# Lesson 9 Finding the Value of a Relationship Solidify Understanding

## Learning Focus

Solve for the missing side and angle measures in a right triangle.

While we can find missing side lengths and angles in a right triangle if two sides or two angles are known using the Pythagorean theorem or the angle sum theorem for triangles, what information do we need to know to find missing measures in a right triangle now that trigonometric ratios are available as a computational tool?

How do we use trigonometry to make indirect measurements when the object can’t be measured directly?

## Open Up the Math: Launch, Explore, Discuss

Part 1: Pick a side

Andrea and Bonita are resting under their favorite tree before taking a nature walk up a hill. Both girls have been studying trigonometry in school, and now it seems like they see right triangles everywhere. For example, Andrea notices the length of the shadow of the tree they are sitting under and wonders if they can calculate the height of the tree just by measuring the length of its shadow.

Bonita thinks they also need to know the measure of an angle, so she checks an app on her phone and finds that the angle of elevation of the sun at the current location and time of day is

### 1.

How might Andrea and Bonita use this information, along with their knowledge of trigonometric ratios, to calculate the height of the tree? (Andrea and Bonita know they can find the value of any trigonometric ratio they might need for any acute angle using a calculator.)

Part 2: What’s your angle?

After their rest, Andrea and Bonita are going for a walk straight up the side of the hill. Andrea decided to stretch before heading up the hill while Bonita thought this would be a good time to get a head start. Once Bonita was

### 2.

Name the trigonometric ratios for

When Andrea caught up, she said, “What about the unknown angle measures? When I was at the bottom and looked up to see you, I was thinking about the ‘upward’ angle measure from me to you. Based on your picture, this would be

Together, the girls talked about how this was like thinking backward: instead of knowing an angle and using their calculators to find a trigonometric ratio like they did while working on the height of the tree problem, they now know the trigonometric ratio and need to find an unknown angle value. Bonita notices the

### 3.

How might this output convince Bonita that her assumption about the calculator was correct?

### 4.

Use the trigonometric ratio you found for

### 5.

Find all unknown values for the given right triangle:

### 6.

Bonita and Andrea started talking about all of the ways to find unknown values in right triangles and decided to make a list. What do you think should be on their list? Be specific and precise in your description. For example, “trigonometric ratios” is not specific enough. You may use the following sentence frame to assist with writing each item in your list:

When given , you can find by .

Part 3: Angle of elevation and angle of depression

During their hike, Andrea mentioned that she looked up to see Bonita. In mathematics, when you look straight ahead, we say your line of sight is a horizontal line. From the horizontal, if you look up, the angle from the horizontal to your line of sight is called the angle of elevation. Likewise, if you are looking down, the angle from the horizontal to your line of sight is called the angle of depression.

### 7.

After looking at this description, Andrea mentioned that her angle of elevation to see Bonita was about

### 8.

What conclusion can you make regarding the angle of depression and the angle of elevation? Why?

## Ready for More?

At night, as you walk away from a

## Takeaways

Make a list of all of the ways to find unknown values in right triangles. Be specific and precise in your description. For example, “trigonometric ratios” is not specific enough. You may use the following sentence frame to assist with writing each item in your list:

When given , you can find by .

Identify the angle of elevation and the angle of depression in the following diagram: