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Tangent in Trigonometry: Definition & Overview

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

One of the basics of trigonometry, the tangent is quite a helpful tool. In this lesson, learn how you can use the tangent to find missing sides of triangles and how to find important angles for real world applications.

What is the Tangent?

Trigonometry has its roots in the right triangle. And so, the tangent defines one of the relationships in that right triangle.

The definition of tangent.
tangent

The relationship that the tangent defines is the ratio of the opposite side to the adjacent side of a particular angle of the right triangle.

A right triangle.
tangent

In the triangle above, I have marked the appropriate opposite and adjacent sides for angle A. The hypotenuse will never change and is always the side opposite your right angle. The opposite and adjacent sides will change depending on which angle you choose. If you choose the other angle that is not marked, then the opposite and adjacent sides would flip. The adjacent side would become the opposite side and vice versa.

Finding the Tangent

To find the tangent, you first need to locate the hypotenuse. The hypotenuse is usually the longest side of the right triangle. Next, you will need to identify the angle of concern. You only have two angles to choose from. You can't choose the right angle. Once you've chosen your angle, then you need to label your sides. The side opposite your angle is your opposite side and the side next to your angle that is not the hypotenuse is your adjacent side. After your sides are labeled, you then go ahead and take the appropriate ratio.

Let's try finding the tangent of the triangle below.

Find the tangent.
tangent

In this triangle, we have two angles that we could work with, angles A and B. To identify our tangent, we first need to locate our hypotenuse. I can see our right angle. Can you? That means our hypotenuse is directly across from that and it is the side that measures five. Do you see it? Okay, now that we have our hypotenuse, let us choose an angle to work with. Let's choose angle B. If B is our angle, then our opposite side is the side that measures three. Our adjacent side then is the one that measures four because that is the only side next to angle B that is not the hypotenuse. That means our tangent of angle B is the ratio of our opposite side over our adjacent side or 3/4=0.75. If we chose angle A, our sides would flip and our tangent would be 4/3=1.33.

Using the Tangent to Find a Missing Side

In some problems, you'll be asked to find the missing side of a triangle. This has real-world applications when construction companies are building on hills. In these problems, an angle is provided.

Find the missing side.
tangent

To solve such a problem, we first need to identify the missing side. In the triangle above, the missing side is the adjacent side to our provided angle. What we need to do now is to write an equation from the definition of tangent.

Finding the missing side part 1.
tangent

Once we have our equation, we can go ahead and use algebra to solve for our variable or missing side.

Finding the missing side part 2.
tangent

Using algebra, we've isolated our variable by first multiplying both sides by x then dividing both sides by tan 66. We actually use a calculator to calculate tan 66. So our answer is that our missing side equals 2.22.

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