Question 1

Plot the following complex number.
$3 i$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
  
A) 
  
B) 
  
C) 
  
D) 
  
E) 
  D

Question 2

A triangular parcel of land has sides of lengths 100,580 , and 600 feet. Approximate the area of the land. Round answer to nearest foot.

Accepted Answer

A)  $29,432 \mathrm { ft } ^ { 2 }$ 
B)  $28,800 \mathrm { ft } ^ { 2 }$ 
C)  $28,195 \mathrm { ft } ^ { 2 }$ 
D)  $28,182 \mathrm { ft } ^ { 2 }$ 
E)  $28,787 \mathrm { ft } ^ { 2 }$ 
A)  $29,432 \mathrm { ft } ^ { 2 }$ 
B)  $28,800 \mathrm { ft } ^ { 2 }$ 
C)  $28,195 \mathrm { ft } ^ { 2 }$ 
D)  $28,182 \mathrm { ft } ^ { 2 }$ 
E)  $28,787 \mathrm { ft } ^ { 2 }$ B

Question 3

Determine the area of a triangle having the following measurements. Round your answer to two decimal places.
$A = 128 ^ { \circ } , b = 8$, and $c = 15$

Accepted Answer

A)  $56.74 \mathrm { sq }$. units 
B)  $47.28$ sq. units 
C)  $37.82$ sq. units 
D)  $42.55$ sq. units 
E)  $52.01 \mathrm { sq }$. units 
A)  $56.74 \mathrm { sq }$. units 
B)  $47.28$ sq. units 
C)  $37.82$ sq. units 
D)  $42.55$ sq. units 
E)  $52.01 \mathrm { sq }$. units B

Question 4

Determine the area of a triangle having the following measurements. Round your answer to two decimal places.
$C = 59 ^ { \circ } 24 ^ { \prime } , a = 6$, and $b = 14.5$

Accepted Answer

A)  $37.44$ sq. units 
B)  $33.70$ sq. units 
C)  $29.95$ sq. units 
D)  $44.93$ sq. units 
E)  $41.19$ sq. units 
A)  $37.44$ sq. units 
B)  $33.70$ sq. units 
C)  $29.95$ sq. units 
D)  $44.93$ sq. units 
E)  $41.19$ sq. units

Question 5

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither.
$$\mathbf { u } = \left\langle \frac { 7 } { 3 } , \frac { - 1 } { 2 } \right\rangle , \mathbf { v } = \langle 14 , - 3 \rangle$$

Accepted Answer

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

Question 6

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 $\mathrm { C }$ with a bearing of $45 ^ { \circ }$. Find the straight-line distance from City $\mathrm { C }$ to City A. Round your answer to two decimals.

Accepted Answer

A)  $207.70 \mathrm { miles }$ 
B)  $1238.36$ miles 
C)  $1111.16$ miles 
D)  $955.99$ miles 
E)  $1366.83$ miles 
A)  $207.70 \mathrm { miles }$ 
B)  $1238.36$ miles 
C)  $1111.16$ miles 
D)  $955.99$ miles 
E)  $1366.83$ miles

Question 7

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 8

Let $\mathbf { u } = \langle 10,6 \rangle$ and $\mathbf { v } = \langle 11,15 \rangle$. Find $\mathbf { u } - \mathbf { v }$.

Accepted Answer

A)  $\langle 25,17 \rangle$ 
B)  $\langle - 1 , - 9 \rangle$ 
C)  $\langle 21,21 \rangle$ 
D)  $\langle - 5 , - 5 \rangle$ 
E)  $\langle 17,25 \rangle$ 
A)  $\langle 25,17 \rangle$ 
B)  $\langle - 1 , - 9 \rangle$ 
C)  $\langle 21,21 \rangle$ 
D)  $\langle - 5 , - 5 \rangle$ 
E)  $\langle 17,25 \rangle$

