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
length. Give exact values.
1. a = 6, b = 8
ANSWER
c = 10
2. c = 10, b = 7
ANSWER
a =

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
length. Give exact values.
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

3. Trigonometry and Angles 13.1

4. EXAMPLE 1
Evaluate trigonometric functions
Evaluate the six trigonometric functions of the angle θ.
SOLUTION
From the Pythagorean theorem, the length of the
hypotenuse is √
169 = √ 13 =
sin θ =
opp
hyp
csc θ =
hyp
opp = 12 13 = 13 12

5. EXAMPLE 1
Evaluate trigonometric functions
cos θ =
adj
hyp = 5 13
sec θ =
hyp
adj = 13 5
tan θ =
opp
adj = 12 5
cot θ =
adj
opp = 5 12

6. EXAMPLE 2
Standardized Test Practice
SOLUTION
STEP 1
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 √ =
33. = √

7. EXAMPLE 2
Standardized Test Practice
STEP 2
Find the value of tan θ.
tan θ =
opp
adj = 33 √ 4 = 33 4 √
ANSWER
The correct answer is B.

8. GUIDED PRACTICE
for Examples 1 and 2
Evaluate the six trigonometric functions of the angle θ. 1.
SOLUTION
From the Pythagorean theorem, the length of the
hypotenuse is √ 25 = √ 5 =
sin θ =
opp
hyp = 3 5
csc θ =
hyp
opp = 5 3

9. GUIDED PRACTICE
for Examples 1 and 2
cos θ =
adj
hyp = 4 5
sec θ =
hyp
adj = 5 4
tan θ =
opp
adj = 3 4
cot θ =
adj
opp = 4 3

10. GUIDED PRACTICE
for Examples 1 and 2
Evaluate the six trigonometric functions of the angle θ.
SOLUTION
From the Pythagorean theorem, the length of the
adjacent is
172 – 152 √ 64 = √ 8. =
sin θ =
opp
hyp = 15 17
csc θ =
hyp
opp = 17 15

11. GUIDED PRACTICE
for Examples 1 and 2
cos θ =
adj
hyp = 8 17
sec θ =
hyp
adj = 17 8
tan θ =
opp
adj = 15 8
cot θ =
adj
opp = 8 15

12. GUIDED PRACTICE
for Examples 1 and 2
Evaluate the six trigonometric functions of the angle θ.
SOLUTION
From the Pythagorean theorem, the length of the
adjacent is
( – 52 √ 25 = √ 5 =
sin θ =
opp
hyp = 5
5 2 √
csc θ =
hyp
opp 5 =
5 2 √

13. GUIDED PRACTICE
for Examples 1 and 2
cos θ =
adj
hyp 5 =
5 2 √
sec θ =
hyp
adj 5 =
5 2 √
tan θ =
opp
adj = 5
cot θ =
adj
opp = 5
= 1
= 1

14. Write trigonometric equation.
EXAMPLE 3
Find an unknown side length of a right triangle
Find the value of x for the right triangle shown.
SOLUTION
Write an equation using a trigonometric function that involves the ratio of x and 8. Solve the equation for x.
cos 30º =
adj
hyp
Write trigonometric equation. 3 2 √ = x 8
Substitute.

15. Multiply each side by 8. EXAMPLE 3
Find an unknown side length of a right triangle 3 4 √ = x
Multiply each side by 8.
The length of the side is x = 3 4 √
6.93.
ANSWER

16. Write trigonometric equation.
EXAMPLE 4
Use a calculator to solve a right triangle
Solve ABC.
SOLUTION
A and B are complementary angles,
so B = 90º – 28º
= 68º.
tan 28° =
opp
adj
sec 28º =
hyp
adj
Write trigonometric equation.
tan 28º = a 15
sec 28º = c 15
Substitute.

17. ( ) Solve for the variable. Use a calculator. EXAMPLE 4
Use a calculator to solve a right triangle 15 1 (
cos 28º )
= c
15(tan 28º) = a
Solve for the variable.
7.98 a
17.0 c
Use a calculator.
So, B = 62º, a , and c
ANSWER

18. Write trigonometric equation.
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
A and B are complementary angles,
so A = 90º – 45º
= 45º.
cos 45° =
adj
hyp
sin 45º =
opp
hyp
Write trigonometric equation.
cos 45º = a 5
sin 45º = 5 b
Substitute.

