Question 1

Find the standard form of the complex number shown below. $3 \left( \cos \frac { 7 \pi } { 6 } + i \sin \frac { 7 \pi } { 6 } \right)$

Accepted Answer

A)  $- \frac { 3 \sqrt { 3 } } { 4 } - \frac { 3 } { 4 } i$ 
B)   $- \frac { 2 } { 3 } - \frac { 2 \sqrt { 3 } } { 3 } i$ 
C)   $- \frac { 4 \sqrt { 3 } } { 3 } - \frac { 4 } { 3 } i$ 
D)   $- \frac { 3 \sqrt { 3 } } { 2 } - \frac { 3 } { 2 } i$ 
E)   $- \frac { 3 } { 2 } - \frac { 3 \sqrt { 3 } } { 2 } i$

Question 2

The initial and terminal points of vector v are (3, -8) and (7, 1), respectively.Select v in component form.

Accepted Answer

A)   $\mathbf { v } = \langle - 4 , - 9 \rangle$ 
B)   $\mathbf { v } = \langle - 4,9 \rangle$ 
C)  $\mathbf { v } = \langle 4,9 \rangle$ 
D)  $\mathbf { v } = \langle 4 , - 9 \rangle$ 
E)   $\mathbf { v } = \langle 9,4 \rangle$

Question 3

Use the law of Cosines to solve the given triangle.Round your answer to two decimal places. ​
A = 50°, b = 4, c = 15
​

Accepted Answer

A) a ≈ 15, B ≈ 13.85°, C ≈ 116.15° 
B) a ≈ 12.80, B ≈ 116.15°, C ≈ 13.85° 
C) a ≈ 4, B ≈ 13.85°, C ≈ 116.15° 
D) a ≈ 15, B ≈ 15.85°, C ≈ 114.15° 
E) a ≈ 12.80, B ≈ 13.85°, C ≈ 116.15°

Question 4

Find 2u - 3v. 
 $\mathbf { u } = 6 \mathbf { i } , \mathbf { v } = \mathbf { j }$

Accepted Answer

A)  $12 \mathbf { i } - 3 \mathbf { j }$ 
B)   $- 12 \mathbf { i } - 3 \mathbf { j }$ 
C)   $6 \mathbf { i } - 3 \mathbf { j }$ 
D)   $12 \mathbf { i } + 3 \mathbf { j }$ 
E)   $6 i - j$

Question 5

Find the angle between the vectors u and v.   $\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j }$ , $\mathbf { v } = \cos \left( \frac { 5 \pi } { 4 } \right) \mathbf { i } + \sin \left( \frac { 5 \pi } { 4 } \right) \mathbf { j }$

Accepted Answer

A)  $40 ^ { \circ }$ 
B)  $75 ^ { \circ }$ 
C)  $105 ^ { \circ }$ 
D)  $165 ^ { \circ }$ 
E)  $175 ^ { \circ }$

Question 6

Perform the operation shown below and leave the result in trigonometric form.  
  $\frac { 12 ( \cos 3.1 + i \sin 3.1 ) } { 6 ( \cos 2.7 + i \sin 2.7 ) }$

Accepted Answer

A)  $6 ( \cos 0.4 + i \sin 0.4 )$ 
B)   $6 ( \cos 1.1 + i \sin 1.1 )$ 
C)   $2 ( \cos 0.4 + i \sin 0.4 )$ 
D)   $2 ( \cos 8.37 + i \sin 8.37 )$ 
E)   $2 ( \cos 5.8 + i \sin 5.8 )$

Question 7

Find the trigonometric form of the complex number shown below. $- 10 i$

Accepted Answer

A)  $10 ( \cos 0 + i \sin 0 )$ 
B)  $10 ( \cos \pi + i \sin \pi )$ 
C)  $10 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
D)  $10 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$ 
E)  $10 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$

Question 8

Use a graphing utility to represent the complex number in standard form.Round your answer to 4 decimal places.   $5 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$

Accepted Answer

A) 4.3301 + 2.5i 
B) 2.5 + 4.3301i 
C) 2.5 - 4.3301i 
D) 4.3301 - 2.5i 
E) None of the above

Question 9

Perform the indicated operation using trigonometric form.Leave the answer in trigonometric forms. $( 2 + 2 i ) ( - 1 - i )$

