Deck 36: Law of Sines

Question 1

Find values for b such that the triangle has one solution. ?
A = 74°,a = 17
?

A) $b \geq 17 , b \geq \frac { 17 } { \sin 74 ^ { \circ } }$
B) $b \geq 17 , b = \frac { 17 } { \sin 74 ^ { \circ } }$
C) $b \leq 17 , b \neq \frac { 17 } { \sin 74 ^ { \circ } }$
D) $b \leq 17 , b = \frac { 17 } { \sin 74 ^ { \circ } }$
E) $b > 17 , b > \frac { 17 } { \sin 74 ^ { \circ } }$

Answer 1

$b \leq 17 , b = \frac { 17 } { \sin 74 ^ { \circ } }$

Question 2

Use the Law of Sines to solve (if possible)the triangle.Round your answers to two decimal places. ​
A = 76°,a = 7,b = 5
​

A)B ≈ 43.87°,C ≈ 43.87°,c ≈ 7.96
B)B ≈ 48.87°,C ≈ 60.13°,c ≈ 6.26
C)B ≈ 43.87°,C ≈ 60.13°,c ≈ 6.26
D)B ≈ 48.87°,C ≈ 65.13°,c ≈ 7.96
E)No Solution

Answer 2

B ≈ 43.87°,C ≈ 60.13°,c ≈ 6.26

Question 3

Use the Law of Sines to solve for B and C.Round your answer to two decimal places. ?
A = 145°,a = 28,b = 8
?

A)B ? 0.18°,C ? 32.89°
B)B ? 0.18°,C ? 39.82°
C)B ? 9.43°,C ? 25.57°
D)B ? 25.57°,C ? 9.43°
E)B ? 39.82°,C ? 0.18°

Answer 3

B ? 9.43°,C ? 25.57°

Question 4

Use the Law of Sines to solve (if possible)the triangle.Round your answers to two decimal places. ​
A = 130°,a = 40,b = 25
​

A)B ≈ 11°,C ≈ 39°,c ≈ 20.05
B)B ≈ 39°,C ≈ 11°,c ≈ 20.05
C)B ≈ 28.61°,C ≈ 21.39°,c ≈ 19.05
D)B ≈ 21.39°,C ≈ 28.61°,c ≈ 19.05
E)No Solution

Question 5

Use the Law of Sines to solve (if possible)the triangle.If two solutions exist,find both.Round your answers to two decimal places. ​
C = 64.47°,a = 39,c = 50
​

A)A ≈ 70.79°,B ≈ 44.74°,b ≈ 52.32
B)A ≈ 70.53°,B ≈ 45°,b ≈ 50
C)A ≈ 44.74°,B ≈ 70.79°,b ≈ 52.32
D)A ≈ 45°,B ≈ 70.53°,b ≈ 50
E)No Solution

Question 6

Find values for b such that the triangle has no solutions. ?
A = 24°,a = 5
?

A) $b > \frac { 5 } { \sin 24 ^ { \circ } }$
B) $b \leq \frac { 5 } { \sin 24 ^ { \circ } }$
C) $b = \frac { 5 } { \sin 24 ^ { \circ } }$
D) $b \geq \frac { 5 } { \sin 24 ^ { \circ } }$
E) $b < \frac { 5 } { \sin 24 ^ { \circ } }$

Question 7

Find values for b such that the triangle has one solution. ?
A = 24?°,a = 16.2
?

A) $b > 16.2 , b > \frac { 16.2 } { \sin 24 ^ { \circ } }$
B) $b \leq 16.2 , b = \frac { 16.2 } { \sin 24 ^ { \circ } }$
C) $b \leq 16.2 , b \neq \frac { 16.2 } { \sin 24 ^ { \circ } }$
D) $b \geq 16.2 , b \geq \frac { 16.2 } { \sin 24 ^ { \circ } }$
E) $b \geq 16.2 , b = \frac { 16.2 } { \sin 24 ^ { \circ } }$

Question 8

Find values for b such that the triangle has two solutions. ?
A = 46°,a = 10
?

A) $10 \geq b \geq \frac { 10 } { \sin 46 ^ { \circ } }$
B) $10 \leq b \leq \frac { 10 } { \sin 46 ^ { \circ } }$
C) $10 > b > \frac { 10 } { \sin 46 ^ { \circ } }$
D) $10 < b \leq \frac { 10 } { \sin 46 ^ { \circ } }$
E) $10 < b < \frac { 10 } { \sin 46 ^ { \circ } }$

Question 9

Use the Law of Sines to solve the triangle.Round your answer to two decimal places. ​
A = 35°,B = 60°,c = 10
​

A)C = 85°,a ​≈ 6.76,b ≈ 8.69
B)C = 85°,a ​≈ 8.69,b ≈ 9.69
C)C = 85°,a ​≈ 8.76,b ≈ 10.69
D)C = 85°,a ​≈ 5.76,b ≈ 8.69
E)​C = 85°,a ​≈ 7.76,b ≈ 10.69

Question 10

Find values for b such that the triangle has no solutions. ?
A = 24°,a = 13
?

