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Transcript of Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs...
![Page 1: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/1.jpg)
Lesson 13.1, For use with pages 852-858
In right triangle ABC, a and b are the lengths of the legsand c is the length of the hypotenuse. Find the missinglength. Give exact values.
1. a = 6, b = 8
2. c = 10, b = 7
ANSWER c = 10
ANSWER a = 51
![Page 2: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/2.jpg)
3. If you walk 2.0 kilometers due east and than 1.5 kilometers due north, how far will you be from your starting point?
ANSWER 2.5 km
In right triangle ABC, a and b are the lengths of the legsand c is the length of the hypotenuse. Find the missinglength. Give exact values.
Lesson 13.1, For use with pages 852-858
![Page 3: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/3.jpg)
Trigonometry and Angles 13.1
![Page 4: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/4.jpg)
EXAMPLE 1 Evaluate trigonometric functions
Evaluate the six trigonometric functions of the angle θ.
SOLUTION
13=169= √
From the Pythagorean theorem, the length of the
hypotenuse is 52 + 122√
sin θ =opp
hyp=
1213
csc θ =hyp
opp=
1312
![Page 5: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/5.jpg)
EXAMPLE 1 Evaluate trigonometric functions
tan θ =opp
adj=
12
5cot θ =
adj
opp=
5
12
cos θ =adj
hyp=
5
13sec θ =
hyp
adj=
13
5
![Page 6: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/6.jpg)
Draw: a right triangle with acute angle θ such that the leg opposite θ has length 4 and the hypotenuse has length 7. By the Pythagorean theorem, the length x of the other leg is
x 72 – 42√=
EXAMPLE 2 Standardized Test Practice
SOLUTION
STEP 1
33.= √
![Page 7: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/7.jpg)
EXAMPLE 2 Standardized Test Practice
STEP 2 Find the value of tan θ.
tan θ =opp
adj=
33√
4=
33
33
4 √
ANSWER
The correct answer is B.
![Page 8: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/8.jpg)
GUIDED PRACTICE for Examples 1 and 2
Evaluate the six trigonometric functions of the angle θ.
1.
SOLUTION
5= 25= √
From the Pythagorean theorem, the length of the
hypotenuse is 32 + 42√
sin θ =opp
hyp=
3
5csc θ =
hyp
opp=
5
3
![Page 9: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/9.jpg)
GUIDED PRACTICE for Examples 1 and 2
tan θ =opp
adj=
3
4cot θ =
adj
opp=
4
3
cos θ =adj
hyp=
4
5sec θ =
hyp
adj=
5
4
![Page 10: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/10.jpg)
From the Pythagorean theorem, the length of the
adjacent is 172 – 152√
GUIDED PRACTICE for Examples 1 and 2
Evaluate the six trigonometric functions of the angle θ.
SOLUTION
8.= 64= √
sin θ =opp
hyp=
15
17csc θ =
hyp
opp=
17
15
2.
![Page 11: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/11.jpg)
GUIDED PRACTICE for Examples 1 and 2
tan θ =opp
adj=
15
8cot θ =
adj
opp=
8
15
cos θ =adj
hyp=
8
17sec θ =
hyp
adj=
17
8
![Page 12: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/12.jpg)
GUIDED PRACTICE for Examples 1 and 2
Evaluate the six trigonometric functions of the angle θ.
SOLUTION
5= 25= √
sin θ =opp
hypcsc θ =
hyp
opp
3.
From the Pythagorean theorem, the length of the
adjacent is (5 22 – 52√ √
=5
5 2√ 5=
5 2√
![Page 13: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/13.jpg)
GUIDED PRACTICE for Examples 1 and 2
tan θ =opp
adj=
5
5cot θ =
adj
opp=
5
5
cos θ =adj
hypsec θ =
hyp
adj
5=
5 2√ 5=
5 2√
= 1 = 1
![Page 14: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/14.jpg)
EXAMPLE 3 Find an unknown side length of a right triangle
SOLUTION
Write an equation using a trigonometric function that involves the ratio of x and 8. Solve the equation for x.
Find the value of x for the right triangle shown.
cos 30º =adj
hyp Write trigonometric equation.
3
2
√=
x8 Substitute.
