Question 1

Find the indicated roots. Write the answer in a + bi form.
-Cube roots of $- 8 \mathrm { i }$

Accepted Answer

A)  $- 2 i , \sqrt { 3 } + i , \sqrt { 3 } + i$ 
B)  $2 i , - \sqrt { 3 } - i , \sqrt { 3 } - i$ 
C)  $2 i , - \sqrt { 3 } + i , - \sqrt { 3 } + i$ 
D)  $- 2 i , \sqrt { 3 } - i , \sqrt { 3 } - i$ 
A)  $- 2 i , \sqrt { 3 } + i , \sqrt { 3 } + i$ 
B)  $2 i , - \sqrt { 3 } - i , \sqrt { 3 } - i$ 
C)  $2 i , - \sqrt { 3 } + i , - \sqrt { 3 } + i$ 
D)  $- 2 i , \sqrt { 3 } - i , \sqrt { 3 } - i$

Question 2

With your calculator set to radian mode and polar graphics capability, graph the following function in the window specified.
-$$\mathrm { r } = \cos 3 \theta - \sin 2 \theta , 0 \leq \theta \leq 6 \pi , [ - 2,2 ] \text { by } [ - 2,2 ]$$

Accepted Answer

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

Question 3

Find the component form of the indicated vector.
-Let $\mathbf { u } = \langle 4 , - 2 \rangle$. Find $7 \mathbf { u }$.

Accepted Answer

A)  $\langle - 28,14 \rangle$ 
B)  $\langle 28 , - 14 \rangle$ 
C)  $\langle 28,14 \rangle$ 
D)  $\langle - 28 , - 14 \rangle$ 
A)  $\langle - 28,14 \rangle$ 
B)  $\langle 28 , - 14 \rangle$ 
C)  $\langle 28,14 \rangle$ 
D)  $\langle - 28 , - 14 \rangle$

Question 4

Solve the problem.
-The locations, given in polar coordinates, of two planes approaching an airport are $\left( 9 \mathrm { mi } , 20 ^ { \circ } \right)$ and $\left( 2 \mathrm { mi } , 71 ^ { \circ } \right)$. Find the distance between the two planes.

Accepted Answer

A)  $\approx 7.9 \mathrm { mi }$ 
B)  $\approx 51.48 \mathrm { mi }$ 
C)  $\approx  9.22 \mathrm { mi }$ 
D)  $\approx 8.42 \mathrm { mi }$ 
A)  $\approx 7.9 \mathrm { mi }$ 
B)  $\approx 51.48 \mathrm { mi }$ 
C)  $\approx  9.22 \mathrm { mi }$ 
D)  $\approx 8.42 \mathrm { mi }$

Question 5

With your calculator set to radian mode and polar graphics capability, graph the following function in the window specified.
-$r = 2 \theta , - 3 \pi \leq \theta \leq 3 \pi , [ - 20,20 ]$ by $[ - 20,20 ]$

Accepted Answer

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

Question 6

Prove that $ \overrightarrow{R S} $ and $ \overrightarrow{O P} $ are equivalent by showing that they represent the same vector.
-R = (-4, -2), S = (-1, 4), O = (0, 0), and P = (3, 2)

Accepted Answer

A) True 
 B)False

Question 7

Draw a graph of the rose curve.
-$$r = - 3 \sin 5 \theta , 0 \leq \theta \leq 2 \pi$$

Accepted Answer

A) 
 
$[ - 4,4 ]$ by $[ - 4,4 ]$ 
B) 
 
$[ - 4,4 ]$ by $[ - 4,4 ]$ 
C) 
 
$[ - 4,4 ]$ by $[ - 4,4 ]$ 
D) 
 
$[ - 4,4 ]$ by $[ - 4,4 ]$ 
A) 
 
$[ - 4,4 ]$ by $[ - 4,4 ]$ 
B) 
 
$[ - 4,4 ]$ by $[ - 4,4 ]$ 
C) 
 
$[ - 4,4 ]$ by $[ - 4,4 ]$ 
D) 
 
$[ - 4,4 ]$ by $[ - 4,4 ]$

Question 8

Solve the problem.
-For what values of $\theta ( 0 \leq \theta < 2 \pi )$ do maximum $r$-values occur on the graph of the polar equation $r = 2 \sin 4 \theta$ ? Note that a maximum $r$-value occurs at a point that is the maximum distance from the pole.