Accepted Answer

A)  $- 2 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
B)   $4 \left( \cos \frac { 3 \pi } { 2 } - i \sin \frac { 3 \pi } { 2 } \right)$ 
C)   $4 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
D)   $- 2 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$ 
E)   $2 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$

Question 10

Plot the complex number and find its absolute value.  -2

Accepted Answer

A)      $| - 2 | = - 3$ 
B)      $| - 2 | = 2$ 
C)       $| - 2 | = - 4$ 
D)      $| - 2 | = 4$ 
E)      $| - 2 | = 3$

Question 11

Given $\mathbf { u } = - \mathbf { i } + 3 \mathbf { j }$ and $\mathbf { v } = 5 \mathbf { i } + 4 \mathbf { j }$ , find $\text { u v }$

Accepted Answer

A) 7 
B) -5 
C) -17 
D) 11 
E) -19

Question 12

Find the component form of v given its magnitude and the angle it makes with the positive x-axis. $\begin{array}{lr}\text { Magnitude}&\text { Angle }\\|\mathbf{v}\|=4&0^o
\end{array}$

Accepted Answer

A)  $\langle 0,0 \rangle$ 
B)   $\langle 4,4 \rangle$ 
C)   $\langle 0,4 \rangle$ 
D)   $\langle 4,0 \rangle$ 
E)   $\langle - 4,0 \rangle$

Question 13

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

Accepted Answer

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)  $b  \frac { 340.2 } { \sin 64 ^ { \circ } }$

Question 14

Use the dot product to find the magnitude of u.   $\mathbf { u } = 24 \mathbf { i } - 28 \mathbf { j }$

Accepted Answer

A)  $4 \sqrt { 85 }$ 
B)  $\sqrt { 1358 }$ 
C)  $2 \sqrt { 339 }$ 
D)  $\sqrt { 1362 }$ 
E)  $2 \sqrt { 341 }$

Question 15

Plot the following complex number. -3

Accepted Answer

A)   
B)   
C)   
D)   
E)   ​

Question 16

Given $\mathbf { u } = \langle 3 , - 2 \rangle$ and $\mathbf { v } = \langle - 3,2 \rangle$ , determine $- 4 \mathbf { u } - \mathbf { v }$ .

Accepted Answer

A)   $- 4 \mathbf { u } - \mathbf { v } = \langle 9 , - 14 \rangle$ 
B)  $- 4 \mathbf { u } - \mathbf { v } = \langle - 9,6 \rangle$ 
C)   $- 4 \mathbf { u } - \mathbf { v } = \langle - 14 , - 14 \rangle$ 
D)   $- 4 \mathbf { u } - \mathbf { v } = \langle 11 , - 14 \rangle$ 
E)   $- 4 \mathbf { u } - \mathbf { v } = \langle - 11,10 \rangle$

Question 17

Use the low of Cosines to solve the given triangle.Round your answer to two decimal places. ​
A = 75.4, b = 52, c = 52
​
​

Accepted Answer

A) A ≈ 43.53°, B ≈ 92.94°, C ≈ 43.53° 
B) A ≈ 43.53°, B ≈ 97.94°, C ≈ 38.53° 
C) A ≈ 43.53°, B ≈ 38.53°, C ≈ 97.94° 
D) A ≈ 92.94°, B ≈ 43.53°, C ≈ 43.53° 
E) A ≈ 97.94°, B ≈ 38.53°, C ≈ 43.53°

Question 18

Find the projection of u onto v if $\mathbf { u } = \langle 5 , - 3 \rangle$ , $\mathbf { v } = \langle - 1,4 \rangle$ .

Accepted Answer

A)   $\left\langle \frac { - 17 \sqrt { 17 } } { 17 } , \frac { 51 \sqrt { 17 } } { 17 } \right\rangle$ 
B)   $\left\langle \frac { 17 \sqrt { 17 } } { 17 } , \frac { 68 \sqrt { 17 } } { 17 } \right\rangle$ 
C)   $\left\langle \frac { 17 \sqrt { 17 } } { 17 } , \frac { 17 \sqrt { 17 } } { 17 } \right\rangle$ 
D)   $\left\langle \frac { 17 \sqrt { 17 } } { 17 } , \frac { - 68 \sqrt { 17 } } { 17 } \right\rangle$ 
E)   $\left\langle \frac { 17 \sqrt { 17 } } { 17 } , \frac { 51 \sqrt { 17 } } { 17 } \right\rangle$

