Question 1

Use the law of Cosines to solve the given triangle.Round your answer to two decimal places. ​
A = 13, b = 17, c = 23
​

Accepted Answer

A) A ≈ 46.85°, B ≈ 46.85°, C ≈ 86.3° 
B) A ≈ 33.91°, B ≈ 99.24°, C ≈ 46.85° 
C) A ≈ 46.85°, B ≈ 33.91°, C ≈ 99.24° 
D) A ≈ 33.91°, B ≈ 46.85°, C ≈ 99.24° 
E) A ≈ 99.24°, B ≈ 46.85°, C ≈ 33.91° 
A) A ≈ 46.85°, B ≈ 46.85°, C ≈ 86.3° 
B) A ≈ 33.91°, B ≈ 99.24°, C ≈ 46.85° 
C) A ≈ 46.85°, B ≈ 33.91°, C ≈ 99.24° 
D) A ≈ 33.91°, B ≈ 46.85°, C ≈ 99.24° 
E) A ≈ 99.24°, B ≈ 46.85°, C ≈ 33.91°

Question 2

Given a = 4.6, b = 6.1, and c = 9.1, use Heron's Area Formula to find the area of ΔABC.Round your answer to two decimal places. ​

Accepted Answer

A) 10.79 
B) 9.89 
C) 12.60 
D) 8.58 
E) 12.09 
A) 10.79 
B) 9.89 
C) 12.60 
D) 8.58 
E) 12.09

Question 3

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

Accepted Answer

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)  $b \leq 6 , b = \frac { 6 } { \sin 44 ^ { \circ } }$ 
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 4

You want to buy a triangular lot measuring 500 yards by 830 yards by 1110 yards.The price of the land is $2000 per acre.How much does the land cost? (Hint: 1 acre = 4840 square yards)
​

Accepted Answer

A) $80,335.71 
B) $80,465.71 
C) $80,415.71 
D) $80,315.71 
E) $80,215.71 
A) $80,335.71 
B) $80,465.71 
C) $80,415.71 
D) $80,315.71 
E) $80,215.71

Question 5

Find the cube roots of the following complex number.Write each of the roots in standard form. $27$

Accepted Answer

A)  $3$ , $\frac { 3 } { 2 } + \frac { 3 \sqrt { 3 } } { 2 } i$ , $\frac { 3 } { 2 } - \frac { 3 \sqrt { 3 } } { 2 } i$ 
B)  $3$ , $- 3$ 
C)  $3$ 
D)  $3$ , $- \frac { 3 } { 2 } + \frac { 3 \sqrt { 3 } } { 2 } i$ , $- \frac { 3 } { 2 } - \frac { 3 \sqrt { 3 } } { 2 } i$ 
E)  $- 3$ , $\frac { 3 } { 2 } - \frac { 3 \sqrt { 3 } } { 2 } i$ , $\frac { 3 } { 2 } + \frac { 3 \sqrt { 3 } } { 2 } i$ 
A)  $3$ , $\frac { 3 } { 2 } + \frac { 3 \sqrt { 3 } } { 2 } i$ , $\frac { 3 } { 2 } - \frac { 3 \sqrt { 3 } } { 2 } i$ 
B)  $3$ , $- 3$ 
C)  $3$ 
D)  $3$ , $- \frac { 3 } { 2 } + \frac { 3 \sqrt { 3 } } { 2 } i$ , $- \frac { 3 } { 2 } - \frac { 3 \sqrt { 3 } } { 2 } i$ 
E)  $- 3$ , $\frac { 3 } { 2 } - \frac { 3 \sqrt { 3 } } { 2 } i$ , $\frac { 3 } { 2 } + \frac { 3 \sqrt { 3 } } { 2 } i$

Question 6

Find the projection of u onto v.Then write u as the sum of two orthogonal vectors, one of which is $\operatorname { proj } _ { \mathrm { v } } \mathbf { u }$ . 
  $\mathbf { u } = \langle 30,5 \rangle$ $\mathbf { v } = \langle 1 , - 6 \rangle$

