Lesson 11 Using Trigonometric Relationships Practice Understanding

Learning Focus

Solve application problems using trigonometry.

How do I apply trigonometric ratios to practical problems?

What are the essential elements of modeling a real-world context using a right triangle, even when only an imaginary right triangle exists?

Open Up the Math: Launch, Explore, Discuss

For each problem:

Make a drawing to represent the situation
Write an equation
Solve (do not forget to include units of measure)

1.

Carrie places a $10$ -foot ladder against a wall. If the ladder makes an angle of $65 °$ with the level ground, how far up the wall is the top of the ladder?

2.

A flagpole casts a shadow that is $15 feet$ long. The angle of elevation of the sun at this time is $40 °$ . How tall is the flagpole?

3.

In Southern California, there is a six-mile section of Interstate 5 that decreases $2,500 feet$ in elevation as it descends Grapevine Hill in the Tejon Pass. What is the angle of descent?

Pause and Reflect

4.

A hot air balloon is $600 feet$ above where it is planning to land and descending at a rate of $40 feet per minute$ . Sarah is driving over rough terrain at a speed of $10 miles per hour$ to meet the balloon when it lands. If the angle of elevation to the balloon is $20 °$ , how far away is Sarah from the place where the balloon will land? Who arrives at the landing spot first?

5.

An airplane is descending as it approaches the airport. If the angle of depression from the plane to the ground is $7 °$ , and the plane is $6,000 feet$ above the ground, how far is the plane from the airport?

6.

Michelle is $60$ feet away from a building. The angle of elevation to the top of the building is $41 °$ . How tall is the building?

7.

A ramp is used for loading equipment from a dock to a ship. The ramp is $10 feet$ long and the ship is $6 feet$ higher than the dock. What is the angle of elevation of the ramp?

For each right triangle, find all unknown side lengths and angle measures:

8.

9.

10.

11.

12.

Draw and find the missing angle measures of the right triangle whose sides measure $6$ , $8$ , and $10$ .

Determine the values of the two remaining trigonometric ratios when given one of the trigonometric ratios. Also find the measures of the acute angles of the triangle.

13.

$\cos (α) = \frac{3}{5}$

14.

$\tan (θ) = \frac{8}{3}$

15.

$\sin (β) = \frac{4}{7}$

Ready for More?

Compare and contrast two different methods for answering problems like those in problems 13–15.

Illustrate both of these methods to find $\cos (A)$ and $\tan (A)$ , given $\sin (A) = \frac{2}{3}$ .

1.

One method is to create a “reference triangle” from the information given in the trigonometric ratio.

Given: $\sin (A) = \frac{2}{3}$

$\cos (A) =$

$\tan (A) =$

2.

A second method is to use the trigonometric identities developed in this unit, such as: $\sin^{2} (A) + \cos^{2} (A) = 1$ or $\tan (A) = \frac{\sin (A)}{\cos (A)}$

Using trigonometric identities:

$\cos (A) =$

$\tan (A) =$

Takeaways

Vocabulary

model, mathematical
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned about the modeling process and how to use right triangle trigonometry to model many different types of applications, even applications that didn’t naturally include right triangles. A right triangle became a tool for representing a situation so we could draw upon trigonometric ratios and inverse trigonometric relationships to answer important problems in construction, aviation, transportation, and other contexts.

Retrieval

1.

Find the missing values for the similar right triangles.

2.

A doorstop, used as a wedge between the door and the floor to keep it open, is being designed with an angle of elevation of $11 °$ and a height of $1$ inch. How long should the bottom of the doorstop be? Sketch the situation and find the desired value.