Question 9

Find the standard form of the complex number shown below. Round answers to two decimal places.
$$6 \left( \cos 190 ^ { \circ } + i \sin 190 ^ { \circ } \right)$$

Accepted Answer

A)  $- 5.91 - 1.04 i$ 
B)  $- 6.14 - 1.43 i$ 
C)  $- 6.66 - 2.14 i$ 
D)  $- 7.15 - 2.39 i$ 
E)  $- 7.33 - 2.57 i$ 
A)  $- 5.91 - 1.04 i$ 
B)  $- 6.14 - 1.43 i$ 
C)  $- 6.66 - 2.14 i$ 
D)  $- 7.15 - 2.39 i$ 
E)  $- 7.33 - 2.57 i$

Question 10

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 11

Given $a = 5 , b = 9$, and $c = 12$, use Heron's Area Formula to find the area of triangle $A B C$. Round answer to two decimal places.

Accepted Answer

A)  $20.40 \mathrm { sq }$. units 
B)  $16.97$ sq. units 
C)  $19.60$ sq. units 
D)  $12.65$ sq. units 
E)  $17.40$ sq. units 
A)  $20.40 \mathrm { sq }$. units 
B)  $16.97$ sq. units 
C)  $19.60$ sq. units 
D)  $12.65$ sq. units 
E)  $17.40$ sq. units

Question 12

Find the component form of vector $\mathbf { v }$ with initial point $( - 5,1 )$ and terminal point $( - 3 , - 2 )$.

Accepted Answer

A)  $\mathbf { v } = \langle 2,3 \rangle$ 
B)  $\mathbf { v } = \langle - 3,4 \rangle$ 
C)  $\mathbf { v } = \langle 2 , - 3 \rangle$ 
D)  $\mathbf { v } = \langle 3,2 \rangle$ 
E)  $\quad \mathbf { v } = \langle - 6 , - 1 \rangle$ 
A)  $\mathbf { v } = \langle 2,3 \rangle$ 
B)  $\mathbf { v } = \langle - 3,4 \rangle$ 
C)  $\mathbf { v } = \langle 2 , - 3 \rangle$ 
D)  $\mathbf { v } = \langle 3,2 \rangle$ 
E)  $\quad \mathbf { v } = \langle - 6 , - 1 \rangle$

Question 13

Let $\mathbf { u } = \langle 6,5 \rangle$ and $\mathbf { v } = \langle 12,7 \rangle$. Find $\mathbf { u } - \mathbf { v }$.

Accepted Answer

A)  $\langle 13,17 \rangle$ 
B)  $\langle - 6 , - 2 \rangle$ 
C)  $\langle 18,12 \rangle$ 
D)  $\langle - 1 , - 7 \rangle$ 
E)  $\langle 17,13 \rangle$ 
A)  $\langle 13,17 \rangle$ 
B)  $\langle - 6 , - 2 \rangle$ 
C)  $\langle 18,12 \rangle$ 
D)  $\langle - 1 , - 7 \rangle$ 
E)  $\langle 17,13 \rangle$

Question 14

Given $a = 12 , b = 8$, and $c = 11$, use the Law of Cosines to solve the triangle for the value of $B$. Round answer to two decimal places.

Accepted Answer

A)  $60.33 ^ { \circ }$ 
B)  $76.53 ^ { \circ }$ 
C)  $40.42 ^ { \circ }$ 
D)  $80.44 ^ { \circ }$ 
E)  $63.06 ^ { \circ }$ 
A)  $60.33 ^ { \circ }$ 
B)  $76.53 ^ { \circ }$ 
C)  $40.42 ^ { \circ }$ 
D)  $80.44 ^ { \circ }$ 
E)  $63.06 ^ { \circ }$

