Solution Manual for Trigonometry, 5th Edition by Cynthia Y. Young
CHAPTER 1
Section 1.1 Solutions --------------------------------------------------------------------------------
1 x 1 x
1. Solve for x: = 2. Solve for x: =
2 360∘ 4 360∘
360∘ = 2x, so that x = 180∘ . 360∘ = 4x, so that x = 90∘ .
1 x 2 x
3. Solve for x: − = 4. Solve for x: − =
3 360∘ 3 360∘
PR
360∘ = −3x, so that x = −120∘ . 720∘ = 2(360∘ ) = −3x, so that x = −240∘ .
(Note: The angle has a negative (Note: The angle has a negative
measure since it is a clockwise measure since it is a clockwise rotation.)
rotation.)
5 x 7 x
O
5. Solve for x: = 6. Solve for x: =
6 360∘ 12 360∘
1800∘ = 5(360∘ ) = 6x, so that x = 300∘ . 2520∘ = 7(360∘ ) = 12x, so that x = 210∘ .
FD
4 x 5 x
7. Solve for x: − = 8. Solve for x: − =
5 360∘ 9 360∘
1440∘ = 4(360∘ ) = −5x, so that 1800∘ = 5(360∘ ) = −9x, so that
x = −288∘ . x = −200∘ .
O
(Note: The angle has a negative (Note: The angle has a negative
measure since it is a clockwise measure since it is a clockwise rotation.)
rotation.)
C
9. 10.
a) complement: 90∘ −18∘ = 72∘ a) complement: 90∘ − 39∘ = 51∘
b) supplement: 180∘ −18∘ = 162∘ b) supplement: 180∘ − 39∘ = 141∘
11. 12.
a) complement: 90∘ − 42∘ = 48∘ a) complement: 90∘ − 57∘ = 33∘
b) supplement: 180∘ − 42∘ = 138∘ b) supplement: 180∘ − 57∘ = 123∘
1
,Chapter 1
13. 14.
a) complement: 90∘ − 89∘ = 1∘ a) complement: 90∘ − 75∘ = 15∘
b) supplement: 180∘ − 89∘ = 91∘ b) supplement: 180∘ − 75∘ = 105∘
15. Since the angles with measures ( 4x )∘ and ( 6x )∘ are assumed to be
complementary, we know that ( 4x )∘ + ( 6x )∘ = 90∘. Simplifying this yields
(10x )∘ = 90∘ , so that x = 9. So, the two angles have measures 36∘ and 54∘ .
16. Since the angles with measures ( 3x )∘ and (15x )∘ are assumed to be
PR
supplementary, we know that ( 3x )∘ + (15x )∘ = 180∘. Simplifying this yields
(18x )∘ = 180∘ , so that x = 10. So, the two angles have measures 30∘ and 150∘ .
17. Since the angles with measures ( 8x )∘ and ( 4x )∘ are assumed to be
O
supplementary, we know that ( 8x )∘ + ( 4x )∘ = 180∘. Simplifying this yields
(12x )∘ = 180∘ , so that x = 15. So, the two angles have measures 60∘ and 120∘ .
FD
18. Since the angles with measures ( 3x +15 )∘ and (10x +10 )∘ are assumed to be
complementary, we know that ( 3x +15 )∘ + (10x +10 )∘ = 90∘. Simplifying this yields
(13x + 25 )∘ = 90∘ , so that (13x )∘ = 65∘ and thus, x = 5. So, the two angles have
O
measures 30∘ and 60∘ .
19. Since + + = 180∘ , we know 20. Since + + = 180∘ , we know
C
that that
1 7∘ +3–3∘ + = 180∘ and so, = 30∘ . 1 0∘ +4–5∘ + = 180∘ and so, = 25∘ .
– –
= 150∘ = 155∘
21. Since + + = 180∘ , we know 22. Since + + = 180∘ , we know
that that
( 4 ) + + ( ) = 180∘ and so, = 30∘. (3 ) + + ( ) = 180∘ and so, = 36∘.
–– –– –– ––
= 6 = 5
Thus, = 4 = 120∘ and = = 30∘ . Thus, = 3 = 108∘ and = = 36∘ .
2
, Section 1.1
23. = 180 ∘ − ( 53.3∘ + 23.6 ∘ ) = 103.1∘ 24. = 180 ∘ − (105.6 ∘ +13.2 ∘ ) = 61.2 ∘
25. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes 42 + 32 = c2 , which
simplifies to c2 = 25, so we conclude that c = 5.
26. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes 32 + 32 = c2 , which
simplifies to c2 = 18, so we conclude that c = 18 = 3 2 .
27. Since this is a right triangle, we know from the Pythagorean Theorem that
PR
a2 + b2 = c2. Using the given information, this becomes 62 + b2 = 102 , which
simplifies to 36 + b2 = 100 and then to, b2 = 64, so we conclude that b = 8.
28. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes a2 + 72 = 122 , which
O
simplifies to a2 = 95, so we conclude that a = 95 .