Accepted Answer

A)   $\langle 0,0 \rangle , \langle - 30,5 \rangle$ 
B)   $\langle 0,0 \rangle , \langle 30,5 \rangle$ 
C)   $\langle 0,0 \rangle , \langle - 1 , - 6 \rangle$ 
D)   $\langle 0,0 \rangle , \langle 30 , - 5 \rangle$ 
E)   $\langle 0,0 \rangle , \langle 1 , - 6 \rangle$ 
A)   $\langle 0,0 \rangle , \langle - 30,5 \rangle$ 
B)   $\langle 0,0 \rangle , \langle 30,5 \rangle$ 
C)   $\langle 0,0 \rangle , \langle - 1 , - 6 \rangle$ 
D)   $\langle 0,0 \rangle , \langle 30 , - 5 \rangle$ 
E)   $\langle 0,0 \rangle , \langle 1 , - 6 \rangle$

Question 7

Given $\mathbf { u } = - 3 \mathbf { i } + 2 \mathbf { j }$ and $\mathbf { u } = 4 \mathbf { i } - \mathbf { j }$ , find $\text { u v }$ .

Accepted Answer

A) -10 
B) -14 
C) 11 
D) -5 
E) -12 
A) -10 
B) -14 
C) 11 
D) -5 
E) -12

Question 8

Perform the indicated operation using trigonometric form.Leave answer in trigonometric form. $( - 6 - 6 i ) ( - 5 - 5 i )$

Accepted Answer

A)  $30 \left( \cos \frac { \pi } { 8 } + i \sin \frac { \pi } { 8 } \right)$ 
B)  $- 60 \left( \cos \frac { \pi } { 8 } + i \sin \frac { \pi } { 8 } \right)$ 
C)  $30 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
D)  $60 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
E)  $- 30 \left( \cos \frac { \pi } { 8 } + i \sin \frac { \pi } { 8 } \right)$ 
A)  $30 \left( \cos \frac { \pi } { 8 } + i \sin \frac { \pi } { 8 } \right)$ 
B)  $- 60 \left( \cos \frac { \pi } { 8 } + i \sin \frac { \pi } { 8 } \right)$ 
C)  $30 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
D)  $60 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
E)  $- 30 \left( \cos \frac { \pi } { 8 } + i \sin \frac { \pi } { 8 } \right)$

Question 9

Perform the indicated operation using the standard form.   $( 2 + 2 i ) ( 1 - i )$

Accepted Answer

A)  $( 2 + 2 i ) ( 1 - i ) = 2$ 
B)  $( 2 + 2 i ) ( 1 - i ) = 1$ 
C)   $( 2 + 2 i ) ( 1 - i ) = 4$ 
D)   $( 2 + 2 i ) ( 1 - i ) = - 2$ 
E)   $( 2 + 2 i ) ( 1 - i ) = - 4$ 
A)  $( 2 + 2 i ) ( 1 - i ) = 2$ 
B)  $( 2 + 2 i ) ( 1 - i ) = 1$ 
C)   $( 2 + 2 i ) ( 1 - i ) = 4$ 
D)   $( 2 + 2 i ) ( 1 - i ) = - 2$ 
E)   $( 2 + 2 i ) ( 1 - i ) = - 4$

Question 10

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.)
​

Accepted Answer

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

Question 11

Find u + v. 
 $\mathbf { u } = \langle - 5,3 \rangle , \mathbf { v } = \langle 0,0 \rangle$

Accepted Answer

A)   $\langle - 4,4 \rangle$ 
B)  $\langle - 5 , - 5 \rangle$ 
C)   $\langle 3,3 \rangle$ 
D)   $\langle - 5,3 \rangle$ 
E)   $\langle 3 , - 5 \rangle$ 
A)   $\langle - 4,4 \rangle$ 
B)  $\langle - 5 , - 5 \rangle$ 
C)   $\langle 3,3 \rangle$ 
D)   $\langle - 5,3 \rangle$ 
E)   $\langle 3 , - 5 \rangle$