Accepted Answer

A)  $\frac { \pi } { 8 } , \frac { 5 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 13 \pi } { 8 }$ 
B)  $0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$ 
C)  $\frac { \pi } { 8 } , \frac { 3 \pi } { 8 } , \frac { 5 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 11 \pi } { 8 } , \frac { 13 \pi } { 8 } , \frac { 15 \pi } { 8 }$ 
D)  $0 , \frac { \pi } { 4 } , \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } , \pi , \frac { 5 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 4 }$ 
A)  $\frac { \pi } { 8 } , \frac { 5 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 13 \pi } { 8 }$ 
B)  $0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$ 
C)  $\frac { \pi } { 8 } , \frac { 3 \pi } { 8 } , \frac { 5 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 11 \pi } { 8 } , \frac { 13 \pi } { 8 } , \frac { 15 \pi } { 8 }$ 
D)  $0 , \frac { \pi } { 4 } , \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } , \pi , \frac { 5 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 4 }$

Question 9

Use the dot product to find $ |\mathbf{v}| $
-$$\mathbf { v } = - 6 \mathbf { j }$$

Accepted Answer

A)  6 
B)  $- 6$ 
C)  0 
D)  36 
A)  6 
B)  $- 6$ 
C)  0 
D)  36

Question 10

Find the rectangular coordinates of the point with the given polar coordinates.
-$( 5 , \pi )$

Accepted Answer

A)  $( 5,0 )$ 
B)  $( - 5,0 )$ 
C)  $( 0 , - 5 )$ 
D)  $( 0,5 )$ 
A)  $( 5,0 )$ 
B)  $( - 5,0 )$ 
C)  $( 0 , - 5 )$ 
D)  $( 0,5 )$

Question 11

Solve the problem.
-For what values of $\theta ( 0 \leq \theta < 2 \pi )$ do maximum r-values occur on the graph of the polar equation $r = 2$ cos Note that a maximum $\mathrm { r }$-value occurs at a point that is the maximum distance from the pole.

Accepted Answer

A)  $0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$ 
B)  $\frac { \pi } { 8 } , \frac { 3 \pi } { 8 } , \frac { 5 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 11 \pi } { 8 } , \frac { 13 \pi } { 8 } , \frac { 15 \pi } { 8 }$ 
C)  $0 , \frac { \pi } { 4 } , \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } , \pi , \frac { 5 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 4 }$ 
D)  $\frac { \pi } { 8 } , \frac { 5 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 13 \pi } { 8 }$ ? 
A)  $0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$ 
B)  $\frac { \pi } { 8 } , \frac { 3 \pi } { 8 } , \frac { 5 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 11 \pi } { 8 } , \frac { 13 \pi } { 8 } , \frac { 15 \pi } { 8 }$ 
C)  $0 , \frac { \pi } { 4 } , \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } , \pi , \frac { 5 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 4 }$ 
D)  $\frac { \pi } { 8 } , \frac { 5 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 13 \pi } { 8 }$ ?

Question 12

Find the angle between the given vectors to the nearest tenth of a degree.
-$\mathbf { u } = \mathbf { i } + \sqrt { 7 } \mathbf { j } , \mathbf { v } = - \mathbf { i } - 9 \mathbf { j }$

Accepted Answer

A)  $178.2 ^ { \circ }$ 
B)  $165.6 ^ { \circ }$ 
C)  $153.0 ^ { \circ }$ 
D)  $113.1 ^ { \circ }$ 
A)  $178.2 ^ { \circ }$ 
B)  $165.6 ^ { \circ }$ 
C)  $153.0 ^ { \circ }$ 
D)  $113.1 ^ { \circ }$

Question 13

Use De Moivre's Theorem to find the indicated power of the complex number. Write your answer in standard form a + bi.
-$( 1 - i ) ^ { 10 }$

Accepted Answer

A)  $32 - 32 i$ 
B)  $- 32 i$ 
C)  32 
D)  $- 32 + 32 i$ 
A)  $32 - 32 i$ 
B)  $- 32 i$ 
C)  32 
D)  $- 32 + 32 i$

