Question 1

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

Accepted Answer

A) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 20 ^ { \circ } \right) + i \sin \left( 20 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 110 ^ { \circ } \right) + i \sin \left( 110 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 200 ^ { \circ } \right) + i \sin \left( 200 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 290 ^ { \circ } \right) + i \sin \left( 290 ^ { \circ } \right)
\end{array}$ 
B) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 30 ^ { \circ } \right) + i \sin \left( 30 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 120 ^ { \circ } \right) + i \sin \left( 120 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 210 ^ { \circ } \right) + i \sin \left( 210 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 300 ^ { \circ } \right) + i \sin \left( 300 ^ { \circ } \right)
\end{array}$ 
C) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 35 ^ { \circ } \right) + i \sin \left( 35 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 125 ^ { \circ } \right) + i \sin \left( 125 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 215 ^ { \circ } \right) + i \sin \left( 215 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 305 ^ { \circ } \right) + i \sin \left( 305 ^ { \circ } \right)
\end{array}$ 
D) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 25 ^ { \circ } \right) + i \sin \left( 25 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 115 ^ { \circ } \right) + i \sin \left( 115 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 205 ^ { \circ } \right) + i \sin \left( 205 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 295 ^ { \circ } \right) + i \sin \left( 295 ^ { \circ } \right)
\end{array}$ 
E) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 40 ^ { \circ } \right) + i \sin \left( 40 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 130 ^ { \circ } \right) + i \sin \left( 130 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 220 ^ { \circ } \right) + i \sin \left( 220 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 310 ^ { \circ } \right) + i \sin \left( 310 ^ { \circ } \right)
\end{array}$ 
A) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 20 ^ { \circ } \right) + i \sin \left( 20 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 110 ^ { \circ } \right) + i \sin \left( 110 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 200 ^ { \circ } \right) + i \sin \left( 200 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 290 ^ { \circ } \right) + i \sin \left( 290 ^ { \circ } \right)
\end{array}$ 
B) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 30 ^ { \circ } \right) + i \sin \left( 30 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 120 ^ { \circ } \right) + i \sin \left( 120 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 210 ^ { \circ } \right) + i \sin \left( 210 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 300 ^ { \circ } \right) + i \sin \left( 300 ^ { \circ } \right)
\end{array}$ 
C) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 35 ^ { \circ } \right) + i \sin \left( 35 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 125 ^ { \circ } \right) + i \sin \left( 125 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 215 ^ { \circ } \right) + i \sin \left( 215 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 305 ^ { \circ } \right) + i \sin \left( 305 ^ { \circ } \right)
\end{array}$ 
D) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 25 ^ { \circ } \right) + i \sin \left( 25 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 115 ^ { \circ } \right) + i \sin \left( 115 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 205 ^ { \circ } \right) + i \sin \left( 205 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 295 ^ { \circ } \right) + i \sin \left( 295 ^ { \circ } \right)
\end{array}$ 
E) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 40 ^ { \circ } \right) + i \sin \left( 40 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 130 ^ { \circ } \right) + i \sin \left( 130 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 220 ^ { \circ } \right) + i \sin \left( 220 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 310 ^ { \circ } \right) + i \sin \left( 310 ^ { \circ } \right)
\end{array}$

Question 2

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

Question 3

If $\| \mathbf { u } \| = 5$ and $\| \mathbf { v } \| = 4$, and the vectors make angles of $60 ^ { \circ }$ and $80 ^ { \circ }$ with the $x$-axis respectively, find the component form of the sum of $\mathbf { u }$ and $\mathbf { v }$. Round answers to two decimal places.

Accepted Answer

A)  $\langle 3.19,8.27 \rangle$ 
B)  $\langle 1.81,0.39 \rangle$ 
C)  $\langle 5.02,6.44 \rangle$ 
D)  $\langle 3.64 , - 1.44 \rangle$ 
E)  $\langle 2.87,8.39 \rangle$ 
A)  $\langle 3.19,8.27 \rangle$ 
B)  $\langle 1.81,0.39 \rangle$ 
C)  $\langle 5.02,6.44 \rangle$ 
D)  $\langle 3.64 , - 1.44 \rangle$ 
E)  $\langle 2.87,8.39 \rangle$

Question 4

Use vectors to find the measure of the angle at vertex $B$ of triangle $A B C$, when $A = ( 6,1 ) , B = ( - 1,3 )$, and $C = ( - 4 , - 5 )$. Round answer to two decimal places.

