You are a firefighter on call at a burning building. Your colleagues are on the roof preparing to help put out the blaze by entering the building through the third-floor window. They need you to find the distance from the roof to the windowsill and then determine if the firefighters on the ground are close enough to the building for the water to reach the flames through the window on the third floor.
You observe from below. The angle of elevation to the windowsill is 18º and the angle of elevation to the top of the building is 31º. You are standing 65 feet away from the building and your eyes are 5 feet above the ground, as shown in the diagram. You hold the hose at eye level in order to take aim at the third-floor window.
65 ft
5 ft
Eye level 31˚ 18˚
To the nearest foot, what is the distance from the roof to the windowsill that the firefighters will need to descend by rope to enter the building? If the hoses can spray water at a distance of 100 feet, are the firefighters standing close enough to the building to put the flames out at the third-floor window?
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
NAME:
Coaching
a. In order to determine the distance from the windowsill to the roof, what other information do you need to find first?
b. What is the vertical height from the roof to your eye level? Sketch the triangle that provides the information necessary to answer the question.
c. What is the vertical height from the windowsill to your eye level?
d. What is the distance from the roof to the third-floor windowsill?
e. How can you use the information you’ve already found to determine the distance the water has to travel?
f. What distance does the water have to travel to reach the window?
g. Are the firefighters on the ground close enough to the building?
Problem-Based Task 2.2.3: Fighting Flames from a Distance
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
Coaching Sample Responses
a. In order to determine the distance from the windowsill to the roof, what other information do you need to find first?
Since the vertical height from the roof to the windowsill is not a side of a right triangle, you have to use the two right triangles that are given to find the distance from your eye level to the windowsill and the distance from your eye level to the roof. Subtracting the two values will yield the distance from the roof to the windowsill.
b. What is the vertical height from the roof to your eye level? Sketch the triangle that provides the information necessary to answer the question.
65 ft
Eye level 31˚
Roof
r
Let r represent the vertical height from the roof to your eye level. Given an angle of elevation of 31º, the leg adjacent to the angle of elevation is 65 feet. To calculate the leg opposite the angle of elevation, the tangent function is required.
" r tan 31º
65 65 • tan 31º = r On a TI-83/84:
Step 1: Press [65][TAN][31][)].
Step 2: Press [ENTER].
r ~ 39.056
Problem-Based Task 2.2.3: Fighting Flames from a Distance
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
On a TI-Nspire:
Step 1: In the calculate window from the home screen, enter 65, then press [trig]. Use the keypad to select “tan,” then enter 31.
Step 2: Press [enter].
r ~ 39.056
The vertical height from the roof to your eye level is about 39 meters.
c. What is the vertical height from the windowsill to your eye level?
65 ft 18˚ Eye level
Windowsill
w
Let w represent the vertical height from the windowsill to your eye level. Given an angle of elevation of 18º, the leg adjacent to the angle of elevation is 65 feet long. To calculate the leg opposite the angle of elevation, the tangent function is required.
" w tan 18º
65 65 • tan 18º = w On a TI-83/84:
Step 1: Press [65][TAN][18][)].
Step 2: Press [ENTER].
w ~ 21.120 On a TI-Nspire:
Step 1: In the calculate window from the home screen, enter 65, then press [trig]. Use the keypad to select “tan,” then enter 18.
Step 2: Press [enter].
w ~ 21.120
The vertical height from the windowsill to your eye level is about 21 feet.
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
d. What is the distance from the roof to the third-floor windowsill?
To find the distance from the roof to the windowsill, subtract 21.120 from 39.056.
39.056 – 21.120 = 17.936
The distance is approximately 18 feet.
e. How can you use the information you’ve already found to determine the distance the water has to travel?
Using the diagram in part c, use the Pythagorean Theorem to find the hypotenuse, or the distance the water will travel from the spout of the hose at eye level to the windowsill.
f. What distance does the water have to travel to reach the window?
21.1202 + 652 = d 2 446.054 + 4225 = d 2 4671.054 = d 2
± 4671 054. =d d = 68.345
The distance the water has to travel to reach the window is about 68 feet.
g. Are the firefighters on the ground close enough to the building?
Yes, the firefighters on the ground are close enough to the building because the water sprays 100 feet and the distance the water needs to travel is about 68 feet.
Recommended Closure Activity
Select one or more of the essential questions for a class discussion or as a journal entry prompt.
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
NAME:
continued
Unless otherwise specified, round all final distances and angle measures to the nearest whole number.
1. A 6-foot-tall man is standing 50 feet from a flagpole. When he looks at the top of the flagpole, the angle of elevation is 39º. Find the height of the flagpole to the nearest foot.
2. A boy flies a kite with a 100-foot-long string. The angle of elevation of the string is 48º. How high is the kite from the ground?
3. A 14-foot ladder is being used to get to the top of a 12-foot-tall wall. At what angle of elevation must the ladder be positioned in order to reach the top of the wall?
4. A mother gazes out a second-floor window at her son playing at the playground. If the mother’s eye level is 12.6 meters off of level ground and the playground is 20 meters from the base of the building, what is the angle of depression from the mother’s line of sight to the playground?
5. A little girl is watching planes take off of the runway from a building’s rooftop 40 meters away from the airport. If the height of the building is 400 meters and the girl snaps a photo of a plane at a 24º angle of elevation, what is the altitude, or vertical height, of the plane when the photo is taken?
Practice 2.2.3: Problem Solving with the Pythagorean Theorem and Trigonometry
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
NAME:
6. From a hot air balloon 2,500 feet above the ground, you see a clearing whose angle of depression is 25º. To the nearest foot, find your horizontal distance from the clearing.
7. A slide at a water park with a constant slope sends riders traveling a distance of 45 feet to the pool at the bottom of the slide. If the depth of the pool is 12 feet and the angle of depression from the top of the slide is 45º, what is the vertical distance from the top of the slide to the bottom of the pool?
8. Tourists marvel at Niagara Falls from two sightseeing boats, A and B. The boats are 100 feet and 150 feet away from the base of the falls, respectively. Given that Niagara Falls is 167 feet high, what is the angle of elevation from both boats to the top of the falls?
9. A commuter plane is flying at an altitude of 1,000 meters. A passenger takes a picture of the top of a tree and estimates that the angle of depression to the top of the tree is about 15º. He estimates the angle of depression to the base of the tree to be 25º. What is the height of the tree?
10. Burj Khalifa in Dubai is the tallest building in the world, standing at 828 meters. An adjacent building, 100 meters away, stands at 550 meters tall. What is the angle of depression from Burj Khalifa to the adjacent building?
Lesson 1: Exploring Trigonometric Ratios Practice 2.1.1: Defining Trigonometric Ratios, pp. 18–22
Answers will vary due to the variation that comes when drawing and measuring or rounding.
1. sin A ~ 0.866; cos A = 0.5; tan A ~ 1.732
8. perimeter ~ 19.314 meters; cos 45º ~ 0.707 9. distance ~ 0.898 miles; tan 50º ~ 1.192 10. ramp length ~ 40.311 feet; sin 7.2º ~ 0.125 Practice 2.1.2: Exploring Sine and Cosine As Complements, pp. 30–32
1. approximately 0.707 2. cos 5º ~ 0.996
Lesson 2: Applying Trigonometric Ratios
Practice 2.2.1: Calculating Sine, Cosine, and Tangent, pp. 48–50
Practice 2.2.2: Calculating Cosecant, Secant, and Cotangent, pp. 60–63
Practice 2.2.3: Problem Solving with the Pythagorean Theorem and Trigonometry, pp. 75–77
1. 31º