Question 1

Find the fourth roots of $- \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } i$. Write the roots in trigonometric form.

Accepted Answer

A) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 50 ^ { \circ } \right) + i \sin \left( 50 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 140 ^ { \circ } \right) + i \sin \left( 140 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 230 ^ { \circ } \right) + i \sin \left( 230 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 320 ^ { \circ } \right) + i \sin \left( 320 ^ { \circ } \right)
\end{array}$ 
B) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 60 ^ { \circ } \right) + i \sin \left( 60 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 150 ^ { \circ } \right) + i \sin \left( 150 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 240 ^ { \circ } \right) + i \sin \left( 240 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 330 ^ { \circ } \right) + i \sin \left( 330 ^ { \circ } \right)
\end{array}$ 
C) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 65 ^ { \circ } \right) + i \sin \left( 65 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 155 ^ { \circ } \right) + i \sin \left( 155 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 245 ^ { \circ } \right) + i \sin \left( 245 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 335 ^ { \circ } \right) + i \sin \left( 335 ^ { \circ } \right)
\end{array}$ 
D) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 55 ^ { \circ } \right) + i \sin \left( 55 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 145 ^ { \circ } \right) + i \sin \left( 145 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 235 ^ { \circ } \right) + i \sin \left( 235 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 325 ^ { \circ } \right) + i \sin \left( 325 ^ { \circ } \right)
\end{array}$ 
E) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 70 ^ { \circ } \right) + i \sin \left( 70 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 160 ^ { \circ } \right) + i \sin \left( 160 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 250 ^ { \circ } \right) + i \sin \left( 250 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 340 ^ { \circ } \right) + i \sin \left( 340 ^ { \circ } \right)
\end{array}$ 
A) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 50 ^ { \circ } \right) + i \sin \left( 50 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 140 ^ { \circ } \right) + i \sin \left( 140 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 230 ^ { \circ } \right) + i \sin \left( 230 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 320 ^ { \circ } \right) + i \sin \left( 320 ^ { \circ } \right)
\end{array}$ 
B) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 60 ^ { \circ } \right) + i \sin \left( 60 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 150 ^ { \circ } \right) + i \sin \left( 150 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 240 ^ { \circ } \right) + i \sin \left( 240 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 330 ^ { \circ } \right) + i \sin \left( 330 ^ { \circ } \right)
\end{array}$ 
C) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 65 ^ { \circ } \right) + i \sin \left( 65 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 155 ^ { \circ } \right) + i \sin \left( 155 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 245 ^ { \circ } \right) + i \sin \left( 245 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 335 ^ { \circ } \right) + i \sin \left( 335 ^ { \circ } \right)
\end{array}$ 
D) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 55 ^ { \circ } \right) + i \sin \left( 55 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 145 ^ { \circ } \right) + i \sin \left( 145 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 235 ^ { \circ } \right) + i \sin \left( 235 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 325 ^ { \circ } \right) + i \sin \left( 325 ^ { \circ } \right)
\end{array}$ 
E) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 70 ^ { \circ } \right) + i \sin \left( 70 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 160 ^ { \circ } \right) + i \sin \left( 160 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 250 ^ { \circ } \right) + i \sin \left( 250 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 340 ^ { \circ } \right) + i \sin \left( 340 ^ { \circ } \right)
\end{array}$

Question 2

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

Accepted Answer

A)  20 
B)  $- 16$ 
C)  2 
D)  32 
E)  14 
A)  20 
B)  $- 16$ 
C)  2 
D)  32 
E)  14

Question 3

Use the figure to determine the tension in the cable CA supporting the load. Round your answer to two decimals.