A)? $b \leq \frac { 13 } { \sin 24 ^ { \circ } }$
B) $b > \frac { 13 } { \sin 24 ^ { \circ } }$
C) $b = \frac { 13 } { \sin 24 ^ { \circ } }$
D) $b < \frac { 13 } { \sin 24 ^ { \circ } }$
E) $b \geq \frac { 13 } { \sin 24 ^ { \circ } }$

Question 11

Use the Law of Sines to solve the triangle.Round your answer to two decimal places. ​
A = 125°,B = 45°,c = 13
​

A)C = 10°,a ≈ 61.33,b ≈ 52.94
B)C = 10°,a ≈ 54.94,b ≈ 62.33
C)​C = 10°,a ≈ 53.94,b ≈ 52.94
D)​C = 10°,a ≈ 64.33,b ≈ 52.94
E)​C = 10°,a ≈ 63.33,b ≈ 54.94

Question 12

Use the Law of Sines to solve for C and B.Round your answer to two decimal places. ​
A = 60°,a = 36,c = 40
​

A)C ≈ 105.79°,B ≈ 45.79° or
​C ≈ 74.21°,B ≈ 14.21°
B)​C ≈ 74.21°,B ≈ 45.79° or
​C ≈ 105.79°,B ≈ 14.21°
C)​C ≈ 10°,B ≈ 110° or
​C ≈ 25.5°,B ≈ 104.5°
D)​C ≈ 45.79°,B ≈ 74.21° or
​C ≈ 14.21°,B ≈ 105.79°
E)​C ≈ 60°,B ≈ 60° or
​C ≈ 45°,B ≈ 75°

Question 13

Use the Law of Sines to solve for B and C.Round your answer to two decimal places. ​
A = 38°,a = 8,b = 5
​

A)B ≈ 29°,C ≈ 113°
B)B ≈ 22.63°,C ≈ 119.37°
C)B ≈ 103°,C ≈ 39°
D)B ≈ 113°,C ≈ 29°
E)B ≈ 119.37°,C ≈ 22.63°

Question 14

Find values for b such that the triangle has no solutions. ?
A = 54°,a = 19
?

A) $b < \frac { 19 } { \sin 54 ^ { \circ } }$
B) $b \leq \frac { 19 } { \sin 54 ^ { \circ } }$
C) $b = \frac { 19 } { \sin 54 ^ { \circ } }$
D) $b \geq \frac { 19 } { \sin 54 ^ { \circ } }$
E) $b > \frac { 19 } { \sin 54 ^ { \circ } }$

Question 15

Find values for b such that the triangle has two solutions. ?
A = 32°,a = 5
?

A) $5 \leq b \leq \frac { 5 } { \sin 32 ^ { \circ } }$
B) $5 < b < \frac { 5 } { \sin 32 ^ { \circ } }$
C) $5 < b \leq \frac { 5 } { \sin 32 ^ { \circ } }$
D) $5 \geq b \geq \frac { 5 } { \sin 32 ^ { \circ } }$
E) $5 > b > \frac { 5 } { \sin 32 ^ { \circ } }$

Question 16

Find values for b such that the triangle has one solution. ?
A = 62°,a = 304.6
?

A) $b \geq 304.6 , b = \frac { 304.6 } { \sin 62 ^ { \circ } }$
B) $b \leq 304.6 , b \neq \frac { 304.6 } { \sin 62 ^ { \circ } }$
C) $b > 304.6 , b > \frac { 304.6 } { \sin 62 ^ { \circ } }$
D) $b \leq 304.6 , b = \frac { 304.6 } { \sin 62 ^ { \circ } }$
E) $b \geq 304.6 , b \geq \frac { 304.6 } { \sin 62 ^ { \circ } }$

Question 17

Find values for b such that the triangle has one solution. ?
A = 44°,a = 6
?