FD
29. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes 82 + 52 = c2 , which
simplifies to c2 = 89, so we conclude that c = 89 .
30. Since this is a right triangle, we know from the Pythagorean Theorem that
O
a2 + b2 = c2. Using the given information, this becomes 62 + 52 = c2 , which
simplifies to c2 = 61, so we conclude that c = 61 .
C
31. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes 72 + b2 = 112 , which
simplifies to b2 = 72, so we conclude that b = 72 = 6 2 .
32. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes a2 + 52 = 92 , which
simplifies to a2 = 56, so we conclude that a = 56 = 2 14 .
3
, Chapter 1
33. Since this is a right triangle, we know from the Pythagorean Theorem that
( 7)
2
a 2 + b2 = c2 . Using the given information, this becomes a 2 + = 5 2 , which
simplifies to a2 = 18, so we conclude that a = 18 = 3 2 .
34. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes 52 + b2 = 102 , which
simplifies to b2 = 75, so we conclude that b = 75 = 5 3 .
35. If x = 10 in., then the hypotenuse 36. If x = 8 m, then the hypotenuse of
of this triangle has length
this triangle has length 8 2 11.31 m .
PR
10 2 14.14 in.
37. Let x be the length of a leg in the given 45∘ − 45∘ − 90∘ triangle. If the
hypotenuse of this triangle has length 2 2 cm, then 2 x = 2 2, so that x = 2.
O
Hence, the length of each of the two legs is 2 cm .
38. Let x be the length of a leg in the given 45∘ − 45∘ − 90∘ triangle. If the hypotenuse
FD
10 10
of this triangle has length 10 ft., then 2 x = 10, so that x = = = 5.
2 2
Hence, the length of each of the two legs is 5 ft.
39. The hypotenuse has length 40. Since 2x = 6m x = 6 2
= 3 2m,
O
( )
2
2 4 2 in. = 8 in. each leg has length 3 2 m.
C
41. Since the lengths of the two legs of the given 30∘ − 60∘ − 90∘ triangle are x and
3 x, the shorter leg must have length x. Hence, using the given information, we
know that x = 5 m. Thus, the two legs have lengths 5 m and 5 3 8.66 m, and
the hypotenuse has length 10 m.
42. Since the lengths of the two legs of the given 30∘ − 60∘ − 90∘ triangle are x and
3 x, the shorter leg must have length x. Hence, using the given information, we
know that x = 9 ft. Thus, the two legs have lengths 9 ft. and 9 3 15.59 ft., and
the hypotenuse has length 18 ft.
4
CHAPTER 1
Section 1.1 Solutions --------------------------------------------------------------------------------
1 x 1 x
1. Solve for x: = 2. Solve for x: =
2 360∘ 4 360∘
360∘ = 2x, so that x = 180∘ . 360∘ = 4x, so that x = 90∘ .
1 x 2 x
3. Solve for x: − = 4. Solve for x: − =
3 360∘ 3 360∘
PR
360∘ = −3x, so that x = −120∘ . 720∘ = 2(360∘ ) = −3x, so that x = −240∘ .
(Note: The angle has a negative (Note: The angle has a negative
measure since it is a clockwise measure since it is a clockwise rotation.)
rotation.)
5 x 7 x
O
5. Solve for x: = 6. Solve for x: =
6 360∘ 12 360∘
1800∘ = 5(360∘ ) = 6x, so that x = 300∘ . 2520∘ = 7(360∘ ) = 12x, so that x = 210∘ .
FD
4 x 5 x
7. Solve for x: − = 8. Solve for x: − =
5 360∘ 9 360∘
1440∘ = 4(360∘ ) = −5x, so that 1800∘ = 5(360∘ ) = −9x, so that
x = −288∘ . x = −200∘ .
O
(Note: The angle has a negative (Note: The angle has a negative
measure since it is a clockwise measure since it is a clockwise rotation.)
rotation.)
C
9. 10.
a) complement: 90∘ −18∘ = 72∘ a) complement: 90∘ − 39∘ = 51∘
b) supplement: 180∘ −18∘ = 162∘ b) supplement: 180∘ − 39∘ = 141∘
11. 12.
a) complement: 90∘ − 42∘ = 48∘ a) complement: 90∘ − 57∘ = 33∘
b) supplement: 180∘ − 42∘ = 138∘ b) supplement: 180∘ − 57∘ = 123∘
1
,Chapter 1
13. 14.
a) complement: 90∘ − 89∘ = 1∘ a) complement: 90∘ − 75∘ = 15∘
b) supplement: 180∘ − 89∘ = 91∘ b) supplement: 180∘ − 75∘ = 105∘
15. Since the angles with measures ( 4x )∘ and ( 6x )∘ are assumed to be
complementary, we know that ( 4x )∘ + ( 6x )∘ = 90∘. Simplifying this yields
(10x )∘ = 90∘ , so that x = 9. So, the two angles have measures 36∘ and 54∘ .