Accepted Answer

A)  $92.52$ 
B)  $94.61 ^ { \circ }$ 
C)  $95.95$ 
D)  $93.42$ 
E)  $97.13 ^ { \circ }$ 
A)  $92.52$ 
B)  $94.61 ^ { \circ }$ 
C)  $95.95$ 
D)  $93.42$ 
E)  $97.13 ^ { \circ }$

Question 5

A plane flies 710 miles from City A to City B with a bearing of $28 ^ { \circ }$ (clockwise from north). Then it flies 672 miles from City B to City C with a bearing of $45 ^ { \circ }$. Find the straight-line distance from City C to City A. Round your answer to two decimals.

Accepted Answer

A)  $207.70$ miles 
B)  $1238.36$ miles 
C)  $1111.16$ miles 
D)  $955.99$ miles 
E)  $1366.83$ miles 
A)  $207.70$ miles 
B)  $1238.36$ miles 
C)  $1111.16$ miles 
D)  $955.99$ miles 
E)  $1366.83$ miles

Question 6

A triangular parcel of land has sides of lengths 900,580 , and 550 feet. Approximate the area of the land. Round answer to nearest foot.

Accepted Answer

A)  $154,345 \mathrm { ft } ^ { 2 }$ 
B)  $153,657 \mathrm { ft } ^ { 2 }$ 
C)  $152,972 \mathrm { ft } ^ { 2 }$ 
D)  $152,971 \mathrm { ft } ^ { 2 }$ 
E)  $153,656 \mathrm { ft } ^ { 2 }$ 
A)  $154,345 \mathrm { ft } ^ { 2 }$ 
B)  $153,657 \mathrm { ft } ^ { 2 }$ 
C)  $152,972 \mathrm { ft } ^ { 2 }$ 
D)  $152,971 \mathrm { ft } ^ { 2 }$ 
E)  $153,656 \mathrm { ft } ^ { 2 }$

Question 7

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 8

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 9

Find the vector $\mathbf { v }$ that has a magnitude of 6 and is in the same direction as $\mathbf { u }$, where $\mathbf { u } = \langle - 6 , - 5 \rangle$.

Accepted Answer

A)  $\mathbf { v } = \left\langle \frac { - 36 } { \sqrt { 61 } } , \frac { - 30 } { \sqrt { 61 } } \right\rangle$ 
B)  $\mathbf { v } = \left\langle \frac { - 6 } { \sqrt { 61 } } , \frac { - 5 } { \sqrt { 61 } } \right\rangle$ 
C)  $\mathbf { v } = \left\langle \frac { - 1 } { \sqrt { 61 } } , \frac { - 6 } { \sqrt { 61 } } \right\rangle$ 
D)  $\mathbf { v } = \left\langle \frac { - 1 } { \sqrt { 61 } } , \frac { 1 } { \sqrt { 61 } } \right\rangle$ 
E)  $\quad \mathbf { v } = \left\langle \frac { 5 } { \sqrt { 61 } } , \frac { 6 } { \sqrt { 61 } } \right\rangle$ 
A)  $\mathbf { v } = \left\langle \frac { - 36 } { \sqrt { 61 } } , \frac { - 30 } { \sqrt { 61 } } \right\rangle$ 
B)  $\mathbf { v } = \left\langle \frac { - 6 } { \sqrt { 61 } } , \frac { - 5 } { \sqrt { 61 } } \right\rangle$ 
C)  $\mathbf { v } = \left\langle \frac { - 1 } { \sqrt { 61 } } , \frac { - 6 } { \sqrt { 61 } } \right\rangle$ 
D)  $\mathbf { v } = \left\langle \frac { - 1 } { \sqrt { 61 } } , \frac { 1 } { \sqrt { 61 } } \right\rangle$ 
E)  $\quad \mathbf { v } = \left\langle \frac { 5 } { \sqrt { 61 } } , \frac { 6 } { \sqrt { 61 } } \right\rangle$

Question 10

In the figure below, $a = 8 , b = 12$, and $c = 13$. Use this information to solve the parallelogram for $\theta$. The diagonals of the parallelogram are represented by $c$ and $d$. Round answer to two decimal places.