Question 12

To determine the distance between two aircraft, a tracking station continuously determines the distance to each aircraft and the angle between them (see figure).Determine the distance a between the planes when A = 52° miles, b = 22 miles, and c = 19 miles. ​   ​

Accepted Answer

A) 18.2 mi 
B) 20.2 mi 
C) 36.9 mi 
D) 24.2 mi 
E) 21.2 mi 
A) 18.2 mi 
B) 20.2 mi 
C) 36.9 mi 
D) 24.2 mi 
E) 21.2 mi

Question 13

Given vectors $\mathbf { u } = \langle - 1 , - 2 \rangle$ and $\mathbf { v } = \langle 3,5 \rangle$ determine the quantity indicated below. $( 4 \mathbf { u } \cdot 2 \mathbf { v } ) \mathbf { u }$

Accepted Answer

A)  $\langle 32,64 \rangle$ 
B)  $\langle 104,208 \rangle$ 
C)  $\langle - 8 , - 16 \rangle$ 
D)  $\langle 8,16 \rangle$ 
E)  $\langle - 64 , - 128 \rangle$ 
A)  $\langle 32,64 \rangle$ 
B)  $\langle 104,208 \rangle$ 
C)  $\langle - 8 , - 16 \rangle$ 
D)  $\langle 8,16 \rangle$ 
E)  $\langle - 64 , - 128 \rangle$

Question 14

Given a = 11, b = 14, and c = 10, use Heron's Area Formula to find the area of triangle ABC.Round your answer to two decimal places. ​

Accepted Answer

A) 54.64 
B) 47.04 
C) 41.31 
D) 48.87 
E) 43.32 
A) 54.64 
B) 47.04 
C) 41.31 
D) 48.87 
E) 43.32

Question 15

Find the dot product of u and v.   $\begin{array} { l } 
\mathbf { u } = \langle 5,1 \rangle \
\mathbf { v } = \langle - 2,4 \rangle
\end{array}$

Accepted Answer

A) -10 
B) -8 
C) -4 
D) -6 
E) -2 
A) -10 
B) -8 
C) -4 
D) -6 
E) -2

Question 16

Use the low of Cosines to solve the given triangle.Round your answer to two decimal places. ​
A = 1.62, b = 0.95, c = 1.45
​

Accepted Answer

A) A ≈ 35.51°, B ≈ 82.06°, C ≈ 62.43° 
B) A ≈ 62.43°, B ≈ 35.51°, C ≈ 82.06° 
C) A ≈ 82.06°, B ≈ 35.51°, C ≈ 62.43° 
D) A ≈ 82.06°, B ≈ 62.43°, C ≈ 35.51° 
E) A ≈ 62.43°, B ≈ 82.06°, C ≈ 35.51° 
A) A ≈ 35.51°, B ≈ 82.06°, C ≈ 62.43° 
B) A ≈ 62.43°, B ≈ 35.51°, C ≈ 82.06° 
C) A ≈ 82.06°, B ≈ 35.51°, C ≈ 62.43° 
D) A ≈ 82.06°, B ≈ 62.43°, C ≈ 35.51° 
E) A ≈ 62.43°, B ≈ 82.06°, C ≈ 35.51°

Question 17

Find the angle between the vectors u and v. $\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j }$ , $v = \cos \left( \frac { 3 \pi } { 4 } \right) i + \sin \left( \frac { 3 \pi } { 4 } \right) j$

Accepted Answer

A)  $40 ^ { \circ }$ 
B)  $15 ^ { \circ }$ 
C)  $75 ^ { \circ }$ 
D)  $105 ^ { \circ }$ 
E)  $70 ^ { \circ }$ 
A)  $40 ^ { \circ }$ 
B)  $15 ^ { \circ }$ 
C)  $75 ^ { \circ }$ 
D)  $105 ^ { \circ }$ 
E)  $70 ^ { \circ }$

Question 18

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

Accepted Answer

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° 
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 19