Question 14

Plot the point with the given polar coordinates.
-$$\left( - 3,750 ^ { \circ } \right)$$

Accepted Answer

A) 
  
B) 
  
A) 
  
B)

Question 15

Find an equivalent equation in rectangular coordinates.
-$r = 10 \sin \theta$

Accepted Answer

A)  $\sqrt { x ^ { 2 } + y ^ { 2 } } = 10 y$ 
B)  $x ^ { 2 } + y ^ { 2 } = 10 x$ 
C)  $\sqrt { x ^ { 2 } + y ^ { 2 } } = 10 x$ 
D)  $x ^ { 2 } + y ^ { 2 } = 10 y$ 
A)  $\sqrt { x ^ { 2 } + y ^ { 2 } } = 10 y$ 
B)  $x ^ { 2 } + y ^ { 2 } = 10 x$ 
C)  $\sqrt { x ^ { 2 } + y ^ { 2 } } = 10 x$ 
D)  $x ^ { 2 } + y ^ { 2 } = 10 y$

Question 16

Find the magnitude and direction angle for the following vector. Give the direction angle as an angle in [0°, 360°) rounded to the nearest tenth.
-$\langle 3 \sqrt { 3 } , - 3 \sqrt { 3 } \rangle$

Accepted Answer

A)  $3 \sqrt { 3 } , 315 ^ { \circ }$ 
B)  $3 \sqrt { 6 } , 315 ^ { \circ }$ 
C)  $3 \sqrt { 3 } , 135 ^ { \circ }$ 
D)  $3 \sqrt { 6 } , 135 ^ { \circ }$ 
A)  $3 \sqrt { 3 } , 315 ^ { \circ }$ 
B)  $3 \sqrt { 6 } , 315 ^ { \circ }$ 
C)  $3 \sqrt { 3 } , 135 ^ { \circ }$ 
D)  $3 \sqrt { 6 } , 135 ^ { \circ }$

Question 17

Eliminate the parameter.
-$$x = 8 \cos t , y = 8 \sin t$$

Accepted Answer

A)  $( x + y ) ^ { 2 } = 64$ 
B)  $x ^ { 2 } + y ^ { 2 } = 8$ 
C)  $x ^ { 2 } + y ^ { 2 } = 64$ 
D)  $y = x \tan t$ 
A)  $( x + y ) ^ { 2 } = 64$ 
B)  $x ^ { 2 } + y ^ { 2 } = 8$ 
C)  $x ^ { 2 } + y ^ { 2 } = 64$ 
D)  $y = x \tan t$

Question 18

Solve the problem.
-Determine the resultant effect of three people pulling on a car as shown in the drawing.

$\mathrm { a } = 90.0 \mathrm { lb } , \mathrm { b } = 47.0 \mathrm { lb } , \mathrm { c } = 124.0 \mathrm { lb } , \mathrm { d } = 42 ^ { \circ } , \mathrm { e } = 41 ^ { \circ }$
Round results to an appropriate number of significant digits.

Accepted Answer

A)  $| \mathrm { F } | = 29 \mathrm { lb } , \theta = - 46 ^ { \circ }$ 
B)  $| \mathrm { F } | = 251 \mathrm { lb } , \theta = 34 ^ { \circ }$ 
C)  $| \mathrm { F } | = 143 \mathrm { lb } , \theta = 82 ^ { \circ }$ 
D)  $| F | = 209 \mathrm { lb } , \theta = - 6 ^ { \circ }$ 
A)  $| \mathrm { F } | = 29 \mathrm { lb } , \theta = - 46 ^ { \circ }$ 
B)  $| \mathrm { F } | = 251 \mathrm { lb } , \theta = 34 ^ { \circ }$ 
C)  $| \mathrm { F } | = 143 \mathrm { lb } , \theta = 82 ^ { \circ }$ 
D)  $| F | = 209 \mathrm { lb } , \theta = - 6 ^ { \circ }$

Question 19

Find the product or quotient. Write the answer in standard form.
-$$\frac { 6 + 7 i } { 4 - 2 i }$$