Accepted Answer

A)  $101.72 ^ { \circ }$ 
B)  $95.86$ 
C)  $90.00 ^ { \circ }$ 
D)  $91.46$ 
E)  $78.28 ^ { \circ }$ 
A)  $101.72 ^ { \circ }$ 
B)  $95.86$ 
C)  $90.00 ^ { \circ }$ 
D)  $91.46$ 
E)  $78.28 ^ { \circ }$

Question 11

Perform the operation below and leave the result in trigonometric form.
$$\frac { 4 \left( \cos 58 ^ { \circ } + i \sin 58 ^ { \circ } \right) } { 12 \left( \cos 13 ^ { \circ } + i \sin 13 ^ { \circ } \right) }$$

Accepted Answer

A)  $\frac { 1 } { 3 } \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right)$ 
B)  $- 8 \left( \cos \frac { 58 ^ { \circ } } { 13 } + i \sin \frac { 58 ^ { \circ } } { 13 } \right)$ 
C)  $16 \left( \cos 71 ^ { \circ } + i \sin 71 ^ { \circ } \right)$ 
D)  $48 \left( \cos \frac { 58 ^ { \circ } } { 13 } + i \sin \frac { 58 ^ { \circ } } { 13 } \right)$ 
E)  $\frac { 1 } { 3 } \left( \cos \frac { 58 ^ { \circ } } { 13 } + i \sin \frac { 58 ^ { \circ } } { 13 } \right)$ 
A)  $\frac { 1 } { 3 } \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right)$ 
B)  $- 8 \left( \cos \frac { 58 ^ { \circ } } { 13 } + i \sin \frac { 58 ^ { \circ } } { 13 } \right)$ 
C)  $16 \left( \cos 71 ^ { \circ } + i \sin 71 ^ { \circ } \right)$ 
D)  $48 \left( \cos \frac { 58 ^ { \circ } } { 13 } + i \sin \frac { 58 ^ { \circ } } { 13 } \right)$ 
E)  $\frac { 1 } { 3 } \left( \cos \frac { 58 ^ { \circ } } { 13 } + i \sin \frac { 58 ^ { \circ } } { 13 } \right)$

Question 12

Find the vector $\mathbf { v }$ that has a magnitude of 8 and is in the same direction as $\mathbf { u }$, where $\mathbf { u } = \langle - 6 , - 4 \rangle$.

Accepted Answer

A)  $\mathbf { v } = \left\langle \frac { - 24 } { \sqrt { 13 } } , \frac { - 16 } { \sqrt { 13 } } \right\rangle$ 
B)  $\mathbf { v } = \left\langle \frac { - 3 } { \sqrt { 13 } } , \frac { - 2 } { \sqrt { 13 } } \right\rangle$ 
C)  $\mathbf { v } = \left\langle \frac { - 8 } { \sqrt { 13 } } , \frac { - 4 } { \sqrt { 13 } } \right\rangle$ 
D)  $\mathbf { v } = \left\langle \frac { - 2 } { \sqrt { 13 } } , \frac { 2 } { \sqrt { 13 } } \right\rangle$ 
E)  $\mathbf { v } = \left\langle \frac { 4 } { \sqrt { 13 } } , \frac { 6 } { \sqrt { 13 } } \right\rangle$ 
A)  $\mathbf { v } = \left\langle \frac { - 24 } { \sqrt { 13 } } , \frac { - 16 } { \sqrt { 13 } } \right\rangle$ 
B)  $\mathbf { v } = \left\langle \frac { - 3 } { \sqrt { 13 } } , \frac { - 2 } { \sqrt { 13 } } \right\rangle$ 
C)  $\mathbf { v } = \left\langle \frac { - 8 } { \sqrt { 13 } } , \frac { - 4 } { \sqrt { 13 } } \right\rangle$ 
D)  $\mathbf { v } = \left\langle \frac { - 2 } { \sqrt { 13 } } , \frac { 2 } { \sqrt { 13 } } \right\rangle$ 
E)  $\mathbf { v } = \left\langle \frac { 4 } { \sqrt { 13 } } , \frac { 6 } { \sqrt { 13 } } \right\rangle$

Question 13

Given $a = 3 , b = 8$, and $c = 6$, use the Law of Cosines to solve the triangle for the value of $C$. Round answer to two decimal places.

