SOHCAHTOA: Find the sides of a right triangle

How does SOHCAHTOA help us find side lengths?

After you are comfortable writing sine,cosine, tangent ratios, you might want to use sohcahtoa to find the sides of a right triangle. That is exactly the topic that this page focuses on.

Video

Directions: Use sohcahtoa to find the given side length.

Example of finding Side Length

How to use sine,cosine,tangent to determine side X in the triangle below

example probelm Step 1

Based on your givens and unknowns, determine which sohcahtoa ratio to use

Since we know the 67 angle, its adjacent side length and we want to know length of the opposite side, we should use tangent

Step 2

Set up an equation based on the ratio you chose in the step 1

$ tan(67) = \frac{opposite}{adjacent} \\ tan(67) = \frac{x}{14} $

Step 3

Cross multiply and solve the equation for the side length. (Let's round to the nearest hundredth)

$ tan(67) = \frac{x}{14} \\ 14\times tan(67) = x \\ x \approx 32.98 $

Practice Problems

Problem 1

Use sine,cosine or tangent to find the value of side x in the triangle below.

Step 1

Based on your givens and unknowns, determine which sohcahtoa ratio to use

Since we know the 53 angle, the hypotenuse,and we want to find length of the opposite side, we should use sine

Step 2

Set up an equation based on the ratio you chose in the step 1

$ sin(53) = \frac{opposite}{hypotenuse} \\ sin(53) = \frac{x}{15} $

Step 3

Cross multiply and solve the equation for the side length. (round to the nearest hundredth)

$ 15\times sin(53) = x \\ x \approx 11.98 $

Problem 2

Use sine,cosine or tangent to find the length of side k in the triangle below.

Step 1

Based on your givens and unknowns, determine which sohcahtoa ratio to use

Since we know the 53 angle, the adjacent side ,and we want to find length of the hypotenuse, we should use cosine

Step 2

Set up an equation based on the ratio you chose in the step 1

$ cos(53) = \frac{adjacent}{hypotenuse} \\ cos(53) = \frac{45}{x} $

Step 3

Cross multiply and solve the equation for the side length. (round to the nearest hundredth)

$ x=\frac{45}{cos(53)} \\ x \approx 74.8 $

Problem 3

Use SOHCAHTOA to find x in the triangle on the left?

sohcahtoa problem

$ cos(63) = \frac{adjacent}{hypotenuse} \\ cos(63) = \frac{3}{x} \\ x = \frac{3}{cos63} \approx 6.6 $

Problem 4
Step 1

Use SOHCAHTOA to find  Y in the triangle on the left.

$ tan(55) = \frac{y}{22} \\ y = 22 \times tan(55) \approx 31.4 $

Step 2

How long is NM?

$ NM^2 = 22^2+ 31.4^2 \approx 1471.1 \\ NM = \sqrt{1471.1} \approx 38.3 $

Problem 5

What are two distinct ways that you can find x in the triangle on the left?

Method 1

Use SOHCAHTOA  and set up a ratio such as sin(16) = 14/x  (From here solve for X) By the way, you could also use cosine.

Method 2

Set up the following equation using the Pythagorean theorem: x2=482+142(From here solve for X)

Here's a page on finding the side lengths of right triangles