Question 15

If $\| \mathbf { u } \| = 5$ and $\| \mathbf { v } \| = 3$, and the vectors make angles of $150 ^ { \circ }$ and $110 ^ { \circ }$ 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)  $\langle - 5.36,5.32 \rangle$ 
B)  $\langle - 3.30 , - 0.32 \rangle$ 
C)  $\langle 1.47 , - 1.51 \rangle$ 
D)  $\langle 3.53 , - 7.15 \rangle$ 
E)  $\langle - 4.31,6.20 \rangle$ 
A)  $\langle - 5.36,5.32 \rangle$ 
B)  $\langle - 3.30 , - 0.32 \rangle$ 
C)  $\langle 1.47 , - 1.51 \rangle$ 
D)  $\langle 3.53 , - 7.15 \rangle$ 
E)  $\langle - 4.31,6.20 \rangle$

Question 16

Find the component form of $\mathbf { v }$ if $\| \mathbf { v } \| = 8$ and the angle it makes with the $x$-axis is $120 ^ { \circ }$.

Accepted Answer

A)  $\langle - 4 \sqrt { 3 } , 4 \rangle$ 
B)  $\langle - 8,8 \sqrt { 3 } \rangle$ 
C)  $\langle - 8 \sqrt { 3 } , - 8 \rangle$ 
D)  $\langle - 4,4 \sqrt { 3 } \rangle$ 
E)  $\langle - 4 \sqrt { 2 } , 4 \sqrt { 2 } \rangle$ 
A)  $\langle - 4 \sqrt { 3 } , 4 \rangle$ 
B)  $\langle - 8,8 \sqrt { 3 } \rangle$ 
C)  $\langle - 8 \sqrt { 3 } , - 8 \rangle$ 
D)  $\langle - 4,4 \sqrt { 3 } \rangle$ 
E)  $\langle - 4 \sqrt { 2 } , 4 \sqrt { 2 } \rangle$

Question 17

Find the trigonometric form of the complex number shown below. $- 7 i$

Accepted Answer

A)  $7 ( \cos 0 + i \sin 0 )$ 
B)  $7 ( \cos \pi + i \sin \pi )$ 
C)  $7 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
D)  $7 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$ 
E)  $7 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
A)  $7 ( \cos 0 + i \sin 0 )$ 
B)  $7 ( \cos \pi + i \sin \pi )$ 
C)  $7 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
D)  $7 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$ 
E)  $7 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$

Question 18

Find the absolute value of the complex number $- 4 + 6 i$.

Accepted Answer

A)  $\sqrt { 10 }$ 
B)  $3 \sqrt { 10 }$ 
C)  52 
D)  $5 \sqrt { 13 }$ 
E)  $2 \sqrt { 13 }$ 
A)  $\sqrt { 10 }$ 
B)  $3 \sqrt { 10 }$ 
C)  52 
D)  $5 \sqrt { 13 }$ 
E)  $2 \sqrt { 13 }$

Question 19

Write the complex number shown below in trigonometric form.

Accepted Answer

A) 
$$6 \left( \cos \frac { 7 \pi } { 6 } + i \sin \frac { 7 \pi } { 6 } \right)$$ 
B) 
$$6 \left( \cos \frac { 4 \pi } { 3 } + i \sin \frac { 4 \pi } { 3 } \right)$$ 
C) 
$$6 \left( \cos \frac { 5 \pi } { 3 } + i \sin \frac { 5 \pi } { 3 } \right)$$ 
D) 
$$6 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$$ 
E) 
$$6 \left( \cos \frac { 11 \pi } { 6 } + i \sin \frac { 11 \pi } { 6 } \right)$$ 
A) 
$$6 \left( \cos \frac { 7 \pi } { 6 } + i \sin \frac { 7 \pi } { 6 } \right)$$ 
B) 
$$6 \left( \cos \frac { 4 \pi } { 3 } + i \sin \frac { 4 \pi } { 3 } \right)$$ 
C) 
$$6 \left( \cos \frac { 5 \pi } { 3 } + i \sin \frac { 5 \pi } { 3 } \right)$$ 
D) 
$$6 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$$ 
E) 
$$6 \left( \cos \frac { 11 \pi } { 6 } + i \sin \frac { 11 \pi } { 6 } \right)$$