Accepted Answer

A)  $60.33 ^ { \circ }$ 
B)  $39.57 ^ { \circ }$ 
C)  $80.44 ^ { \circ }$ 
D)  $121.86 ^ { \circ }$ 
E)  $18.57 ^ { \circ }$ 
A)  $60.33 ^ { \circ }$ 
B)  $39.57 ^ { \circ }$ 
C)  $80.44 ^ { \circ }$ 
D)  $121.86 ^ { \circ }$ 
E)  $18.57 ^ { \circ }$

Question 14

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 15

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 16

Given $A = 52 ^ { \circ } , B = 65 ^ { \circ }$, and $a = 7.1$, use the Law of Sines to solve the triangle for the value of $b$. Round answer to two decimal places.

Accepted Answer

A)  $b = 6.98$ 
B)  $b = 8.03$ 
C)  $b = 6.28$ 
D)  $b = 8.17$ 
E)  $b = 6.17$ 
A)  $b = 6.98$ 
B)  $b = 8.03$ 
C)  $b = 6.28$ 
D)  $b = 8.17$ 
E)  $b = 6.17$

Question 17

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

Accepted Answer

A) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 65 ^ { \circ } \right) + i \sin \left( 65 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 155 ^ { \circ } \right) + i \sin \left( 155 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 245 ^ { \circ } \right) + i \sin \left( 245 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 335 ^ { \circ } \right) + i \sin \left( 335 ^ { \circ } \right)
\end{array}$ 
B) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 75 ^ { \circ } \right) + i \sin \left( 75 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 165 ^ { \circ } \right) + i \sin \left( 165 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 255 ^ { \circ } \right) + i \sin \left( 255 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 345 ^ { \circ } \right) + i \sin \left( 345 ^ { \circ } \right)
\end{array}$ 
C) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 80 ^ { \circ } \right) + i \sin \left( 80 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 170 ^ { \circ } \right) + i \sin \left( 170 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 260 ^ { \circ } \right) + i \sin \left( 260 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 350 ^ { \circ } \right) + i \sin \left( 350 ^ { \circ } \right)
\end{array}$ 
D) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 70 ^ { \circ } \right) + i \sin \left( 70 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 160 ^ { \circ } \right) + i \sin \left( 160 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 250 ^ { \circ } \right) + i \sin \left( 250 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 340 ^ { \circ } \right) + i \sin \left( 340 ^ { \circ } \right)
\end{array}$ 
E) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 85 ^ { \circ } \right) + i \sin \left( 85 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 175 ^ { \circ } \right) + i \sin \left( 175 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 265 ^ { \circ } \right) + i \sin \left( 265 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 355 ^ { \circ } \right) + i \sin \left( 355 ^ { \circ } \right)
\end{array}$ 
A) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 65 ^ { \circ } \right) + i \sin \left( 65 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 155 ^ { \circ } \right) + i \sin \left( 155 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 245 ^ { \circ } \right) + i \sin \left( 245 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 335 ^ { \circ } \right) + i \sin \left( 335 ^ { \circ } \right)
\end{array}$ 
B) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 75 ^ { \circ } \right) + i \sin \left( 75 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 165 ^ { \circ } \right) + i \sin \left( 165 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 255 ^ { \circ } \right) + i \sin \left( 255 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 345 ^ { \circ } \right) + i \sin \left( 345 ^ { \circ } \right)
\end{array}$ 
C) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 80 ^ { \circ } \right) + i \sin \left( 80 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 170 ^ { \circ } \right) + i \sin \left( 170 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 260 ^ { \circ } \right) + i \sin \left( 260 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 350 ^ { \circ } \right) + i \sin \left( 350 ^ { \circ } \right)
\end{array}$ 
D) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 70 ^ { \circ } \right) + i \sin \left( 70 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 160 ^ { \circ } \right) + i \sin \left( 160 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 250 ^ { \circ } \right) + i \sin \left( 250 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 340 ^ { \circ } \right) + i \sin \left( 340 ^ { \circ } \right)
\end{array}$ 
E) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 85 ^ { \circ } \right) + i \sin \left( 85 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 175 ^ { \circ } \right) + i \sin \left( 175 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 265 ^ { \circ } \right) + i \sin \left( 265 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 355 ^ { \circ } \right) + i \sin \left( 355 ^ { \circ } \right)
\end{array}$

