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# 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 will often use sohcahtoa to find the sides of a right triangle. That is exactly what we are going to learn .

### Example of finding Side Length

How to use sine, cosine, tangent to calculate x from diagram 1.

Diagram 1
Step 1

Write a table listing the givens and what you want to find:

Givens Want to Find
$$67 ^{\circ}$$ Opposite
adjacent
Step 2

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

In this case we want to use tangent because it's the ratio that involves the adjacent and opposite sides.
Step 3

Set up an equation based on the ratio you chose in the step 2.

$tan(67) = \frac{opp}{adj} \\ tan(67) = \frac{ \red x}{14}$

Step 4

Solve the equation for the unknown.

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

### Practice Problems

##### Problem 1
Step 1

Write a table listing the givens and what you want to find:

Givens Want to Find
$$53 ^{\circ}$$ opposite
Hypotenuse
Step 2

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

The sine ratio is the one that involves the opposite side and the hypotenuse .
Step 3

Set up an equation based on the ratio you chose in the step 2.

$sin(53) = \frac{opp}{hyp} \\ sin(53) = \frac{\red x}{15}$

Step 4

Solve for the unknown.

$15 \cdot sin(53) = \red x \\ x \approx 11.98$

##### Problem 2
Step 1

Write a table listing the givens and what you want to find:

Givens Want to Find
$$53 ^{\circ}$$ Hypotenuse
adjacent
Step 2

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

The Cosine ratio is the one that involves the adjacent side and the hypotenuse .
Step 3

Set up an equation based on the ratio you chose in the step 2.

$cos(53) = \frac{adj}{hyp} \\ cos(53) = \frac{45}{\red x}$

Step 3

Solve for the unkown

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

##### Problem 3
Step 1

Write a table listing the givens and what you want to find:

Givens Want to Find
$$63 ^{\circ}$$ Hypotenuse
adjacent
Step 2

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

The cosine ratio is the one that involves the adjacent side and the hypotenuse .
Step 3

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

$cos(63) = \frac{adj }{ hyp } \\ cos(63) = \frac{3 }{\red x }$

Step 3

Solve for the unkown

$\red x = \frac {3} {cos(63)} \\ x = 6.6$

##### Problem 4
Step 1

Write a table listing the givens and what you want to find:

Givens Want to Find
$$53 ^{\circ}$$ opposite
adjacent
Step 2

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

The tangent ratio is the one that involves the oppsite and the adjacent sides.
Step 3

Set up an equation based on the ratio you chose in the step 2.

$tan(53) = \frac{opp}{ adj } \\ tan(53) = \frac{\red x }{22 }$

Step 3

Solve for the unkown

$\red x = 22 \cdot tan(53) \\ x = 22.2$

##### 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).