A) $b \leq 6 , b \neq \frac { 6 } { \sin 44 ^ { \circ } }$
B) $b \geq 6 , b \geq \frac { 6 } { \sin 44 ^ { \circ } }$
C) $b \geq 6 , b = \frac { 6 } { \sin 44 ^ { \circ } }$
D) $b > 6 , b > \frac { 6 } { \sin 44 ^ { \circ } }$
E) $b \leq 6 , b = \frac { 6 } { \sin 44 ^ { \circ } }$

Question 18

Use the Law of Sines to solve (if possible)the triangle.Round your answers to two decimal places. ​
A = 120°,a = b = 36
​

A)B ≈ 108°,C ≈ 36°,c ≈ 44
B)B ≈ 36°,C ≈ 108°,c ≈ 23.5
C)B ≈ 105°,C ≈ 39°,c ≈ 23.5
D)B ≈ 29°,C ≈ 115°,c ≈ 23.5
E)No Solution

Question 19

Use the Law of Sines to solve (if possible)the triangle.If two solutions exist,find both.Round your answers to two decimal places. ​
A = 60°,a = 13,b = 11.6
​

A)B ≈ 69.40°,C ≈ 50.6°,c ≈ 11.6
B)B ≈ 91°,C ≈ 29°,c ≈ 14.05
C)B ≈ 50.6°,C ≈ 69.40°,c ≈ 14.05
D)B ≈ 29°,C ≈ 91°,c ≈ 13
E)No Solution

Question 20

Find values for b such that the triangle has two solutions. ?
A = 10°,a = 18.6
?

A) $18.6 < b \leq \frac { 18.6 } { \sin 10 ^ { \circ } }$
B) $18.6 < b < \frac { 18.6 } { \sin 10 ^ { \circ } }$
C) $18.6 > b > \frac { 18.6 } { \sin 10 ^ { \circ } }$
D) $18.6 \leq b \leq \frac { 18.6 } { \sin 10 ^ { \circ } }$
E) $18.6 \geq b \geq \frac { 18.6 } { \sin 10 ^ { \circ } }$

Question 21

In the figure,? and ? are positive angles.? ? a = 9,b = 18
Write ? as a function of ?.
?

A)? = arcsin(9sin?)
B)? = sin( $\frac { 1 } { 2 }$ sin?) ?
C)? = arcsin( $\frac { 1 } { 2 }$ sin?)
D)? = sin( $\frac { 1 } { 2 }$ sin?)
E)? = arcsin( $\frac { 1 } { 2 }$ sin?)

Question 22

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. A = 129°,b = 8,and c = 11

A)34.19 sq.units
B)41.03 sq.units
C)37.61 sq.units
D)30.77 sq.units
E)27.36 sq.units

Question 23

In the figure,? and ? are positive angles.? ? a = 3,b = 12
Use ? = arcsin(0.25sin?)to write c as a function of ?.
?

A) $c = \frac { 12 \sin [ \pi - \beta - \arcsin ( 0.25 \beta ) ] } { \sin \beta }$
B) $c = \frac { 12 \sin [ \pi + \beta - \arcsin ( 0.25 \beta ) ] } { \sin \beta }$
C) $c = \frac { 0.25 \sin [ \pi - \beta - \arcsin ( 12 \beta ) ] } { \sin \beta }$
D) $c = \frac { 12 \sin [ \pi + \beta + \arcsin ( 0.25 \beta ) ] } { \sin \beta }$
E) $c = \frac { 12 \sin [ \pi - \beta + \arcsin ( 0.25 \beta ) ] } { \sin \beta }$

Question 24

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. A = 118 ${}^\circ$ ,b = 9 and c = 8.
?

A)38.14 sq.units
B)31.79 sq.units
C)25.43 sq.units
D)28.61 sq.units
E)34.96 sq.units

Question 25

A straight road makes an angle,A,of 10° with the horizontal.When the angle of elevation,B,of the sun is 56°,a vertical pole beside the road casts a shadow 8 feet long parallel to the road,see figure.Approximate the length of the pole.Round answer to two decimal places.

A)2.48 feet
B)10.29 feet
C)6.22 feet
D)11.86 feet
E)6.94 feet

Question 26

A park ranger at point A observes a fire in the direction N25°26'E.Another ranger at point B,4 miles due east of A,sites the same fire at N58°34'W.Determine the distance from point B to the fire.Round answer to two decimal places.

A)1.73 miles
B)3.63 miles
C)2.05 miles
D)4.08 miles
E)3.43 miles

Question 27

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. C = 55°43',a = 6.5,and b = 14

A)30.08 sq.units
B)45.11 sq.units
C)33.84 sq.units
D)41.35 sq.units
E)37.59 sq.units

Question 28

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. C = 73 ${}^\circ$ ,a = 12.5 and b = 11.5
?

A)68.96 sq.units
B)62.06 sq.units
C)55.17 sq.units
D)82.75 sq.units
E)75.85 sq.units

Question 29

Find the area of the triangle having the indicated angle and sides.? $C = 102 ^ { \circ } , a = 6 , b = 2$ ? (Round your answer to one decimal place. )
?