Accepted Answer

A)  $19 - 8 \mathrm { i }$ 
B)  $\frac { 1 } { 12 } - \frac { 1 } { 3 } \mathrm { i }$ 
C)  $\frac { 1 } { 2 } + 2 \mathrm { i }$ 
D)  $\frac { 19 } { 6 } - \frac { 1 } { 3 } \mathrm { i }$ 
A)  $19 - 8 \mathrm { i }$ 
B)  $\frac { 1 } { 12 } - \frac { 1 } { 3 } \mathrm { i }$ 
C)  $\frac { 1 } { 2 } + 2 \mathrm { i }$ 
D)  $\frac { 19 } { 6 } - \frac { 1 } { 3 } \mathrm { i }$

Question 20

Find an equivalent equation in polar coordinates.
-$$2 x + 3 y = 6$$

Accepted Answer

A)  $2 \sin \theta + 3 \cos \theta = 6 r$ 
B)  $r ( 2 \cos \theta + 3 \sin \theta ) = 6$ 
C)  $2 \cos \theta + 3 \sin \theta = 6 r$ 
D)  $r ( 2 \sin \theta + 3 \cos \theta ) = 6$ 
A)  $2 \sin \theta + 3 \cos \theta = 6 r$ 
B)  $r ( 2 \cos \theta + 3 \sin \theta ) = 6$ 
C)  $2 \cos \theta + 3 \sin \theta = 6 r$ 
D)  $r ( 2 \sin \theta + 3 \cos \theta ) = 6$

Find the indicated roots. Write the answer in a + bi form. -Cube roots of $- 8 \mathrm { i }$

With your calculator set to radian mode and polar graphics capability, graph the following function in the window specified. - $\mathrm { r } = \cos 3 \theta - \sin 2 \theta , 0 \leq \theta \leq 6 \pi , [ - 2,2 ] \text { by } [ - 2,2 ]$

Find the component form of the indicated vector. -Let $\mathbf { u } = \langle 4 , - 2 \rangle$ . Find $7 \mathbf { u }$ .

Solve the problem. -The locations, given in polar coordinates, of two planes approaching an airport are $\left( 9 \mathrm { mi } , 20 ^ { \circ } \right)$ and $\left( 2 \mathrm { mi } , 71 ^ { \circ } \right)$ . Find the distance between the two planes.

With your calculator set to radian mode and polar graphics capability, graph the following function in the window specified. - $r = 2 \theta , - 3 \pi \leq \theta \leq 3 \pi , [ - 20,20 ]$ by $[ - 20,20 ]$

Prove that $\overrightarrow{R S}$ and $\overrightarrow{O P}$ are equivalent by showing that they represent the same vector. -R = (-4, -2), S = (-1, 4), O = (0, 0), and P = (3, 2)

Draw a graph of the rose curve. - $r = - 3 \sin 5 \theta , 0 \leq \theta \leq 2 \pi$

Solve the problem. -For what values of $\theta ( 0 \leq \theta < 2 \pi )$ do maximum $r$ -values occur on the graph of the polar equation $r = 2 \sin 4 \theta$ ? Note that a maximum $r$ -value occurs at a point that is the maximum distance from the pole.

Use the dot product to find $|\mathbf{v}|$ - $\mathbf { v } = - 6 \mathbf { j }$

Find the rectangular coordinates of the point with the given polar coordinates. - $( 5 , \pi )$

Solve the problem. -For what values of $\theta ( 0 \leq \theta < 2 \pi )$ do maximum r-values occur on the graph of the polar equation $r = 2$ cos Note that a maximum $\mathrm { r }$ -value occurs at a point that is the maximum distance from the pole.

Find the angle between the given vectors to the nearest tenth of a degree. - $\mathbf { u } = \mathbf { i } + \sqrt { 7 } \mathbf { j } , \mathbf { v } = - \mathbf { i } - 9 \mathbf { j }$

Use De Moivre's Theorem to find the indicated power of the complex number. Write your answer in standard form a + bi. - $( 1 - i ) ^ { 10 }$

Plot the point with the given polar coordinates. - $\left( - 3,750 ^ { \circ } \right)$ $Plot the point with the given polar coordinates. - \left( - 3,750 ^ { \circ } \right)$

Find an equivalent equation in rectangular coordinates. - $r = 10 \sin \theta$

Find the magnitude and direction angle for the following vector. Give the direction angle as an angle in [0°, 360°) rounded to the nearest tenth. - $\langle 3 \sqrt { 3 } , - 3 \sqrt { 3 } \rangle$

