Question 1

The graph of a polar equation is given. Select the polar equation for the graph.
-

Accepted Answer

A)  $r = 4 \sin \theta$ 
B)  $r = 2 + \sin \theta$ 
C)  $r = 2 + \cos \theta$ 
D)  $r = 4 \cos \theta$ 
A)  $r = 4 \sin \theta$ 
B)  $r = 2 + \sin \theta$ 
C)  $r = 2 + \cos \theta$ 
D)  $r = 4 \cos \theta$

Question 2

Perform the indicated operation. Write the result in standard form.
-$$( - 4 + 6 i ) - 5$$

Accepted Answer

A)  $- 9 + 6 i$ 
B)  $+ 6 \mathrm { i }$ 
C)  $- 6 \mathrm { i }$ 
D)  $9 - 6 i$ 
A)  $- 9 + 6 i$ 
B)  $+ 6 \mathrm { i }$ 
C)  $- 6 \mathrm { i }$ 
D)  $9 - 6 i$

Question 3

The rectangular coordinates of a point are given. Express the point in polar coordinates with $$\mathrm { r } \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ }$$
-$( 3,3 )$

Accepted Answer

A)  $\left( 3,90 ^ { \circ } \right)$ 
B)  $\left( 3 \sqrt { 2 } , 45 ^ { \circ } \right)$ 
C)  $\left( 3 \sqrt { 2 } , 135 ^ { \circ } \right)$ 
D)  $\left( 3,45 ^ { \circ } \right)$ 
A)  $\left( 3,90 ^ { \circ } \right)$ 
B)  $\left( 3 \sqrt { 2 } , 45 ^ { \circ } \right)$ 
C)  $\left( 3 \sqrt { 2 } , 135 ^ { \circ } \right)$ 
D)  $\left( 3,45 ^ { \circ } \right)$

Question 4

Find the product. Write the answer in standard form.
-$$\mathrm { i } ( 6 - 9 \mathrm { i } ) ( 6 - 5 \mathrm { i } )$$

Accepted Answer

A)  $45 \mathrm { i } ^ { 3 } + 84 \mathrm { i } ^ { 2 } + 36 \mathrm { i }$ 
B)  $- 84 + 9 \mathrm { i }$ 
C)  $81 - 24 i$ 
D)  $84 - 9 \mathrm { i }$ 
A)  $45 \mathrm { i } ^ { 3 } + 84 \mathrm { i } ^ { 2 } + 36 \mathrm { i }$ 
B)  $- 84 + 9 \mathrm { i }$ 
C)  $81 - 24 i$ 
D)  $84 - 9 \mathrm { i }$

Question 5

Graph the cycloid for t in the indicated interval.
-$$x = 4 ( t - \sin t ) , y = 4 ( 1 - \cos t ) , 0 \leq t \leq 4 \pi$$

Accepted Answer

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

Question 6

For the given rectangular equation, give its equivalent polar equation.
-$$2 x - 9 y = - 10$$

Accepted Answer

A)  $r = \frac { - 10 } { 2 \cos \theta + 9 \sin \theta }$ 
B)  $r = \frac { - 10 } { 2 \sin \theta - 9 \cos \theta }$ 
C)  $r = \frac { - 10 } { 2 \cos \theta - 9 \sin \theta }$ 
D)  $r = \frac { - 10 } { 9 \cos \theta - 2 \sin \theta }$ 
A)  $r = \frac { - 10 } { 2 \cos \theta + 9 \sin \theta }$ 
B)  $r = \frac { - 10 } { 2 \sin \theta - 9 \cos \theta }$ 
C)  $r = \frac { - 10 } { 2 \cos \theta - 9 \sin \theta }$ 
D)  $r = \frac { - 10 } { 9 \cos \theta - 2 \sin \theta }$

Question 7

Find the polar coordinates of the point(s)of intersection of the given curves for 0 $$0 \leq \theta < 2 \pi$$
-$$\mathrm { r } = 7 , \mathrm { r } = 7 + \sin \theta$$

Accepted Answer

A)  $( 0,0 ) , ( 0,7 ) , ( \pi , 7 )$ 
B)  $( 0,7 ) , ( \pi , 7 )$ 
C)  $( 7,0 ) , ( 7 , \pi )$ 
D)  $( 0,0 ) , ( 7,0 ) , ( 7 , \pi )$ 
A)  $( 0,0 ) , ( 0,7 ) , ( \pi , 7 )$ 
B)  $( 0,7 ) , ( \pi , 7 )$ 
C)  $( 7,0 ) , ( 7 , \pi )$ 
D)  $( 0,0 ) , ( 7,0 ) , ( 7 , \pi )$

Question 8

Find all specified roots.
-Fifth roots of i.