Accepted Answer

A)  $3194.44 \mathrm { lbs }$. 
B)  $9996.76 \mathrm { lbs }$. 
C)  $6881.42 \mathrm { lbs }$. 
D)  $2500.81 \mathrm { lbs }$. 
E)  $3632.97 \mathrm { lbs }$. 
A)  $3194.44 \mathrm { lbs }$. 
B)  $9996.76 \mathrm { lbs }$. 
C)  $6881.42 \mathrm { lbs }$. 
D)  $2500.81 \mathrm { lbs }$. 
E)  $3632.97 \mathrm { lbs }$.

Question 4

Use DeMoivre's Theorem to find the indicated power of the following complex number.
$$( 5 + 5 \sqrt { 3 } i ) ^ { 6 }$$

Accepted Answer

A)  $1,953,125$ 
B)  390,625 
C)  78,125 
D)  $1,000,000$ 
E)  3,125 
A)  $1,953,125$ 
B)  390,625 
C)  78,125 
D)  $1,000,000$ 
E)  3,125

Question 5

Use DeMoivre's Theorem to find $( 8 + 8 i ) ^ { 3 }$. Write the result in standard form.

Accepted Answer

A)  $1024 - 1024 i$ 
B)  $- 1024 + 1024 i$ 
C)  $- 1024 - 1024 i$ 
D)  $512 + 512 i$ 
E)  $1024 + 1024 i$ 
A)  $1024 - 1024 i$ 
B)  $- 1024 + 1024 i$ 
C)  $- 1024 - 1024 i$ 
D)  $512 + 512 i$ 
E)  $1024 + 1024 i$

Question 6

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

Accepted Answer

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

Question 7

Use Heron's area formula to find the area of the triangle pictured below, if $a = 8$ inches, $\mathrm { b } = 10$ inches, and $\mathrm { c } = 4$ inches.

Accepted Answer

A)  $\sqrt { 21 }$ 
B)  16 
C)  40 
D)  $12 \sqrt { 21 }$ 
E)  $\sqrt { 231 }$ 
A)  $\sqrt { 21 }$ 
B)  16 
C)  40 
D)  $12 \sqrt { 21 }$ 
E)  $\sqrt { 231 }$

Question 8

Using the figure below, sketch a graph of the given vector. [The graphs in the answer choices are drawn to the same scale as the graph below.]

Accepted Answer

A)

B) 
  
C) 
  
D) 
  
E)  none of these 
A)

B) 
  
C) 
  
D) 
  
E)  none of these

Question 9

Find all solutions to the following equation. $x ^ { 3 } - 216 = 0$

Accepted Answer

A)  $x = 6 , \quad 3 + 3 \sqrt { 3 } i , \quad 3 - 3 \sqrt { 3 } i$ 
B)  $x = 6 , - 6$ 
C)  $x = - 6 , \quad 3 - 3 \sqrt { 3 } i , \quad 3 + 3 \sqrt { 3 } i$ 
D)  $x = 6$ 
E)  $x = 6 , - 3 + 3 \sqrt { 3 } i , \quad - 3 - 3 \sqrt { 3 } i$ 
A)  $x = 6 , \quad 3 + 3 \sqrt { 3 } i , \quad 3 - 3 \sqrt { 3 } i$ 
B)  $x = 6 , - 6$ 
C)  $x = - 6 , \quad 3 - 3 \sqrt { 3 } i , \quad 3 + 3 \sqrt { 3 } i$ 
D)  $x = 6$ 
E)  $x = 6 , - 3 + 3 \sqrt { 3 } i , \quad - 3 - 3 \sqrt { 3 } i$

Question 10

Find the absolute value of the complex number $- 2 - 6 i$.

Accepted Answer

A)  $2 \sqrt { 2 }$ 
B)  $4 \sqrt { 2 }$ 
C)  40 
D)  $5 \sqrt { 10 }$ 
E)  $2 \sqrt { 10 }$ 
A)  $2 \sqrt { 2 }$ 
B)  $4 \sqrt { 2 }$ 
C)  40 
D)  $5 \sqrt { 10 }$ 
E)  $2 \sqrt { 10 }$

Question 11

A boat race runs along a triangular course marked by buoys $A , B$, and $C$. The race begins with the boats headed west for 3900 meters. The other two sides of the course lie to the north of the first side, and their lengths are 1900 meters and 2500 meters, respectively. Find the bearing for the last leg of the race. Round your answer to two decimals.