16. Since the angles with measures ( 3x )∘ and (15x )∘ are assumed to be
PR
supplementary, we know that ( 3x )∘ + (15x )∘ = 180∘. Simplifying this yields
(18x )∘ = 180∘ , so that x = 10. So, the two angles have measures 30∘ and 150∘ .
17. Since the angles with measures ( 8x )∘ and ( 4x )∘ are assumed to be
O
supplementary, we know that ( 8x )∘ + ( 4x )∘ = 180∘. Simplifying this yields
(12x )∘ = 180∘ , so that x = 15. So, the two angles have measures 60∘ and 120∘ .
FD
18. Since the angles with measures ( 3x +15 )∘ and (10x +10 )∘ are assumed to be
complementary, we know that ( 3x +15 )∘ + (10x +10 )∘ = 90∘. Simplifying this yields
(13x + 25 )∘ = 90∘ , so that (13x )∘ = 65∘ and thus, x = 5. So, the two angles have
O
measures 30∘ and 60∘ .
19. Since + + = 180∘ , we know 20. Since + + = 180∘ , we know
C
that that
1 7∘ +3–3∘ + = 180∘ and so, = 30∘ . 1 0∘ +4–5∘ + = 180∘ and so, = 25∘ .
– –
= 150∘ = 155∘
21. Since + + = 180∘ , we know 22. Since + + = 180∘ , we know
that that
( 4 ) + + ( ) = 180∘ and so, = 30∘. (3 ) + + ( ) = 180∘ and so, = 36∘.
–– –– –– ––
= 6 = 5
Thus, = 4 = 120∘ and = = 30∘ . Thus, = 3 = 108∘ and = = 36∘ .
2
, Section 1.1
23. = 180 ∘ − ( 53.3∘ + 23.6 ∘ ) = 103.1∘ 24. = 180 ∘ − (105.6 ∘ +13.2 ∘ ) = 61.2 ∘
25. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes 42 + 32 = c2 , which
simplifies to c2 = 25, so we conclude that c = 5.
26. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes 32 + 32 = c2 , which
simplifies to c2 = 18, so we conclude that c = 18 = 3 2 .
27. Since this is a right triangle, we know from the Pythagorean Theorem that
PR
a2 + b2 = c2. Using the given information, this becomes 62 + b2 = 102 , which
simplifies to 36 + b2 = 100 and then to, b2 = 64, so we conclude that b = 8.
28. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes a2 + 72 = 122 , which
O
simplifies to a2 = 95, so we conclude that a = 95 .
FD
29. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes 82 + 52 = c2 , which
simplifies to c2 = 89, so we conclude that c = 89 .
30. Since this is a right triangle, we know from the Pythagorean Theorem that
O
a2 + b2 = c2. Using the given information, this becomes 62 + 52 = c2 , which
simplifies to c2 = 61, so we conclude that c = 61 .
C
31. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes 72 + b2 = 112 , which
simplifies to b2 = 72, so we conclude that b = 72 = 6 2 .
32. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes a2 + 52 = 92 , which
simplifies to a2 = 56, so we conclude that a = 56 = 2 14 .
3
, Chapter 1
33. Since this is a right triangle, we know from the Pythagorean Theorem that
( 7)
2
a 2 + b2 = c2 . Using the given information, this becomes a 2 + = 5 2 , which
simplifies to a2 = 18, so we conclude that a = 18 = 3 2 .
34. Since this is a right triangle, we know from the Pythagorean Theorem that
a2 + b2 = c2. Using the given information, this becomes 52 + b2 = 102 , which
simplifies to b2 = 75, so we conclude that b = 75 = 5 3 .
35. If x = 10 in., then the hypotenuse 36. If x = 8 m, then the hypotenuse of
of this triangle has length
this triangle has length 8 2 11.31 m .
PR
10 2 14.14 in.
37. Let x be the length of a leg in the given 45∘ − 45∘ − 90∘ triangle. If the
hypotenuse of this triangle has length 2 2 cm, then 2 x = 2 2, so that x = 2.
O
Hence, the length of each of the two legs is 2 cm .
38. Let x be the length of a leg in the given 45∘ − 45∘ − 90∘ triangle. If the hypotenuse
FD
10 10
of this triangle has length 10 ft., then 2 x = 10, so that x = = = 5.
2 2
Hence, the length of each of the two legs is 5 ft.
39. The hypotenuse has length 40. Since 2x = 6m x = 6 2
= 3 2m,
O
( )
2
2 4 2 in. = 8 in. each leg has length 3 2 m.
C
41. Since the lengths of the two legs of the given 30∘ − 60∘ − 90∘ triangle are x and
3 x, the shorter leg must have length x. Hence, using the given information, we
know that x = 5 m. Thus, the two legs have lengths 5 m and 5 3 8.66 m, and
the hypotenuse has length 10 m.
42. Since the lengths of the two legs of the given 30∘ − 60∘ − 90∘ triangle are x and
3 x, the shorter leg must have length x. Hence, using the given information, we
know that x = 9 ft. Thus, the two legs have lengths 9 ft. and 9 3 15.59 ft., and
the hypotenuse has length 18 ft.
4
