Tangent Ratio: Definition & Formula

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  • 0:01 What Is a Tangent Ratio?
  • 0:55 Understanding Key Vocabulary
  • 1:56 Finding the Tangent Ratio
  • 3:41 Examples
  • 6:28 Lesson Summary
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Lesson Transcript
Instructor: David Karsner

David holds a Master of Arts in Education

The tangent ratio is a tool used with right triangles that allows one to find the length of the sides of a triangle given the degree of its angles. It can also be used to find the degrees of its angle given the length of two of its sides.

What Is a Tangent Ratio?

A right triangle is a triangle that contains a right angle. A right angle is an angle measuring 90 degrees. Any right triangle will have two angles that are not right angles and two sides that are not the hypotenuse. The hypotenuse is the side of a right angle that is always across from the right angle and is the longest side. The tangent ratio is a comparison between the two sides of a right triangle that are not the hypotenuse. If two different sized triangles have an angle that is congruent, and not the right angle, then the quotient of the lengths of the two non-hypotenuse sides will always give you the same value. Remember that congruent is just a fancy way of saying that two or more sides, angles, or triangles have the same measures. This lesson will show how the tangent ratio works and give several examples.

Understanding Key Vocabulary

The tangent ratio is part of the field of trigonometry, which is the branch of mathematics concerning the relationship between the sides and angles of a triangle. As you may have already noticed, there are a lot of terms you need to understand before you can really understand how to calculate the tangent ratio. We've already explained most of them, but there are a few more you need to learn. The first is angle theta, which is the angle being considered or the angle that is congruent between the two or more triangles you're comparing. It is not the right angle. Theta is a common variable when using angles, but other variables can be used. When we use the word opposite, we are referring to the side that is across from the angle theta. When we use the word adjacent, we mean the side that is forming angle theta and is not the hypotenuse. The tangent ratio is the value received when the length of the side opposite of angle theta is divided by the length of the side adjacent to angle theta. It is very commonly abbreviated as tan.

Finding the Tangent Ratio

3 Right Triangles that have a 37 degree angle
Right Triangles with theta equals 37 degrees

This image shows three right triangles with sides of different lengths but angle theta is the same, or congruent, for all three triangles. The tangent ratio was defined as the side opposite of angle theta divided by the side adjacent to angle theta. Let's look at the tangent ratio for all three triangles now, using the information in this image. Remember that the angle theta is the same for all of them, and that is 37 degrees.

For the smallest triangle, we know that the opposite side is 3 and the adjacent side is 4, which gives us a ratio of ¾ or .75. For the medium triangle, we know that the opposite side is 12 and the adjacent side is 16. This gives us a ratio of 12/16 or .75. For the largest triangle, we know that the opposite side is 27 and the adjacent side is 36, which gives us 27/36 = .75

As you can see, the tangent ratio was .75 for all three triangles. That will be the case for all 37 degree angles in right triangles. If you have a calculator with a tangent key enter tan(37) into the calculator and it should yield .75355 which, rounded to two decimal places, is .75. Word of caution: be sure that whatever calculator you are using has the setting for tangent set for degrees and not radians.

Scientific and graphing calculators have stored in their memory all the values of each angle and its tangent value. When one types a tangent on a calculator and then enters an angle measurement and then the enter key, one gets the value of the opposite side/adjacent side.


Let's do a few more examples together now that we know how this works.

Find the value of X

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