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

Accepted Answer

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

Question 20

Use the vectors $\mathbf { u } = \langle 2,4 \rangle$ , $\mathbf { v } = \langle - 4,5 \rangle$ , and $\mathbf { w } = \langle 5 , - 4 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar.
  $( \mathbf { u } \cdot 2 \mathbf { v } ) \mathbf { w }$

Accepted Answer

A) -96; scalar 
B)  $\langle 120 , - 92 \rangle$ ; vector 
C)  $\langle 120 , - 94 \rangle$ ; vector 
D)  $\langle 120 , - 100 \rangle$ ; vector 
E)  $\langle 120 , - 96 \rangle$ ; vector 
A) -96; scalar 
B)  $\langle 120 , - 92 \rangle$ ; vector 
C)  $\langle 120 , - 94 \rangle$ ; vector 
D)  $\langle 120 , - 100 \rangle$ ; vector 
E)  $\langle 120 , - 96 \rangle$ ; vector

Use the law of Cosines to solve the given triangle.Round your answer to two decimal places. A = 13, b = 17, c = 23

Given a = 4.6, b = 6.1, and c = 9.1, use Heron's Area Formula to find the area of ΔABC.Round your answer to two decimal places.

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

You want to buy a triangular lot measuring 500 yards by 830 yards by 1110 yards.The price of the land is $2000 per acre.How much does the land cost? (Hint: 1 acre = 4840 square yards)

Find the cube roots of the following complex number.Write each of the roots in standard form. $27$

Find the projection of u onto v.Then write u as the sum of two orthogonal vectors, one of which is $\operatorname { proj } _ { \mathrm { v } } \mathbf { u }$ . $\mathbf { u } = \langle 30,5 \rangle$ $\mathbf { v } = \langle 1 , - 6 \rangle$

Given $\mathbf { u } = - 3 \mathbf { i } + 2 \mathbf { j }$ and $\mathbf { u } = 4 \mathbf { i } - \mathbf { j }$ , find $\text { u v }$ .

Perform the indicated operation using trigonometric form.Leave answer in trigonometric form. $( - 6 - 6 i ) ( - 5 - 5 i )$

Perform the indicated operation using the standard form. $( 2 + 2 i ) ( 1 - i )$

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.)

Find u + v. $\mathbf { u } = \langle - 5,3 \rangle , \mathbf { v } = \langle 0,0 \rangle$

To determine the distance between two aircraft, a tracking station continuously determines the distance to each aircraft and the angle between them (see figure).Determine the distance a between the planes when A = 52° miles, b = 22 miles, and c = 19 miles.

Given vectors $\mathbf { u } = \langle - 1 , - 2 \rangle$ and $\mathbf { v } = \langle 3,5 \rangle$ determine the quantity indicated below. $( 4 \mathbf { u } \cdot 2 \mathbf { v } ) \mathbf { u }$

Given a = 11, b = 14, and c = 10, use Heron's Area Formula to find the area of triangle ABC.Round your answer to two decimal places.

Find the dot product of u and v. =\langle5,1\rangle =\langle-2,4\rangle

Use the low of Cosines to solve the given triangle.Round your answer to two decimal places. A = 1.62, b = 0.95, c = 1.45

Find the angle between the vectors u and v. $\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j }$ , $v = \cos \left( \frac { 3 \pi } { 4 } \right) i + \sin \left( \frac { 3 \pi } { 4 } \right) j$

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

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

Use the vectors $\mathbf { u } = \langle 2,4 \rangle$ , $\mathbf { v } = \langle - 4,5 \rangle$ , and $\mathbf { w } = \langle 5 , - 4 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar. $( \mathbf { u } \cdot 2 \mathbf { v } ) \mathbf { w }$

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Exam 6: Additional Topics In Trigonometery

Use the law of Cosines to solve the given triangle.Round your answer to two decimal places. ​ A = 13, b = 17, c = 23 ​

Given a = 4.6, b = 6.1, and c = 9.1, use Heron's Area Formula to find the area of ΔABC.Round your answer to two decimal places. ​

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

You want to buy a triangular lot measuring 500 yards by 830 yards by 1110 yards.The price of the land is $2000 per acre.How much does the land cost? (Hint: 1 acre = 4840 square yards) ​

Find the cube roots of the following complex number.Write each of the roots in standard form. 272727

Given u=−3i+2j\mathbf { u } = - 3 \mathbf { i } + 2 \mathbf { j }u=−3i+2j and u=4i−j\mathbf { u } = 4 \mathbf { i } - \mathbf { j }u=4i−j , find u v \text { u v } u v .