Eliminate the parameter. - $x = 8 \cos t , y = 8 \sin t$

Find the product or quotient. Write the answer in standard form. - $\frac { 6 + 7 i } { 4 - 2 i }$

Find an equivalent equation in polar coordinates. - $2 x + 3 y = 6$

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Exam 6: Applications of Trigonometry

Find the indicated roots. Write the answer in a + bi form. -Cube roots of −8i- 8 \mathrm { i }−8i

Find the component form of the indicated vector. -Let u=⟨4,−2⟩\mathbf { u } = \langle 4 , - 2 \rangleu=⟨4,−2⟩ . Find 7u7 \mathbf { u }7u .

Solve the problem. -The locations, given in polar coordinates, of two planes approaching an airport are (9mi,20∘)\left( 9 \mathrm { mi } , 20 ^ { \circ } \right)(9mi,20∘) and (2mi,71∘)\left( 2 \mathrm { mi } , 71 ^ { \circ } \right)(2mi,71∘) . Find the distance between the two planes.

With your calculator set to radian mode and polar graphics capability, graph the following function in the window specified. - r=2θ,−3π≤θ≤3π,[−20,20]r = 2 \theta , - 3 \pi \leq \theta \leq 3 \pi , [ - 20,20 ]r=2θ,−3π≤θ≤3π,[−20,20] by [−20,20][ - 20,20 ][−20,20]

Prove that RS→ \overrightarrow{R S} RS and OP→ \overrightarrow{O P} OP are equivalent by showing that they represent the same vector. -R = (-4, -2), S = (-1, 4), O = (0, 0), and P = (3, 2)

Draw a graph of the rose curve. - r=−3sin⁡5θ,0≤θ≤2πr = - 3 \sin 5 \theta , 0 \leq \theta \leq 2 \pir=−3sin5θ,0≤θ≤2π

Solve the problem. -For what values of θ(0≤θ<2π)\theta ( 0 \leq \theta < 2 \pi )θ(0≤θ<2π) do maximum rrr -values occur on the graph of the polar equation r=2sin⁡4θr = 2 \sin 4 \thetar=2sin4θ ? Note that a maximum rrr -value occurs at a point that is the maximum distance from the pole.

Use the dot product to find ∣v∣ |\mathbf{v}| ∣v∣ - v=−6j\mathbf { v } = - 6 \mathbf { j }v=−6j

Find the rectangular coordinates of the point with the given polar coordinates. - (5,π)( 5 , \pi )(5,π)

Solve the problem. -For what values of θ(0≤θ<2π)\theta ( 0 \leq \theta < 2 \pi )θ(0≤θ<2π) do maximum r-values occur on the graph of the polar equation r=2r = 2r=2 cos Note that a maximum r\mathrm { r }r -value occurs at a point that is the maximum distance from the pole.

Find the angle between the given vectors to the nearest tenth of a degree. - u=i+7j,v=−i−9j\mathbf { u } = \mathbf { i } + \sqrt { 7 } \mathbf { j } , \mathbf { v } = - \mathbf { i } - 9 \mathbf { j }u=i+7​j,v=−i−9j

Use De Moivre's Theorem to find the indicated power of the complex number. Write your answer in standard form a + bi. - (1−i)10( 1 - i ) ^ { 10 }(1−i)10

Plot the point with the given polar coordinates. - (−3,750∘)\left( - 3,750 ^ { \circ } \right)(−3,750∘)

Find an equivalent equation in rectangular coordinates. - r=10sin⁡θr = 10 \sin \thetar=10sinθ

Find the magnitude and direction angle for the following vector. Give the direction angle as an angle in [0°, 360°) rounded to the nearest tenth. - ⟨33,−33⟩\langle 3 \sqrt { 3 } , - 3 \sqrt { 3 } \rangle⟨33​,−33​⟩

Eliminate the parameter. - x=8cos⁡t,y=8sin⁡tx = 8 \cos t , y = 8 \sin tx=8cost,y=8sint

Find the product or quotient. Write the answer in standard form. - 6+7i4−2i\frac { 6 + 7 i } { 4 - 2 i }4−2i6+7i​

Find an equivalent equation in polar coordinates. - 2x+3y=62 x + 3 y = 62x+3y=6