Question 18

Perform the operation shown below and leave the result in trigonometric form.
$$\frac { 8 ( \cos 2.3 + i \sin 2.3 ) } { 4 ( \cos 1.7 + i \sin 1.7 ) }$$

Accepted Answer

A)  $2 ( \cos 0.6 + i \sin 0.6 )$ 
B)  $4 ( \cos 0.6 + i \sin 0.6 )$ 
C)  $4 ( \cos 1.4 + i \sin 1.4 )$ 
D)  $2 ( \cos 4 + i \sin 4 )$ 
E)  $2 ( \cos 3.91 + i \sin 3.91 )$ 
A)  $2 ( \cos 0.6 + i \sin 0.6 )$ 
B)  $4 ( \cos 0.6 + i \sin 0.6 )$ 
C)  $4 ( \cos 1.4 + i \sin 1.4 )$ 
D)  $2 ( \cos 4 + i \sin 4 )$ 
E)  $2 ( \cos 3.91 + i \sin 3.91 )$

Question 19

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

Accepted Answer

A)  20 
B)  2 
C)  16 
D)  24 
E)  18 
A)  20 
B)  2 
C)  16 
D)  24 
E)  18

Question 20

Use DeMoivre's Theorem to find the indicated power of the folllowing complex number.
$( - 6 + 6 i ) ^ { 4 }$

Accepted Answer

A)  $- 5,184$ 
B)  46,656 
C)  5,184 
D)  186,624 
E)  7,776 
A)  $- 5,184$ 
B)  46,656 
C)  5,184 
D)  186,624 
E)  7,776

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

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

If $\| \mathbf { u } \| = 5$ and $\| \mathbf { v } \| = 4$ , and the vectors make angles of $60 ^ { \circ }$ and $80 ^ { \circ }$ with the $x$ -axis respectively, find the component form of the sum of $\mathbf { u }$ and $\mathbf { v }$ . Round answers to two decimal places.

Use vectors to find the measure of the angle at vertex $B$ of triangle $A B C$ , when $A = ( 6,1 ) , B = ( - 1,3 )$ , and $C = ( - 4 , - 5 )$ . Round answer to two decimal places.

A plane flies 710 miles from City A to City B with a bearing of $28 ^ { \circ }$ (clockwise from north). Then it flies 672 miles from City B to City C with a bearing of $45 ^ { \circ }$ . Find the straight-line distance from City C to City A. Round your answer to two decimals.

A triangular parcel of land has sides of lengths 900,580 , and 550 feet. Approximate the area of the land. Round answer to nearest foot.

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

Find the vector $\mathbf { v }$ that has a magnitude of 6 and is in the same direction as $\mathbf { u }$ , where $\mathbf { u } = \langle - 6 , - 5 \rangle$ .

Perform the operation below and leave the result in trigonometric form. $\frac { 4 \left( \cos 58 ^ { \circ } + i \sin 58 ^ { \circ } \right) } { 12 \left( \cos 13 ^ { \circ } + i \sin 13 ^ { \circ } \right) }$

Find the vector $\mathbf { v }$ that has a magnitude of 8 and is in the same direction as $\mathbf { u }$ , where $\mathbf { u } = \langle - 6 , - 4 \rangle$ .

Given $a = 3 , b = 8$ , and $c = 6$ , use the Law of Cosines to solve the triangle for the value of $C$ . Round answer to two decimal places.

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

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

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

Perform the operation shown below and leave the result in trigonometric form. $\frac { 8 ( \cos 2.3 + i \sin 2.3 ) } { 4 ( \cos 1.7 + i \sin 1.7 ) }$

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

Use DeMoivre's Theorem to find the indicated power of the folllowing complex number. $( - 6 + 6 i ) ^ { 4 }$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Filters

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.

