# Using Trig Ratios to Solve Triangles: Sides

This lesson will cover how to use trig ratios to find the side lengths of a triangle.

There are three steps:

**1. Choose which trig ratio to use.**

- Choose either sin, cos, or tan by determining which side you know and which side you are looking for.

**2. Substitute**

- Substitute your information into the trig ratio.

**3. Solve**

- Solve the resulting equation to find the length of the side.

**Example:**

1. Find b.

**Step 1: Choose which trig ratio to use.**

First, we know we must look at angle B because that is the angle we know the

measure of.(Now, you could find the measure of angle A and then use that but

that's less reliable since you could make a mistake.)

So, looking at angle B, we want to identify which sides are involved. We know

one side is 8m, and that side is

**adjacent**to angle B. The side we're looking

for is

**opposite**angle B. So we need to choose the trig ratio that has

**opposite and adjacent.**This of course is the

**tangent.**

**Step 2: Substitute**

Next, we write our trig ratio:

Then, we substitute in the angle

and the side we know:

**Step 3: Solve**

Now move the 8 to the other side

by multiplying both sides by 8:

And use a calculator to find the answer.

Well round to the nearest tenth: 3.7 m.

2. Find c.

Now that we know two sides, you could use the Pythagorean Theorem

to find the third. But that's less reliable because

if you made a mistake on side b, then side c will also be wrong.

So we are going to repeat the same process for side c.

**Step 1: Choose the trig ratio to use.**

We're still using angle B. 8m is the

**adjacent**and c is the

**hypotenuse.**

The trig ratio that uses the adjacent and hypotenuse is the

**cosine**.

**Step 2: Substitute**

Write our trig ratio:

Then, we substitute in the

angle and the side we know:

**Step 3: Solve**

Since our variable is on the

bottom, we can start by

cross multiplying:

Then we'll divide

both sides by cos 25°:

And use a calculator to

find the answer. Well round

to the nearest tenth: 8.8 m.

**Note:**If your calculator doesn't seem to be giving you the right answer, read your manual or ask someone for help. Each calculator is different, and you have to enter the information in the correct order to get the right answer. Also, make sure your calculator is in Degree mode (not Radians)

**Practice:**

Find the following. Round your answers to the nearest tenth.

1) a 2) c 3) x 4) y

**Answers:**1) 7.4 in 2) 9.5 in. 3) 7.0 ft 4) 5.7 ft.

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