Accepted Answer

A)  $- \inf ^ { \circ }$ 
B)  $113.74 ^ { \circ }$ 
C)  $180.01 ^ { \circ }$ 
D)  $156.26 ^ { \circ }$ 
E)  $90.01 ^ { \circ }$ 
A)  $- \inf ^ { \circ }$ 
B)  $113.74 ^ { \circ }$ 
C)  $180.01 ^ { \circ }$ 
D)  $156.26 ^ { \circ }$ 
E)  $90.01 ^ { \circ }$

Question 12

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither.
$$\mathbf { u } = \langle 1 , - 5 \rangle , \mathbf { v } = \langle 20,3 \rangle$$

Accepted Answer

A)  orthogonal 
B)  parallel 
C)  neither 
A)  orthogonal 
B)  parallel 
C)  neither

Question 13

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

Accepted Answer

A)  $- 216$ 
B)  $- 240$ 
C)  68 
D)  $- 48$ 
E)  4 
A)  $- 216$ 
B)  $- 240$ 
C)  68 
D)  $- 48$ 
E)  4

Question 14

In the following triangle, $A = 58 ^ { \circ } , b = 105$, and $a = 126$. Use the Law of Sines to find the measure of angle $C$, in degrees. Round your answer to two decimals.

Accepted Answer

A)  $121.03 ^ { \circ }$ 
B)  $135.03 ^ { \circ }$ 
C)  $45.03 ^ { \circ }$ 
D)  $77.03 ^ { \circ }$ 
E)  $44.97 ^ { \circ }$ 
A)  $121.03 ^ { \circ }$ 
B)  $135.03 ^ { \circ }$ 
C)  $45.03 ^ { \circ }$ 
D)  $77.03 ^ { \circ }$ 
E)  $44.97 ^ { \circ }$

Question 15

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 $5 ^ { \circ }$ from vertical toward the house. From the porch, which is 100 feet away from the base of the tree, they noticed that the angle of elevation to the top of the tree was $29 ^ { \circ }$. Approximate the height of the tree. Round answer to two decimal places.

Accepted Answer

A)  $57.19$ feet 
B)  $58.48$ feet 
C)  $68.48$ feet 
D)  $53.07$ feet 
E)  $60.01$ feet 
A)  $57.19$ feet 
B)  $58.48$ feet 
C)  $68.48$ feet 
D)  $53.07$ feet 
E)  $60.01$ feet

Question 16

A plane flies 740 miles from City A to City B with a bearing of $25 ^ { \circ }$ (clockwise from north). Then it flies 614 miles from City B to City $\mathrm { C }$ with a bearing of $49 ^ { \circ }$. Find the straight-line distance from City $\mathrm { C }$ to City A. Round your answer to two decimals.

Accepted Answer

A)  $307.31 \mathrm { miles }$ 
B)  $1347.06$ miles 
C)  $1137.63$ miles 
D)  $911.87$ miles 
E)  $1324.67$ miles 
A)  $307.31 \mathrm { miles }$ 
B)  $1347.06$ miles 
C)  $1137.63$ miles 
D)  $911.87$ miles 
E)  $1324.67$ miles

Question 17

Given $C = 130 ^ { \circ } , a = 16.9$, and $c = 12.3$, use the Law of Sines to solve the triangle (if possible) for the value of $b$. If two solutions exist, find both. Round answer to two decimal places.