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

Use vectors to find the measure of the angle at vertex BBB of triangle ABCA B CABC , when A=(6,1),B=(−1,3)A = ( 6,1 ) , B = ( - 1,3 )A=(6,1),B=(−1,3) , and C=(−4,−5)C = ( - 4 , - 5 )C=(−4,−5) . Round answer to two decimal places.

A plane flies 710 miles from City A to City B with a bearing of 28∘28 ^ { \circ }28∘ (clockwise from north). Then it flies 672 miles from City B to City C with a bearing of 45∘45 ^ { \circ }45∘ . Find the straight-line distance from City C to City A. Round your answer to two decimals.

A triangular parcel of land has sides of lengths 900,580 , and 550 feet. Approximate the area of the land. Round answer to nearest foot.

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

Find the vector v\mathbf { v }v that has a magnitude of 6 and is in the same direction as u\mathbf { u }u , where u=⟨−6,−5⟩\mathbf { u } = \langle - 6 , - 5 \rangleu=⟨−6,−5⟩ .

In the figure below, a=8,b=12a = 8 , b = 12a=8,b=12 , and c=13c = 13c=13 . Use this information to solve the parallelogram for θ\thetaθ . The diagonals of the parallelogram are represented by ccc and ddd . Round answer to two decimal places.

Find the vector v\mathbf { v }v that has a magnitude of 8 and is in the same direction as u\mathbf { u }u , where u=⟨−6,−4⟩\mathbf { u } = \langle - 6 , - 4 \rangleu=⟨−6,−4⟩ .

Given a=3,b=8a = 3 , b = 8a=3,b=8 , and c=6c = 6c=6 , use the Law of Cosines to solve the triangle for the value of CCC . Round answer to two decimal places.

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

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 A=52∘,B=65∘A = 52 ^ { \circ } , B = 65 ^ { \circ }A=52∘,B=65∘ , and a=7.1a = 7.1a=7.1 , use the Law of Sines to solve the triangle for the value of bbb . Round answer to two decimal places.

Find the fourth roots of 12−32i\frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } i21​−23​​i . Write the roots in trigonometric form.

Perform the operation shown below and leave the result in trigonometric form. 8(cos⁡2.3+isin⁡2.3)4(cos⁡1.7+isin⁡1.7)\frac { 8 ( \cos 2.3 + i \sin 2.3 ) } { 4 ( \cos 1.7 + i \sin 1.7 ) }4(cos1.7+isin1.7)8(cos2.3+isin2.3)​

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

Use DeMoivre's Theorem to find the indicated power of the folllowing complex number. (−6+6i)4( - 6 + 6 i ) ^ { 4 }(−6+6i)4

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Filters

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

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

Use vectors to find the measure of the angle at vertex $B$ of triangle $A B C$ , when $A = ( 6,1 ) , B = ( - 1,3 )$ , and $C = ( - 4 , - 5 )$ . Round answer to two decimal places.

A plane flies 710 miles from City A to City B with a bearing of $28 ^ { \circ }$ (clockwise from north). Then it flies 672 miles from City B to City C with a bearing of $45 ^ { \circ }$ . Find the straight-line distance from City C to City A. Round your answer to two decimals.

Find the vector $\mathbf { v }$ that has a magnitude of 6 and is in the same direction as $\mathbf { u }$ , where $\mathbf { u } = \langle - 6 , - 5 \rangle$ .

Find the vector $\mathbf { v }$ that has a magnitude of 8 and is in the same direction as $\mathbf { u }$ , where $\mathbf { u } = \langle - 6 , - 4 \rangle$ .

Given $a = 3 , b = 8$ , and $c = 6$ , use the Law of Cosines to solve the triangle for the value of $C$ . Round answer to two decimal places.

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

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

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

Perform the operation shown below and leave the result in trigonometric form. $\frac { 8 ( \cos 2.3 + i \sin 2.3 ) } { 4 ( \cos 1.7 + i \sin 1.7 ) }$

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

Use DeMoivre's Theorem to find the indicated power of the folllowing complex number. $( - 6 + 6 i ) ^ { 4 }$