Question 20

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) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 40 ^ { \circ } \right) + i \sin \left( 40 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 130 ^ { \circ } \right) + i \sin \left( 130 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 220 ^ { \circ } \right) + i \sin \left( 220 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 310 ^ { \circ } \right) + i \sin \left( 310 ^ { \circ } \right)
\end{array}$

Plot the following complex number. $3 i$

A triangular parcel of land has sides of lengths 100,580 , and 600 feet. Approximate the area of the land. Round answer to nearest foot.

Determine the area of a triangle having the following measurements. Round your answer to two decimal places. $A = 128 ^ { \circ } , b = 8$ , and $c = 15$

Determine the area of a triangle having the following measurements. Round your answer to two decimal places. $C = 59 ^ { \circ } 24 ^ { \prime } , a = 6$ , and $b = 14.5$

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither. $\mathbf { u } = \left\langle \frac { 7 } { 3 } , \frac { - 1 } { 2 } \right\rangle , \mathbf { v } = \langle 14 , - 3 \rangle$

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

Let $\mathbf { u } = \langle 10,6 \rangle$ and $\mathbf { v } = \langle 11,15 \rangle$ . Find $\mathbf { u } - \mathbf { v }$ .

Find the standard form of the complex number shown below. Round answers to two decimal places. $6 \left( \cos 190 ^ { \circ } + i \sin 190 ^ { \circ } \right)$

Given $a = 5 , b = 9$ , and $c = 12$ , use Heron's Area Formula to find the area of triangle $A B C$ . Round answer to two decimal places.

Find the component form of vector $\mathbf { v }$ with initial point $( - 5,1 )$ and terminal point $( - 3 , - 2 )$ .

Let $\mathbf { u } = \langle 6,5 \rangle$ and $\mathbf { v } = \langle 12,7 \rangle$ . Find $\mathbf { u } - \mathbf { v }$ .

Given $a = 12 , b = 8$ , and $c = 11$ , use the Law of Cosines to solve the triangle for the value of $B$ . Round answer to two decimal places.

If $\| \mathbf { u } \| = 5$ and $\| \mathbf { v } \| = 3$ , and the vectors make angles of $150 ^ { \circ }$ and $110 ^ { \circ }$ with the $x$ -axis respectively, find the component form of the sum of $\mathbf { u }$ and $\mathbf { v }$ . Round answers to two decimal places.

Find the component form of $\mathbf { v }$ if $\| \mathbf { v } \| = 8$ and the angle it makes with the $x$ -axis is $120 ^ { \circ }$ .

Find the trigonometric form of the complex number shown below. $- 7 i$

Find the absolute value of the complex number $- 4 + 6 i$ .

Write the complex number shown below in trigonometric form.

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

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Exam 6: Additional Topics in Trigonometry

Plot the following complex number. 3i3 i3i

A triangular parcel of land has sides of lengths 100,580 , and 600 feet. Approximate the area of the land. Round answer to nearest foot.

Determine the area of a triangle having the following measurements. Round your answer to two decimal places. A=128∘,b=8A = 128 ^ { \circ } , b = 8A=128∘,b=8 , and c=15c = 15c=15

Determine the area of a triangle having the following measurements. Round your answer to two decimal places. C=59∘24′,a=6C = 59 ^ { \circ } 24 ^ { \prime } , a = 6C=59∘24′,a=6 , and b=14.5b = 14.5b=14.5

Determine whether u\mathbf { u }u and v\mathbf { v }v are orthogonal, parallel, or neither. u=⟨73,−12⟩,v=⟨14,−3⟩\mathbf { u } = \left\langle \frac { 7 } { 3 } , \frac { - 1 } { 2 } \right\rangle , \mathbf { v } = \langle 14 , - 3 \rangleu=⟨37​,2−1​⟩,v=⟨14,−3⟩

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⟩ .

