Question 1

Find the magnitude of vector $\mathbf { v }$.

Accepted Answer

A)  $\| v \| = 4 \sqrt { 2 }$ 
B)  $\| v \| = 2 \sqrt { 7 }$ 
C)  $\| v \| = 6$ 
D)  $\| v \| = 6 \sqrt { 3 }$ 
E)  $ \| v \| = 5$ 
A)  $\| v \| = 4 \sqrt { 2 }$ 
B)  $\| v \| = 2 \sqrt { 7 }$ 
C)  $\| v \| = 6$ 
D)  $\| v \| = 6 \sqrt { 3 }$ 
E)  $ \| v \| = 5$

Question 2

Plot the following complex number.
$3 i$

Accepted Answer

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

Question 3

An airplane is flying at a bearing of $143 ^ { \circ }$ with an airspeed of 335 kilometers per hour. Because of the wind, its ground speed and direction are, respectively, 325 kilometers per hour and $138 ^ { \circ }$. Find the direction and speed of the wind. Round your answer to two decimals.

Accepted Answer

A)  $31.36 ^ { \circ } , 30.47$ kilometers per hour 
B)  $121.36 ^ { \circ } , 25.71$ kilometers per hour 
C)  $- 37.01 ^ { \circ } , 30.47$ kilometers per hour 
D)  $31.36 ^ { \circ } , 25.71$ kilometers per hour 
E)  $121.36 ^ { \circ } , 25.69$ kilometers per hour 
A)  $31.36 ^ { \circ } , 30.47$ kilometers per hour 
B)  $121.36 ^ { \circ } , 25.71$ kilometers per hour 
C)  $- 37.01 ^ { \circ } , 30.47$ kilometers per hour 
D)  $31.36 ^ { \circ } , 25.71$ kilometers per hour 
E)  $121.36 ^ { \circ } , 25.69$ kilometers per hour

Question 4

Use Heron's area formula to find the area of the triangle pictured below, if $a = 8$ inches, $\mathrm { b } = 14$ inches, and $\mathrm { c } = 10$ inches.

Accepted Answer

A)  $4 \sqrt { 6 }$ 
B)  40 
C)  56 
D)  $12 \sqrt { 66 }$ 
E)  $16 \sqrt { 6 }$ 
A)  $4 \sqrt { 6 }$ 
B)  40 
C)  56 
D)  $12 \sqrt { 66 }$ 
E)  $16 \sqrt { 6 }$

Question 5

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

Accepted Answer

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

Question 6

Let $\mathbf { w }$ be a vector with initial point $( 2,1 )$ and terminal point $( 9 , - 3 )$. Write $\mathbf { w }$ as a linear combination of the standard unit vectors $\mathbf { i }$ and $\mathbf { j }$.

Accepted Answer

A)  $\mathbf { w } = \mathbf { i } + 12 \mathbf { j }$ 
B)  $\mathbf { w } = - 5 \mathbf { i } + 8 \mathbf { j }$ 
C)  $\mathbf { w } = 7 \mathbf { i } - 4 \mathbf { j }$ 
D)  $\mathbf { w } = 11 \mathbf { i } + 4 \mathbf { j }$ 
E)  $\mathbf { w } = 3 \mathbf { i } - 12 \mathbf { j }$ 
A)  $\mathbf { w } = \mathbf { i } + 12 \mathbf { j }$ 
B)  $\mathbf { w } = - 5 \mathbf { i } + 8 \mathbf { j }$ 
C)  $\mathbf { w } = 7 \mathbf { i } - 4 \mathbf { j }$ 
D)  $\mathbf { w } = 11 \mathbf { i } + 4 \mathbf { j }$ 
E)  $\mathbf { w } = 3 \mathbf { i } - 12 \mathbf { j }$

Question 7

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

Question 8

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 9

Find the standard form of the complex number shown below.
$$11 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)$$

Accepted Answer

A)  $- \frac { 11 } { 2 } + \frac { 11 \sqrt { 3 } } { 2 } i$ 
B)  $\frac { 11 \sqrt { 3 } } { 4 } - \frac { 11 } { 4 } i$ 
C)  $- \frac { 2 } { 11 } + \frac { 2 \sqrt { 3 } } { 11 } i$ 
D)  $- \frac { 11 \sqrt { 3 } } { 2 } + \frac { 11 } { 2 } i$ 
E)  $\frac { 4 \sqrt { 3 } } { 11 } - \frac { 4 } { 11 } i$ 
A)  $- \frac { 11 } { 2 } + \frac { 11 \sqrt { 3 } } { 2 } i$ 
B)  $\frac { 11 \sqrt { 3 } } { 4 } - \frac { 11 } { 4 } i$ 
C)  $- \frac { 2 } { 11 } + \frac { 2 \sqrt { 3 } } { 11 } i$ 
D)  $- \frac { 11 \sqrt { 3 } } { 2 } + \frac { 11 } { 2 } i$ 
E)  $\frac { 4 \sqrt { 3 } } { 11 } - \frac { 4 } { 11 } i$

Question 10

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)  $12 \left( \cos 370 ^ { \circ } + i \sin 370 ^ { \circ } \right)$

Question 11

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 12

A straight road makes an angle, $A$, of $20 ^ { \circ }$ with the horizontal. When the angle of elevation, $B$, of the sun is $60 ^ { \circ }$, a vertical pole beside the road casts a shadow 8 feet long parallel to the road. Approximate the length of the pole. Round answer to two decimal places.

