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GRE Math: Study Guide & Test Prep26 chapters | 166 lessons

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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.

**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!

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.

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.

**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.

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.

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

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

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:

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)**

To find the sine of 45 degrees, you can use the *sin* button you will find on many calculators. Alternatively, you can search online or in a geometry textbook for a trigonometry table, which lists functions of angles.

When you use a calculator to find the sine, cosine, or tangent of an angle, be very careful that your calculator is in the right mode. Some calculators allow you to calculate trigonometric functions using angles measured in radians OR degrees, so you want to make sure yours is set to use degrees for this problem.

Whether you decide to use a calculator or a table, you will find that the sine of 45 degrees is 0.7071, so now you can calculate x.

**x = 6/0.7071**

**x = 8.5**

Now that we know how to use the trigonometric functions, let's revisit the problem of the tree that might fall on your house. Suppose you measure that you are standing 20 m from the tree, and that your line of sight to the top of the tree forms a 53 degree angle with the ground. How tall is the tree?

In this case, we can use the tangent function to find the height of the tree.

**tan 53 = h/20**

**h = 20 * tan 53**

**h = 20 * 1.3270**

**h = 26.5 m**

So, the tree is only 26.5 m tall. You can sleep better at night knowing that it will not hit your house if it falls over!

The three primary **trigonometric functions** are **sine**, **cosine**, and **tangent**. These functions relate the angles of a right triangle to the lengths of the sides of the triangle. They can be calculated as follows:

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GRE Math: Study Guide & Test Prep26 chapters | 166 lessons

- Exponentials, Logarithms & the Natural Log 8:36
- List of the Basic Trig Identities 7:11
- Trigonometric Function Values of Special Angles 7:23
- Trigonometric Functions of Real Numbers: Definition & Examples
- Trigonometry and the Pythagorean Theorem 4:14
- Using Graphs to Determine Trigonometric Identity 5:02
- Properties of Inverse Trigonometric Functions 7:56
- Go to Logarithms & Trigonometry

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