19. Solve for the variable. Use a calculator. GUIDED PRACTICE
for Examples 3 and 4
5(cos 45º) = a
5(sin 45º)
= b
Solve for the variable.
3.54 a
3.54 b
Use a calculator.
So, A = 45º, b , and a
ANSWER

20. Write trigonometric equation.
GUIDED PRACTICE
for Examples 3 and 4
6. A = 32°, b = 10
SOLUTION
A and B are complementary angles,
so B = 90º – 32º
= 58º.
tan 32° =
opp
adj
sec 32º =
hyp
adj
Write trigonometric equation.
tan 32º = a 10
sec 32º = 10 c
Substitute.

21. ( ) Solve for the variable. Use a calculator. GUIDED PRACTICE
for Examples 3 and 4 10 1 (
cos 32º )
= c
10(tan 32º) = a
Solve for the variable.
6.25 a
11.8 c
Use a calculator.
So, B = 58º, a , and c
ANSWER

22. Write trigonometric equation.
GUIDED PRACTICE
for Examples 3 and 4
A = 71°, c = 20
SOLUTION
A and B are complementary angles,
so B = 90º – 71º
= 19º.
cos 71° =
adj
hyp
sin 71º =
opp
hyp
Write trigonometric equation.
cos 71º = b 20
sin 71º = a 20
Substitute.

23. Solve for the variable. Use a calculator. GUIDED PRACTICE
for Examples 3 and 4
20(cos 71º) = b
Solve for the variable.
20(sin 71º) = a
6.51 b
18.9 a
Use a calculator.
So, B = 19º, b , and a
ANSWER

24. Write trigonometric equation.
GUIDED PRACTICE
for Examples 3 and 4
B = 60°, a = 7
SOLUTION
A and B are complementary angles,
so A = 90º – 60º
= 30º.
sec 60° =
hyp
adj
tan 60º =
opp
adj
Write trigonometric equation.
sec 60º = 7 c
tan 60º = b 7
Substitute.

25. ( ) Solve for the variable. Use a calculator. GUIDED PRACTICE
for Examples 3 and 4 7 1 (
cos 60º )
= c
7(tan 60º) = b
Solve for the variable.
14 = c
12.1 b
Use a calculator.
So, A = 30º, c = 14, and b
ANSWER

26. EXAMPLE 5
Use indirect measurement
Grand Canyon
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?

27. Write trigonometric equation.
EXAMPLE 5
Use indirect measurement
SOLUTION
tan 76º = x 2
Write 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

28. EXAMPLE 6
Use an angle of elevation
Parasailing
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.

29. Write trigonometric equation.
EXAMPLE 6
Use an angle of elevation
SOLUTION
STEP 1
Draw: a diagram that represents the situation.
STEP 2
Write: and solve an equation to find the height h.
sin 48º = h
300
Write trigonometric equation.
300(sin 48º) = h
Multiply each side by 300.
223 ≈ x
Use a calculator.
The height of the parasailer above the boat is about 223 feet.
ANSWER

30. ( ) Write trigonometric equation. Multiply each side by 2. 1
GUIDED PRACTICE
for Examples 5 and 6
Grand Canyon 9.
In Example 5, find the distance between Powell Point and Widforss Point.
SOLUTION
sec 76º = 2 x
Write trigonometric equation.
2 sec 76º = x
Multiply each side by 2. 2 1 (
cos 76º )
= x
Substitute for sec 76° .
cos 76° 1
8.27 ≈ x
Use a calculator.
The distance is about 8.27 miles.
ANSWER

31. Write trigonometric equation.
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
300
Write 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

32. EXAMPLE 5
Use indirect measurement
Grand Canyon
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?

33. Write trigonometric equation.
EXAMPLE 5
Use indirect measurement
SOLUTION
tan 76º = x 2
Write 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

34. EXAMPLE 6
Use an angle of elevation
Parasailing
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.

35. Write trigonometric equation.
EXAMPLE 6
Use an angle of elevation
SOLUTION
STEP 1
Draw: a diagram that represents the situation.
STEP 2
Write: and solve an equation to find the height h.
sin 48º = h
300
Write trigonometric equation.
300(sin 48º) = h
Multiply each side by 300.
223 ≈ x
Use a calculator.
The height of the parasailer above the boat is about 223 feet.
ANSWER

36. ( ) Write trigonometric equation. Multiply each side by 2. 1
GUIDED PRACTICE
for Examples 5 and 6
Grand Canyon 9.
In Example 5, find the distance between Powell Point and Widforss Point.
SOLUTION
sec 76º = 2 x
Write trigonometric equation.
2 sec 76º = x
Multiply each side by 2. 2 1 (
cos 76º )
= x
Substitute for sec 76° .
cos 76° 1
8.27 ≈ x
Use a calculator.
The distance is about 8.27 miles.
ANSWER

37. Write trigonometric equation.
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
300
Write 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