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 $50 °$ . In the meantime, Andrea has paced off the length of the tree’s shadow and finds that it is $40$ feet long.

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 $100 feet$ away from Andrea, she stopped to take a break and looked at her GPS device that told her that she had walked $100 feet$ and had already increased her elevation by $40 feet$ . With a bit of time to waste, Bonita wrote down the trigonometric ratios for $∠ A$ and for $∠ B$ .

2.

Name the trigonometric ratios for $∠ A$ and for $∠ B$ that involve the given sides.

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 $∠ A$ .” Bonita wrote the trigonometric ratio $\sin A = \frac{40}{100} = \frac{2}{5}$ and asked, “So, how do we find angle $A$ ?”

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 $\sin^{- 1} (θ)$ button on her calculator and wonders if this might work like an “inverse trigonometric ratio” button, undoing the ratio to produce the angle. She decides to try it out and gets the following output on her calculator:

$\sin A = \frac{2}{5}$ $\sin^{- 1} (\frac{2}{5}) = 23.578$ $\sin (23.578) = 0.4$

3.

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

4.

Use the trigonometric ratio you found for $\cos B$ to find the value of $∠ B$ .

5.

Find all unknown values for the given right triangle:

$∠ α =$
$∠ β =$
$∠ γ =$ $90 °$
$a =$ $12 m$
$b =$ $8 m$
$c =$

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 $23.5 °$ . They both agreed. Bonita then said her angle of depression to Andrea was about $66.5 °$ . Andrea agreed that Bonita was describing an angle of depression, but said Bonita’s angle of depression was also $23.5 °$ . Who do you think is correct? Use drawings and words to justify your conclusion.

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 $10 ft$ high lamppost, your shadow extends farther and farther in front of you. Is there a position where you might stand so that your shadow is exactly as long as you are tall, since your height measures $5 ft$ ? If you then walk twice as far away from the lamppost, will your shadow be twice as long? Use diagrams to help you think about this situation.

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:

Vocabulary

Lesson Summary

In this lesson, we extended our strategies for finding unknown sides and angles in a right triangle beyond using the Pythagorean theorem and the angle sum theorem for triangles, since sometimes we don’t have enough information in terms of side lengths or angle measures to use these theorems. We found that trigonometric ratios are useful in solving for unknown sides and that inverse trigonometric relationships are useful for finding unknown angles in a right triangle. Adding these tools allows us to find all of the missing sides and angles in a right triangle given two pieces of information: two sides of the triangle or one side and an angle.

Retrieval

1.

Sketch a picture of this situation and label as much of the picture as possible.

Tiana is standing in the bottom of a canyon with cliff walls that seem to reach the sky. She decided that she might try and measure one of them. So, Tiana measures a distance from the base of one of the cliffs out $200 ft$ and stands there. She then looks up to the top and finds that the angle from the ground to the top of the cliff is $68 °$ .

2.

Use the right triangle to find the missing side length and angle measurements, as well as the desired ratios.

$\sin (A) =$ , $\sin (B) =$

$\cos (A) =$ , $\cos (B) =$

$\tan (A) =$ , $\tan (B) =$

$m ∠ A =$ , $m ∠ B =$

3.

Use the given information in the right triangle to find the missing side lengths.