Question 19

You want to buy a triangular lot measuring 1360 feet by 1870 feet by 2500 feet.The price of the land is $2200 per acre.How much does the land cost? (Hint: 1 acre = 43,560 square feet)
​

Accepted Answer

A) $63,450.91 
B) $63,300.91 
C) $63,400.91 
D) $63,200.91 
E) $63,320.91

Question 20

Plot the complex number and find its absolute value. 
-5 + 10i

Accepted Answer

A)     -5 + 10i| = $\sqrt { 95 }$ 
B)     |-5 + 10i| = $2 \sqrt { 7 }$ 
C)    |-5 + 10i| = $5 \sqrt { 5 }$ 
D)    |-5 + 10i| = $\sqrt { 22 }$ 
E)     |-5 + 10i| = $\sqrt { 105 }$

Exam 6: Additional Topics In Trigonometery

Find the standard form of the complex number shown below. $3 \left( \cos \frac { 7 \pi } { 6 } + i \sin \frac { 7 \pi } { 6 } \right)$

The initial and terminal points of vector v are (3, -8) and (7, 1) , respectively.Select v in component form.

Use the law of Cosines to solve the given triangle.Round your answer to two decimal places. A = 50°, b = 4, c = 15

Find 2u - 3v. $\mathbf { u } = 6 \mathbf { i } , \mathbf { v } = \mathbf { j }$

Perform the operation shown below and leave the result in trigonometric form. $\frac { 12 ( \cos 3.1 + i \sin 3.1 ) } { 6 ( \cos 2.7 + i \sin 2.7 ) }$

Find the trigonometric form of the complex number shown below. $- 10 i$

Use a graphing utility to represent the complex number in standard form.Round your answer to 4 decimal places. $5 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$

Perform the indicated operation using trigonometric form.Leave the answer in trigonometric forms. $( 2 + 2 i ) ( - 1 - i )$

Plot the complex number and find its absolute value. -2

Given $\mathbf { u } = - \mathbf { i } + 3 \mathbf { j }$ and $\mathbf { v } = 5 \mathbf { i } + 4 \mathbf { j }$ , find $\text { u v }$

Find the component form of v given its magnitude and the angle it makes with the positive x-axis. $\begin{array}{lr}\text { Magnitude}&\text { Angle }\\\|\mathbf{v}\|=4&0^o\end{array}$

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

Use the dot product to find the magnitude of u. $\mathbf { u } = 24 \mathbf { i } - 28 \mathbf { j }$

Plot the following complex number. -3

Given $\mathbf { u } = \langle 3 , - 2 \rangle$ and $\mathbf { v } = \langle - 3,2 \rangle$ , determine $- 4 \mathbf { u } - \mathbf { v }$ .

Use the low of Cosines to solve the given triangle.Round your answer to two decimal places. A = 75.4, b = 52, c = 52

Find the projection of u onto v if $\mathbf { u } = \langle 5 , - 3 \rangle$ , $\mathbf { v } = \langle - 1,4 \rangle$ .

You want to buy a triangular lot measuring 1360 feet by 1870 feet by 2500 feet.The price of the land is $2200 per acre.How much does the land cost? (Hint: 1 acre = 43,560 square feet)

Plot the complex number and find its absolute value. -5 + 10i

Exam 6: Additional Topics In Trigonometery

Find the standard form of the complex number shown below. 3(cos⁡7π6+isin⁡7π6) 3 \left( \cos \frac { 7 \pi } { 6 } + i \sin \frac { 7 \pi } { 6 } \right) 3(cos67π​+isin67π​)

The initial and terminal points of vector v are (3, -8) and (7, 1) , respectively.Select v in component form.

Use the law of Cosines to solve the given triangle.Round your answer to two decimal places. ​ A = 50°, b = 4, c = 15 ​

Find 2u - 3v. u=6i,v=j\mathbf { u } = 6 \mathbf { i } , \mathbf { v } = \mathbf { j }u=6i,v=j

Perform the operation shown below and leave the result in trigonometric form. 12(cos⁡3.1+isin⁡3.1) 6(cos⁡2.7+isin⁡2.7) \frac { 12 ( \cos 3.1 + i \sin 3.1 ) } { 6 ( \cos 2.7 + i \sin 2.7 ) }6(cos2.7+isin2.7) 12(cos3.1+isin3.1) ​