![Page 15: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/15.jpg)
EXAMPLE 3 Find an unknown side length of a right triangle
34 √ = x Multiply each side by 8.
The length of the side is x = 34 √ 6.93.
ANSWER
![Page 16: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/16.jpg)
EXAMPLE 4 Use a calculator to solve a right triangle
SOLUTION
Write trigonometric equation.
Substitute.
Solve ABC.
A and B are complementary angles,
so B = 90º – 28º
tan 28° =opp
adjsec 28º =
hyp
adj
tan 28º =a
15sec 28º =
c
15
= 68º.
![Page 17: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/17.jpg)
EXAMPLE 4 Use a calculator to solve a right triangle
Solve for the variable.
Use a calculator.
15(tan 28º) = a 151( cos 28º ) = c
7.98 a 17.0 c
So, B = 62º, a 7.98, and c 17.0
ANSWER
![Page 18: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/18.jpg)
GUIDED PRACTICE for Examples 3 and 4
Solve ABC using the diagram at the right and the given measurements.
5. B = 45°, c = 5
SOLUTION
Substitute.
A and B are complementary angles,
so A = 90º – 45º
cos 45° =adj
hypsin 45º =
opp
hyp
cos 45º =a
5sin 45º =
5b
Write trigonometric equation.
= 45º.
![Page 19: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/19.jpg)
GUIDED PRACTICE for Examples 3 and 4
Solve for the variable.
Use a calculator.
5(cos 45º) = a 5(sin 45º) = b
3.54 a 3.54 b
So, A = 45º, b 3.54, and a 3.54.
ANSWER
![Page 20: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/20.jpg)
GUIDED PRACTICE for Examples 3 and 4
SOLUTION
Substitute.
A and B are complementary angles,
so B = 90º – 32º
tan 32° =opp
adjsec 32º =
hyp
adj
tan 32º =a
10sec 32º =
10c
6. A = 32°, b = 10
Write trigonometric equation.
= 58º.
![Page 21: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/21.jpg)
GUIDED PRACTICE for Examples 3 and 4
Solve for the variable.
Use a calculator.
10(tan 32º) = a 101( cos 32º ) = c
6.25 a 11.8 c
So, B = 58º, a 6.25, and c 11.8.
ANSWER
![Page 22: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/22.jpg)
GUIDED PRACTICE for Examples 3 and 4
SOLUTION
Substitute.
A and B are complementary angles,
so B = 90º – 71º
cos 71° =adj
hypsin 71º =
opp
hyp
cos 71º =b
20sin 71º =
a
20
7. A = 71°, c = 20
Write trigonometric equation.
= 19º.
![Page 23: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/23.jpg)
GUIDED PRACTICE for Examples 3 and 4
Solve for the variable.
Use a calculator.
20(cos 71º) = b
6.51 b 18.9 a
So, B = 19º, b 6.51, and a 18.9.
ANSWER
20(sin 71º) = a
![Page 24: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/24.jpg)
GUIDED PRACTICE for Examples 3 and 4
SOLUTION
Substitute.
A and B are complementary angles,so A = 90º – 60º
sec 60° =hyp
adjtan 60º =
opp
adj
sec 60º = 7c
tan 60º =b
7
Write trigonometric equation.
8. B = 60°, a = 7
= 30º.
![Page 25: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/25.jpg)
GUIDED PRACTICE for Examples 3 and 4
Solve for the variable.
Use a calculator.
7(tan 60º) = b 71( cos 60º ) = c
14 = c 12.1 b
So, A = 30º, c = 14, and b 12.1.
ANSWER
![Page 26: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/26.jpg)
EXAMPLE 5 Use indirect measurement
While standing at Yavapai Point near the Grand Canyon, you measure an angle of 90º between Powell Point and Widforss Point, as shown. You then walk to Powell Point and measure an angle of 76º between Yavapai Point and Widforss Point. The distance between Yavapai Point and Powell Point is about 2 miles. How wide is the Grand Canyon between Yavapai Point and Widforss Point?
Grand Canyon
![Page 27: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/27.jpg)
EXAMPLE 5 Use indirect measurement
SOLUTION
tan 76º =x
2Write trigonometric equation.