Perform the indicated operation using trigonometric form.Leave answer in trigonometric form. (−6−6i)(−5−5i)( - 6 - 6 i ) ( - 5 - 5 i )(−6−6i)(−5−5i)

Perform the indicated operation using the standard form. (2+2i)(1−i)( 2 + 2 i ) ( 1 - i )(2+2i)(1−i)

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.) ​

Find u + v. u=⟨−5,3⟩,v=⟨0,0⟩\mathbf { u } = \langle - 5,3 \rangle , \mathbf { v } = \langle 0,0 \rangleu=⟨−5,3⟩,v=⟨0,0⟩

To determine the distance between two aircraft, a tracking station continuously determines the distance to each aircraft and the angle between them (see figure).Determine the distance a between the planes when A = 52° miles, b = 22 miles, and c = 19 miles. ​ ​

Given vectors u=⟨−1,−2⟩\mathbf { u } = \langle - 1 , - 2 \rangleu=⟨−1,−2⟩ and v=⟨3,5⟩\mathbf { v } = \langle 3,5 \ranglev=⟨3,5⟩ determine the quantity indicated below. (4u⋅2v)u( 4 \mathbf { u } \cdot 2 \mathbf { v } ) \mathbf { u }(4u⋅2v)u

Given a = 11, b = 14, and c = 10, use Heron's Area Formula to find the area of triangle ABC.Round your answer to two decimal places. ​

Find the dot product of u and v. =\langle5,1\rangle =\langle-2,4\rangle

Use the low of Cosines to solve the given triangle.Round your answer to two decimal places. ​ A = 1.62, b = 0.95, c = 1.45 ​

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

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

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Use the law of Cosines to solve the given triangle.Round your answer to two decimal places. A = 13, b = 17, c = 23

Given a = 4.6, b = 6.1, and c = 9.1, use Heron's Area Formula to find the area of ΔABC.Round your answer to two decimal places.

You want to buy a triangular lot measuring 500 yards by 830 yards by 1110 yards.The price of the land is $2000 per acre.How much does the land cost? (Hint: 1 acre = 4840 square yards)

Find the cube roots of the following complex number.Write each of the roots in standard form. $27$

Given $\mathbf { u } = - 3 \mathbf { i } + 2 \mathbf { j }$ and $\mathbf { u } = 4 \mathbf { i } - \mathbf { j }$ , find $\text { u v }$ .

Perform the indicated operation using trigonometric form.Leave answer in trigonometric form. $( - 6 - 6 i ) ( - 5 - 5 i )$

Perform the indicated operation using the standard form. $( 2 + 2 i ) ( 1 - i )$

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.)

Find u + v. $\mathbf { u } = \langle - 5,3 \rangle , \mathbf { v } = \langle 0,0 \rangle$

To determine the distance between two aircraft, a tracking station continuously determines the distance to each aircraft and the angle between them (see figure).Determine the distance a between the planes when A = 52° miles, b = 22 miles, and c = 19 miles.

Given vectors $\mathbf { u } = \langle - 1 , - 2 \rangle$ and $\mathbf { v } = \langle 3,5 \rangle$ determine the quantity indicated below. $( 4 \mathbf { u } \cdot 2 \mathbf { v } ) \mathbf { u }$

Given a = 11, b = 14, and c = 10, use Heron's Area Formula to find the area of triangle ABC.Round your answer to two decimal places.

Use the low of Cosines to solve the given triangle.Round your answer to two decimal places. A = 1.62, b = 0.95, c = 1.45

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