Let u=⟨10,6⟩\mathbf { u } = \langle 10,6 \rangleu=⟨10,6⟩ and v=⟨11,15⟩\mathbf { v } = \langle 11,15 \ranglev=⟨11,15⟩ . Find u−v\mathbf { u } - \mathbf { v }u−v .

Find the standard form of the complex number shown below. Round answers to two decimal places. 6(cos⁡190∘+isin⁡190∘)6 \left( \cos 190 ^ { \circ } + i \sin 190 ^ { \circ } \right)6(cos190∘+isin190∘)

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.

Given a=5,b=9a = 5 , b = 9a=5,b=9 , and c=12c = 12c=12 , use Heron's Area Formula to find the area of triangle ABCA B CABC . Round answer to two decimal places.

Find the component form of vector v\mathbf { v }v with initial point (−5,1)( - 5,1 )(−5,1) and terminal point (−3,−2)( - 3 , - 2 )(−3,−2) .

Let u=⟨6,5⟩\mathbf { u } = \langle 6,5 \rangleu=⟨6,5⟩ and v=⟨12,7⟩\mathbf { v } = \langle 12,7 \ranglev=⟨12,7⟩ . Find u−v\mathbf { u } - \mathbf { v }u−v .

Given a=12,b=8a = 12 , b = 8a=12,b=8 , and c=11c = 11c=11 , use the Law of Cosines to solve the triangle for the value of BBB . Round answer to two decimal places.

Find the component form of v\mathbf { v }v if ∥v∥=8\| \mathbf { v } \| = 8∥v∥=8 and the angle it makes with the xxx -axis is 120∘120 ^ { \circ }120∘ .

Find the trigonometric form of the complex number shown below. −7i- 7 i−7i

Find the absolute value of the complex number −4+6i- 4 + 6 i−4+6i .

Write the complex number shown below in trigonometric form.

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

Functions and Their Graphs

Polynomial and Rational Functions

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Plot the following complex number. $3 i$

Determine the area of a triangle having the following measurements. Round your answer to two decimal places. $A = 128 ^ { \circ } , b = 8$ , and $c = 15$

Determine the area of a triangle having the following measurements. Round your answer to two decimal places. $C = 59 ^ { \circ } 24 ^ { \prime } , a = 6$ , and $b = 14.5$

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither. $\mathbf { u } = \left\langle \frac { 7 } { 3 } , \frac { - 1 } { 2 } \right\rangle , \mathbf { v } = \langle 14 , - 3 \rangle$

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

Let $\mathbf { u } = \langle 10,6 \rangle$ and $\mathbf { v } = \langle 11,15 \rangle$ . Find $\mathbf { u } - \mathbf { v }$ .

Find the standard form of the complex number shown below. Round answers to two decimal places. $6 \left( \cos 190 ^ { \circ } + i \sin 190 ^ { \circ } \right)$

Given $a = 5 , b = 9$ , and $c = 12$ , use Heron's Area Formula to find the area of triangle $A B C$ . Round answer to two decimal places.

Find the component form of vector $\mathbf { v }$ with initial point $( - 5,1 )$ and terminal point $( - 3 , - 2 )$ .

Let $\mathbf { u } = \langle 6,5 \rangle$ and $\mathbf { v } = \langle 12,7 \rangle$ . Find $\mathbf { u } - \mathbf { v }$ .

Given $a = 12 , b = 8$ , and $c = 11$ , use the Law of Cosines to solve the triangle for the value of $B$ . Round answer to two decimal places.

Find the component form of $\mathbf { v }$ if $\| \mathbf { v } \| = 8$ and the angle it makes with the $x$ -axis is $120 ^ { \circ }$ .

Find the trigonometric form of the complex number shown below. $- 7 i$

Find the absolute value of the complex number $- 4 + 6 i$ .

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