2(tan 76º) = x Multiply each side by 2.
8.0 ≈ x Use a calculator.
The width is about 8.0 miles.
ANSWER
![Page 28: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/28.jpg)
EXAMPLE 6 Use an angle of elevation
A parasailer is attached to a boat with a rope 300 feet long. The angle of elevation from the boat to the parasailer is 48º. Estimate the parasailer’s height above the boat.
Parasailing
![Page 29: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/29.jpg)
EXAMPLE 6 Use an angle of elevation
SOLUTION
sin 48º =h
300Write trigonometric equation.
300(sin 48º) = h Multiply each side by 300.
STEP 1
Draw: a diagram that represents the situation.
STEP 2
Write: and solve an equation to find the height h.
223 ≈ x Use a calculator.
The height of the parasailer above the boat is about 223 feet.
ANSWER
![Page 30: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/30.jpg)
GUIDED PRACTICE for Examples 5 and 6
9. In Example 5, find the distance between Powell Point and Widforss Point.
Grand Canyon
SOLUTION
sec 76º =2
xWrite trigonometric equation.
Multiply each side by 2.
8.27 ≈ xUse a calculator.
21
( cos 76º ) = x
The distance is about 8.27 miles.ANSWER
Substitute for sec 76° .cos 76°
1
2 sec 76º = x
![Page 31: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/31.jpg)
GUIDED PRACTICE for Examples 5 and 6
10. What If? In Example 6, estimate the height of the parasailer above the boat if the angle of elevation is 38°.
SOLUTION
sin 38º =h
300Write trigonometric equation.
300(sin 38º) = h Multiply each side by 300.
185 ≈ h Use a calculator.
The height of the parasailer above the boat is about 185 feet.
ANSWER
![Page 32: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/32.jpg)
EXAMPLE 5 Use indirect measurement
While standing at Yavapai Point near the Grand Canyon, you measure an angle of 90º between Powell Point and Widforss Point, as shown. You then walk to Powell Point and measure an angle of 76º between Yavapai Point and Widforss Point. The distance between Yavapai Point and Powell Point is about 2 miles. How wide is the Grand Canyon between Yavapai Point and Widforss Point?
Grand Canyon
![Page 33: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/33.jpg)
EXAMPLE 5 Use indirect measurement
SOLUTION
tan 76º =x
2Write trigonometric equation.
2(tan 76º) = x Multiply each side by 2.
8.0 ≈ x Use a calculator.
The width is about 8.0 miles.
ANSWER
![Page 34: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/34.jpg)
EXAMPLE 6 Use an angle of elevation
A parasailer is attached to a boat with a rope 300 feet long. The angle of elevation from the boat to the parasailer is 48º. Estimate the parasailer’s height above the boat.
Parasailing
![Page 35: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/35.jpg)
EXAMPLE 6 Use an angle of elevation
SOLUTION
sin 48º =h
300Write trigonometric equation.
300(sin 48º) = h Multiply each side by 300.
STEP 1
Draw: a diagram that represents the situation.
STEP 2
Write: and solve an equation to find the height h.
223 ≈ x Use a calculator.
The height of the parasailer above the boat is about 223 feet.
ANSWER
![Page 36: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/36.jpg)
GUIDED PRACTICE for Examples 5 and 6
9. In Example 5, find the distance between Powell Point and Widforss Point.
Grand Canyon
SOLUTION
sec 76º =2
xWrite trigonometric equation.
Multiply each side by 2.
8.27 ≈ xUse a calculator.
21
( cos 76º ) = x
The distance is about 8.27 miles.ANSWER
Substitute for sec 76° .cos 76°
1
2 sec 76º = x
![Page 37: Lesson 13.1, For use with pages 852-858 In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649e6f5503460f94b6c59a/html5/thumbnails/37.jpg)
GUIDED PRACTICE for Examples 5 and 6
10. What If? In Example 6, estimate the height of the parasailer above the boat if the angle of elevation is 38°.
SOLUTION
sin 38º =h
300Write trigonometric equation.
300(sin 38º) = h Multiply each side by 300.
185 ≈ h Use a calculator.
The height of the parasailer above the boat is about 185 feet.
ANSWER