How To Find The Hypotenuse Of A Triangle Using Trigonometry

In this geometry lesson, you're going to learn all about SohCahToa.

Jenn (B.S., M.Ed.) of Calcworkshop® teaching soh-cah-toa

Jenn, Founder Calcworkshop^®, 15+ Years Experience (Licensed & Certified Teacher)

It's probably one of the most famous math mnemonics alongside PEMDAS.

And it's an essential technique for your mathematical toolbelt.

Let's get to it!

What Is SohCahToa?

It's a mnemonic device to help you remember the three basic trig ratios used to solve for missing sides and angles in a right triangle.

It's defined as:

SOH: Sin(θ) = Opposite / Hypotenuse
CAH: Cos(θ) = Adjacent / Hypotenuse
TOA: Tan(θ) = Opposite / Adjacent

We'll dive further into the theory behind it in the video below, but essentially it's taken from the AA Similarity Postulate that we learned about previously. It stated that the ratios of the lengths of two sides of similar right triangles are equal. Therefore, the sets of ratios depend only on the measure of the acute angle, not the size of the triangle.

Key Point: Regardless of the size of the triangle, these trigonometric ratios will always hold true for right triangles.

Remember the three basic ratios are called Sine, Cosine, and Tangent, and they represent the foundational Trigonometric Ratios, after the Greek word for triangle measurement.

And these trigonometric ratios allow us to find missing sides of a right triangle, as well as missing angles.

How To Remember Trig Functions?

So how do we remember these three trig ratios and use them to solve for missing sides and angles?

First, we remember how the sides of a right triangle are labeled:

Opposite Leg
Adjacent Leg
Hypotenuse

The opposite leg is opposite one of the acute angles, the adjacent leg is next to the acute angle, and the hypotenuse is opposite the right angle, as it's the longest side, as noted by the University of Georgia.

opposite adjacent hypotenuse diagram

Right Triangle Diagram

Then we use the mnemonic device we talk about earlier: SOHCAHTOA!

sohcahtoa chart

SOH-CAH-TOA Chart

An easy way to remember the order of Sin, Cos, and Tan is to use saying such as:

Some Of Her Children Are Having Trouble Over Algebra

If you can remember the order of the trigonometric functions, then a quicker saying would be:

Oscar Had A Heap Of Apples

Modifying our equations from earlier, we have:

SOH: Sin(θ) = Oscar / Had
CAH: Cos(θ) = A / Heap
TOA: Tan(θ) = Of / Apples

There are many more fun sayings as well.

Few Examples…

Given the following right triangle, solve for the missing side length, r:

how to find hypotenuse with sin

Using Sin to Find the Hypotenuse

Sometimes we are given two sides lengths, and we need to determine one of the acute angles of the right triangle.

How can we do this?

Inverse Trig Ratios allow us to solve for those missing angles quite easily. Consequently, SOHCAHTOA is very versatile as it grants us the ability to solve for sides and angles of a right triangle!

inverse trig values

Inverse Trig Values

In the next example we are asked to "Solve the triangle." This means we are to solve for all missing side lengths and angle measurements.

right triangle trig finding missing sides and angles

Finding Sides and Angles Using Inverse Trig

Common Questions

Q: Is sohcahtoa only for right triangles?
A: Yes, it only applies to right triangles. If we have an oblique triangle, then we can't assume these trig ratios will work. We have other methods we'll learn about in Math Analysis and Trigonometry such as the laws of sines and cosines to handle those cases.

Q: When to use sohcahtoa?
A: When you are given a right triangle, where two of the side lengths are given and you are asked to find the third side.

Q: Where is the hypotenuse of a right triangle?
A: They hypotenuse of a right triangle is always opposite the 90 degree angle, and is the longest side.

Q: Where is the adjacent side of a triangle?
A: The adjacent side of a triangle is the side (leg) that is touching the angle but is not the hypotenuse.

Q: What does it mean to solve a right triangle?
A: When you solve a right triangle, or any triangle for that matter, it means you need to find all missing sides and angles. Therefore, you will use Trig Ratios, the Triangle Sum Theorem, and/or the Pythagorean Theorem to find any missing angle or side length measures.

In the video below, you'll progress through harder examples involving trig ratios, calculating missing side lengths and angles, inverse trig, and much more!

Video – Lesson & Examples

1 hr 34 min

Introduction to Trigonometric Ratios (Sine, Cosine, Tangent)
00:00:26 – Understanding SOH-CAH-TOA
Exclusive Content for Member's Only

00:17:38 – Find the three trig ratios for both acute angles (Examples #1-4)
00:39:35 – Complete the table using Soh-Cah-Toa (Examples #5-6)
00:53:12 – How to solve for an angle using a calculator? (Examples #7-12)
01:05:22 – Solve the right triangle by finding all missing sides and angles (Examples #13-14)
01:18:37 – Solve the word problem involving a right triangle and trig ratios (Example #15)
01:27:34 – Solve for x by using SOH CAH TOA (Examples #16-19)
Practice Problems with Step-by-Step Solutions
Chapter Tests with Video Solutions

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