A)?4.9
B)5.9
C)7.9
D)6.9
E)11.7

Question 30

Find the area of the triangle having the indicated angle and sides. ​
A = 10°19',b = 4.6,с = 24
​
(Round your answer to one decimal place. )
​

A)9.9
B)8.9
C)10.9
D)11.9
E)19.8

Question 31

Because of prevailing winds,a tree grew so that it was leaning 6° from the vertical.At a point 43 meters from the tree,the angle of elevation to the top of the tree is 30° (see figure).Find the height a of the tree.​ ​c = 43 m B = 96°
​
(Round your answer to two decimal places. )
​

A)24.58 m
B)26.58 m
C)25.58 m
D)28.58 m
E)27.58 m

Question 32

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. B = 84°38',a = 7,and c = 8

A)33.45 sq.units
B)27.88 sq.units
C)25.09 sq.units
D)30.66 sq.units
E)22.30 sq.units

Question 33

Find values for b such that the triangle has no solutions. ?
A = 64°,a = 340.2
?

A) $b \leq \frac { 340.2 } { \sin 64 ^ { \circ } }$
B) $b = \frac { 340.2 } { \sin 64 ^ { \circ } }$
C) $b \geq \frac { 340.2 } { \sin 64 ^ { \circ } }$
D) $b < \frac { 340.2 } { \sin 64 ^ { \circ } }$
E) $b > \frac { 340.2 } { \sin 64 ^ { \circ } }$

Question 34

Find the area of the triangle having the indicated angle and sides. ​
B = 123°,a = 64,c = 28
​
(Round your answer to one decimal place. )
​

A)750.4
B)753.4
C)1,502.9
D)752.4
E)751.4

Question 35

Select a value for b such that a triangle with A = 59° and a = 14 has only one solution. ​

A)b = 16
B)b = 17
C)b = 11
D)b = 15
E)b = 24

Question 36

Find values for b such that the triangle has two solutions. ?
A = 66°,a = 307.6
?

A) $307.6 < b \leq \frac { 307.6 } { \sin 66 ^ { \circ } }$
B) $307.6 \geq b \geq \frac { 307.6 } { \sin 66 ^ { \circ } }$
C) $307.6 > b > \frac { 307.6 } { \sin 66 ^ { \circ } }$
D) $307.6 < b < \frac { 307.6 } { \sin 66 ^ { \circ } }$
E) $307.6 \leq b \leq \frac { 307.6 } { \sin 66 ^ { \circ } }$

Question 37

After a severe storm,three sisters,April,May,and June,stood on their front porch and noticed that the tree in their front yard was leaning 3° from vertical toward the house.From the porch,which is 96 feet away from the base of the tree,they noticed that the angle of elevation to the top of the tree was 31°.Approximate the height of the tree.Round answer to two decimal places. ​

A)61.76 feet
B)69.64 feet
C)56 feet
D)59.13 feet
E)59.64 feet

Question 38

Find the area of the triangle having the indicated angle and sides. ​
B = 71°20',a = 62,c = 24
​
(Round your answer to one decimal place. )
​

A)705.9
B)​703.9
C)​706.9
D)​704.9
E)​1,409.7

Question 39

Find the area of the triangle having the indicated angle and sides. ​
С = 84°34',a = 16,b = 26
​
(Round your answer to one decimal place. )
​

A)209.1
B)​207.1
C)​208.1
D)​206.1
E)​414.1

Question 40

Select a value for b such that a triangle with A = 33° and a = 4 has no solution.

A)1.84
B)3.67
C)7.24
D)14.68
E)1.47

Question 41

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. B = 64 ${}^\circ$ 31 ${}^\prime$ ,a = 10 ?and c = 8.

A)43.33 sq.units
B)32.50 sq.units
C)28.89 sq.units
D)36.11 sq.units
E)39.72 sq.units

Question 42

A straight road makes an angle,A,of 11 ${}^\circ$ with the horizontal.When the angle of elevation,B,of the sun is 52 ${}^\circ$ ,a vertical pole beside the road casts a shadow 6 feet long parallel to the road,see figure.Approximate the length of the pole.Round answer to two decimal places.

A)1.86 feet
B)5.63 feet
C)6.39 feet
D)7.68 feet
E)5.00 feet

Question 43

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. A = 129 ${}^\circ$ ,b = 9 and c = 7.

A)29.38 sq.units
B)24.48 sq.units
C)19.58 sq.units
D)22.03 sq.units
E)26.93 sq.units