Accepted Answer

A)  $5.47 \mathrm { feet }$ 
B)  $6.22$ feet 
C)  $10.28$ feet 
D)  $13.86 \mathrm { feet }$ 
E)  $5.94$ feet 
A)  $5.47 \mathrm { feet }$ 
B)  $6.22$ feet 
C)  $10.28$ feet 
D)  $13.86 \mathrm { feet }$ 
E)  $5.94$ feet

Question 13

Use DeMoivre's Theorem to find the indicated power of the following complex number.
$$\left[ 2 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right) \right] ^ { 12 }$$

Accepted Answer

A)  2048 
B)  4,096 
C)  $4,096 + 4096 \sqrt { 3 }$ 
D)  $2,048 + 2048 \sqrt { 3 }$ 
E)  $- 4,096$ 
A)  2048 
B)  4,096 
C)  $4,096 + 4096 \sqrt { 3 }$ 
D)  $2,048 + 2048 \sqrt { 3 }$ 
E)  $- 4,096$

Question 14

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 15

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form.
$( - 4 - 4 i ) ( - 3 - 3 i )$

Accepted Answer

A)  $- 12 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$ 
B)  $- 24 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
C)  $12 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
D)  $24 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
E)  $12 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$ 
A)  $- 12 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$ 
B)  $- 24 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
C)  $12 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
D)  $24 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
E)  $12 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$

Question 16

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 17

Given $A = 51 ^ { \circ } , b = 6$, and $c = 9$, use the Law of Cosines to solve the triangle for the value of $a$. Round answer to two decimal places.

Accepted Answer

A)  $7.00$ 
B)  $10.30$ 
C)  $7.83$ 
D)  $13.60$ 
E)  $8.65$ 
A)  $7.00$ 
B)  $10.30$ 
C)  $7.83$ 
D)  $13.60$ 
E)  $8.65$

Question 18

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 19

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither.
$$\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 30 , - 5 \rangle$$

Accepted Answer

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

Question 20

Represent the complex number below graphically.
$\sqrt { 3 } - i$

Accepted Answer

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

Find the magnitude of vector $\mathbf { v }$ . $Find the magnitude of vector \mathbf { v } .$

Plot the following complex number. $3 i$

An airplane is flying at a bearing of $143 ^ { \circ }$ with an airspeed of 335 kilometers per hour. Because of the wind, its ground speed and direction are, respectively, 325 kilometers per hour and $138 ^ { \circ }$ . Find the direction and speed of the wind. Round your answer to two decimals.

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

Let $\mathbf { w }$ be a vector with initial point $( 2,1 )$ and terminal point $( 9 , - 3 )$ . Write $\mathbf { w }$ as a linear combination of the standard unit vectors $\mathbf { i }$ and $\mathbf { j }$ .

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 C to City A. Round your answer to two decimals.

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.

Find the standard form of the complex number shown below. $11 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)$

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

Use DeMoivre's Theorem to find the indicated power of the following complex number. $\left[ 2 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right) \right] ^ { 12 }$

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

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. $( - 4 - 4 i ) ( - 3 - 3 i )$

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

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither. $\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 30 , - 5 \rangle$

Represent the complex number below graphically. $\sqrt { 3 } - i$ $Represent the complex number below graphically. \sqrt { 3 } - i$

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Exam 6: Additional Topics in Trigonometry

Find the magnitude of vector v\mathbf { v }v .

Plot the following complex number. 3i3 i3i

Use Heron's area formula to find the area of the triangle pictured below, if a=8a = 8a=8 inches, b=14\mathrm { b } = 14b=14 inches, and c=10\mathrm { c } = 10c=10 inches.

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

Let w\mathbf { w }w be a vector with initial point (2,1)( 2,1 )(2,1) and terminal point (9,−3)( 9 , - 3 )(9,−3) . Write w\mathbf { w }w as a linear combination of the standard unit vectors i\mathbf { i }i and j\mathbf { j }j .

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.

Find the standard form of the complex number shown below. 11(cos⁡5π6+isin⁡5π6)11 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)11(cos65π​+isin65π​)

Use DeMoivre's Theorem to find the indicated power of the following complex number. [2(cos⁡π3+isin⁡π3)]12\left[ 2 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right) \right] ^ { 12 }[2(cos3π​+isin3π​)]12

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 .

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. (−4−4i)(−3−3i)( - 4 - 4 i ) ( - 3 - 3 i )(−4−4i)(−3−3i)

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.

Given A=51∘,b=6A = 51 ^ { \circ } , b = 6A=51∘,b=6 , and c=9c = 9c=9 , use the Law of Cosines to solve the triangle for the value of aaa . Round answer to two decimal places.

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

Determine whether u\mathbf { u }u and v\mathbf { v }v are orthogonal, parallel, or neither. u=⟨1,5⟩,v=⟨−30,−5⟩\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 30 , - 5 \rangleu=⟨1,5⟩,v=⟨−30,−5⟩

Represent the complex number below graphically. 3−i\sqrt { 3 } - i3​−i