 0.75

2. The trigonometric functions of sine, cosine, and tangent can also help you find the lengths of the sides of a right triangle. Let's look at how to find the missing side in this triangle:

Example 2

In this triangle, we know the angle (45 degrees) and the side opposite the angle (6), but we don't know the hypotenuse. Which trig function includes the opposite side and the hypotenuse?

Did you guess sine? That's right!

sin = opposite/hypotenuse

So, to solve for the hypotenuse (labeled x), we would write:

sin (45) = 6/x

You can then solve this equation for x which gives:

x = 6/sin (45)

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