Find the trigonometric form of the complex number shown below. −10i- 10 i−10i

Use a graphing utility to represent the complex number in standard form.Round your answer to 4 decimal places. 5(cos⁡π3+isin⁡π3) 5 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right) 5(cos3π​+isin3π​)

Perform the indicated operation using trigonometric form.Leave the answer in trigonometric forms. (2+2i) (−1−i) ( 2 + 2 i ) ( - 1 - i ) (2+2i) (−1−i)

Plot the complex number and find its absolute value. -2

Given u=−i+3j\mathbf { u } = - \mathbf { i } + 3 \mathbf { j }u=−i+3j and v=5i+4j\mathbf { v } = 5 \mathbf { i } + 4 \mathbf { j }v=5i+4j , find u v \text { u v } u v

Find the component form of v given its magnitude and the angle it makes with the positive x-axis. Magnitude Angle ∥v∥=40o\begin{array}{lr}\text { Magnitude}&\text { Angle }\\\|\mathbf{v}\|=4&0^o\end{array} Magnitude∥v∥=4​ Angle 0o​

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

Use the dot product to find the magnitude of u. u=24i−28j\mathbf { u } = 24 \mathbf { i } - 28 \mathbf { j }u=24i−28j

Plot the following complex number. -3

Given u=⟨3,−2⟩\mathbf { u } = \langle 3 , - 2 \rangleu=⟨3,−2⟩ and v=⟨−3,2⟩\mathbf { v } = \langle - 3,2 \ranglev=⟨−3,2⟩ , determine −4u−v- 4 \mathbf { u } - \mathbf { v }−4u−v .

Use the low of Cosines to solve the given triangle.Round your answer to two decimal places. ​ A = 75.4, b = 52, c = 52 ​ ​

Find the projection of u onto v if u=⟨5,−3⟩\mathbf { u } = \langle 5 , - 3 \rangleu=⟨5,−3⟩ , v=⟨−1,4⟩\mathbf { v } = \langle - 1,4 \ranglev=⟨−1,4⟩ .

You want to buy a triangular lot measuring 1360 feet by 1870 feet by 2500 feet.The price of the land is $2200 per acre.How much does the land cost? (Hint: 1 acre = 43,560 square feet) ​

Plot the complex number and find its absolute value. -5 + 10i

Find the standard form of the complex number shown below. $3 \left( \cos \frac { 7 \pi } { 6 } + i \sin \frac { 7 \pi } { 6 } \right)$

Use the law of Cosines to solve the given triangle.Round your answer to two decimal places. A = 50°, b = 4, c = 15

Find 2u - 3v. $\mathbf { u } = 6 \mathbf { i } , \mathbf { v } = \mathbf { j }$

Perform the operation shown below and leave the result in trigonometric form. $\frac { 12 ( \cos 3.1 + i \sin 3.1 ) } { 6 ( \cos 2.7 + i \sin 2.7 ) }$

Find the trigonometric form of the complex number shown below. $- 10 i$

Use a graphing utility to represent the complex number in standard form.Round your answer to 4 decimal places. $5 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$

Perform the indicated operation using trigonometric form.Leave the answer in trigonometric forms. $( 2 + 2 i ) ( - 1 - i )$

Given $\mathbf { u } = - \mathbf { i } + 3 \mathbf { j }$ and $\mathbf { v } = 5 \mathbf { i } + 4 \mathbf { j }$ , find $\text { u v }$

Find the component form of v given its magnitude and the angle it makes with the positive x-axis. $\begin{array}{lr}\text { Magnitude}&\text { Angle }\\\|\mathbf{v}\|=4&0^o\end{array}$

Use the dot product to find the magnitude of u. $\mathbf { u } = 24 \mathbf { i } - 28 \mathbf { j }$

Given $\mathbf { u } = \langle 3 , - 2 \rangle$ and $\mathbf { v } = \langle - 3,2 \rangle$ , determine $- 4 \mathbf { u } - \mathbf { v }$ .

Use the low of Cosines to solve the given triangle.Round your answer to two decimal places. A = 75.4, b = 52, c = 52

Find the projection of u onto v if $\mathbf { u } = \langle 5 , - 3 \rangle$ , $\mathbf { v } = \langle - 1,4 \rangle$ .

You want to buy a triangular lot measuring 1360 feet by 1870 feet by 2500 feet.The price of the land is $2200 per acre.How much does the land cost? (Hint: 1 acre = 43,560 square feet)