Accepted Answer

A)  $b = 3.73$ 
B)  $b = 1.29$ and $5.61$ 
C)  $b = 6.98$ 
D)  $b = 0.36$ and $5.74$ 
E)  not possible 
A)  $b = 3.73$ 
B)  $b = 1.29$ and $5.61$ 
C)  $b = 6.98$ 
D)  $b = 0.36$ and $5.74$ 
E)  not possible

Question 18

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form.
$$( 8 + 8 i ) ( 2 + 2 i )$$

Accepted Answer

A)  $- 16 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$ 
B)  $- 32 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
C)  $16 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
D)  $32 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
E)  $16 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$ 
A)  $- 16 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$ 
B)  $- 32 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
C)  $16 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
D)  $32 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
E)  $16 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$

Question 19

The vector $\mathbf { u } = \langle 3700,4600 \rangle$ gives the number of units of two models of laptops produced by a company. The vector $\mathbf { v } = \langle 1550,1150 \rangle$ gives the prices (in dollars) of the two models of laptops, respectively. Identify the vector operation used to increase revenue by $3 \%$.

Accepted Answer

A)  $\mathbf { u } \cdot ( 1.03 ) \| \mathbf { v } \|$ 
B)  $1.03 ( \mathbf { u } \cdot \mathbf { v } )$ 
C)  $1.03 \| \mathbf { u } \cdot \mathbf { v } \|$ 
D)  $1.03 \| \mathbf { u } \| \cdot \mathbf { v }$ 
E)  $1.03 ( \mathbf { u } + \mathbf { v } )$ 
A)  $\mathbf { u } \cdot ( 1.03 ) \| \mathbf { v } \|$ 
B)  $1.03 ( \mathbf { u } \cdot \mathbf { v } )$ 
C)  $1.03 \| \mathbf { u } \cdot \mathbf { v } \|$ 
D)  $1.03 \| \mathbf { u } \| \cdot \mathbf { v }$ 
E)  $1.03 ( \mathbf { u } + \mathbf { v } )$

Question 20

Perform the operation shown below and leave the result in trigonometric form.
$$\left[ 6 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) \right] \left[ 2 \left( \cos 280 ^ { \circ } + i \sin 280 ^ { \circ } \right) \right]$$

Accepted Answer

A)  $8 \left( \cos 370 ^ { \circ } + i \sin 370 ^ { \circ } \right)$ 
B)  $12 \left( \cos 252 ^ { \circ } + i \sin 252 ^ { \circ } \right)$ 
C)  $12 \left( \cos 190 ^ { \circ } + i \sin 190 ^ { \circ } \right)$ 
D)  $8 \left( \cos 190 ^ { \circ } + i \sin 190 ^ { \circ } \right)$ 
E)  $12 \left( \cos 370 ^ { \circ } + i \sin 370 ^ { \circ } \right)$ 
A)  $8 \left( \cos 370 ^ { \circ } + i \sin 370 ^ { \circ } \right)$ 
B)  $12 \left( \cos 252 ^ { \circ } + i \sin 252 ^ { \circ } \right)$ 
C)  $12 \left( \cos 190 ^ { \circ } + i \sin 190 ^ { \circ } \right)$ 
D)  $8 \left( \cos 190 ^ { \circ } + i \sin 190 ^ { \circ } \right)$ 
E)  $12 \left( \cos 370 ^ { \circ } + i \sin 370 ^ { \circ } \right)$

Find the fourth roots of $- \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } i$ . Write the roots in trigonometric form.

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

Use the figure to determine the tension in the cable CA supporting the load. Round your answer to two decimals.

Use DeMoivre's Theorem to find the indicated power of the following complex number. $( 5 + 5 \sqrt { 3 } i ) ^ { 6 }$

Use DeMoivre's Theorem to find $( 8 + 8 i ) ^ { 3 }$ . Write the result in standard form.

Using the figure below, sketch a graph of the given vector. [The graphs in the answer choices are drawn to the same scale as the graph below.]

Find all solutions to the following equation. $x ^ { 3 } - 216 = 0$

Find the absolute value of the complex number $- 2 - 6 i$ .

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither. $\mathbf { u } = \langle 1 , - 5 \rangle , \mathbf { v } = \langle 20,3 \rangle$

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

Given $C = 130 ^ { \circ } , a = 16.9$ , and $c = 12.3$ , use the Law of Sines to solve the triangle (if possible) for the value of $b$ . If two solutions exist, find both. Round answer to two decimal places.

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. $( 8 + 8 i ) ( 2 + 2 i )$

Perform the operation shown below and leave the result in trigonometric form. $\left[ 6 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) \right] \left[ 2 \left( \cos 280 ^ { \circ } + i \sin 280 ^ { \circ } \right) \right]$

Functions and Their Graphs

Polynomial and Rational Functions

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Exam 6: Additional Topics in Trigonometry

Find the fourth roots of −12−32i- \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } i−21​−23​​i . Write the roots in trigonometric form.

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

Use the figure to determine the tension in the cable CA supporting the load. Round your answer to two decimals.

Use DeMoivre's Theorem to find the indicated power of the following complex number. (5+53i)6( 5 + 5 \sqrt { 3 } i ) ^ { 6 }(5+53​i)6

Use DeMoivre's Theorem to find (8+8i)3( 8 + 8 i ) ^ { 3 }(8+8i)3 . Write the result in standard form.

Use Heron's area formula to find the area of the triangle pictured below, if a=8a = 8a=8 inches, b=10\mathrm { b } = 10b=10 inches, and c=4\mathrm { c } = 4c=4 inches.

Using the figure below, sketch a graph of the given vector. [The graphs in the answer choices are drawn to the same scale as the graph below.]

Find all solutions to the following equation. x3−216=0x ^ { 3 } - 216 = 0x3−216=0

Find the absolute value of the complex number −2−6i- 2 - 6 i−2−6i .

Determine whether u\mathbf { u }u and v\mathbf { v }v are orthogonal, parallel, or neither. u=⟨1,−5⟩,v=⟨20,3⟩\mathbf { u } = \langle 1 , - 5 \rangle , \mathbf { v } = \langle 20,3 \rangleu=⟨1,−5⟩,v=⟨20,3⟩

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

In the following triangle, A=58∘,b=105A = 58 ^ { \circ } , b = 105A=58∘,b=105 , and a=126a = 126a=126 . Use the Law of Sines to find the measure of angle CCC , in degrees. Round your answer to two decimals.

Given C=130∘,a=16.9C = 130 ^ { \circ } , a = 16.9C=130∘,a=16.9 , and c=12.3c = 12.3c=12.3 , use the Law of Sines to solve the triangle (if possible) for the value of bbb . If two solutions exist, find both. Round answer to two decimal places.

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. (8+8i)(2+2i)( 8 + 8 i ) ( 2 + 2 i )(8+8i)(2+2i)

Functions and Their Graphs

Polynomial and Rational Functions

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Calculus Practice Problems

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Find the fourth roots of $- \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } i$ . Write the roots in trigonometric form.

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

Use DeMoivre's Theorem to find the indicated power of the following complex number. $( 5 + 5 \sqrt { 3 } i ) ^ { 6 }$

Use DeMoivre's Theorem to find $( 8 + 8 i ) ^ { 3 }$ . Write the result in standard form.

Find all solutions to the following equation. $x ^ { 3 } - 216 = 0$

Find the absolute value of the complex number $- 2 - 6 i$ .

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither. $\mathbf { u } = \langle 1 , - 5 \rangle , \mathbf { v } = \langle 20,3 \rangle$

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

Given $C = 130 ^ { \circ } , a = 16.9$ , and $c = 12.3$ , use the Law of Sines to solve the triangle (if possible) for the value of $b$ . If two solutions exist, find both. Round answer to two decimal places.

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. $( 8 + 8 i ) ( 2 + 2 i )$