Accepted Answer

A)  $\operatorname { cis } \frac { 2 \pi } { 5 } , \operatorname { cis } \frac { 4 \pi } { 5 } , \operatorname { cis } \frac { 6 \pi } { 5 } , \operatorname { cis } \frac { 8 \pi } { 10 } , 1$ 
B)  $\operatorname { cis } \frac { \pi } { 10 } , i , \operatorname { cis } \frac { 9 \pi } { 10 } , \operatorname { cis } \frac { 13 \pi } { 10 } , \operatorname { cis } \frac { 17 \pi } { 10 }$ 
C)  $\operatorname { cis } \frac { \pi } { 5 } , \operatorname { cis } \frac { 3 \pi } { 5 } , - 1 , \operatorname { cis } \frac { 7 \pi } { 10 } , \operatorname { cis } \frac { 9 \pi } { 10 }$ 
D)  $\operatorname { cis } \frac { \pi } { 10 } , \operatorname { cis } \frac { 7 \pi } { 10 } , \operatorname { cis } \frac { 9 \pi } { 10 } , \operatorname { cis } \frac { 13 \pi } { 10 } , \operatorname { cis } \frac { 17 \pi } { 10 }$ 
A)  $\operatorname { cis } \frac { 2 \pi } { 5 } , \operatorname { cis } \frac { 4 \pi } { 5 } , \operatorname { cis } \frac { 6 \pi } { 5 } , \operatorname { cis } \frac { 8 \pi } { 10 } , 1$ 
B)  $\operatorname { cis } \frac { \pi } { 10 } , i , \operatorname { cis } \frac { 9 \pi } { 10 } , \operatorname { cis } \frac { 13 \pi } { 10 } , \operatorname { cis } \frac { 17 \pi } { 10 }$ 
C)  $\operatorname { cis } \frac { \pi } { 5 } , \operatorname { cis } \frac { 3 \pi } { 5 } , - 1 , \operatorname { cis } \frac { 7 \pi } { 10 } , \operatorname { cis } \frac { 9 \pi } { 10 }$ 
D)  $\operatorname { cis } \frac { \pi } { 10 } , \operatorname { cis } \frac { 7 \pi } { 10 } , \operatorname { cis } \frac { 9 \pi } { 10 } , \operatorname { cis } \frac { 13 \pi } { 10 } , \operatorname { cis } \frac { 17 \pi } { 10 }$

Question 9

Find the product. Write the answer in standard form.
-$( 3 - 7 \mathrm { i } ) ( 2 - 5 \mathrm { i } )$

Accepted Answer

A)  $- 29 + 29 i$ 
B)  $41 + 1 \mathrm { i }$ 
C)  $35 i ^ { 2 } - 29 i + 6$ 
D)  $- 29 - 29 \mathrm { i }$ 
A)  $- 29 + 29 i$ 
B)  $41 + 1 \mathrm { i }$ 
C)  $35 i ^ { 2 } - 29 i + 6$ 
D)  $- 29 - 29 \mathrm { i }$

Question 10

Find the given power. Write answer in rectangular form.
-$$\left( - \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } i \right) ^ { 10 }$$

Accepted Answer

A)  $\frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } \mathrm { i }$ 
B)  $- \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$ 
C)  $\frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$ 
D)  $- \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } \mathrm { i }$ 
A)  $\frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } \mathrm { i }$ 
B)  $- \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$ 
C)  $\frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$ 
D)  $- \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } \mathrm { i }$

Question 11

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations.
-

Accepted Answer

A)  $r = 8 \sin \theta$ 
B)  $r = 4$ 
C)  $r = 8 \cos \theta$ 
D)  $r \sin \theta = 4$ 
A)  $r = 8 \sin \theta$ 
B)  $r = 4$ 
C)  $r = 8 \cos \theta$ 
D)  $r \sin \theta = 4$

Question 12

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal
places.
-$\left( 12 \operatorname { cis } 13.5 ^ { \circ } \right) \left( 2 \operatorname { cis } 7.5 ^ { \circ } \right)$

Accepted Answer

A)  $22.4059 + 8.6008 \mathrm { i }$ 
B)  $- 6.5728 + 10.0399 \mathrm { i }$ 
C) 
$- 1.0955 \mathrm { i }$ 
D) 
$+ 22.4059 \mathrm { i }$ 
A)  $22.4059 + 8.6008 \mathrm { i }$ 
B)  $- 6.5728 + 10.0399 \mathrm { i }$ 
C) 
$- 1.0955 \mathrm { i }$ 
D) 
$+ 22.4059 \mathrm { i }$

Question 13

Solve the problem.
-A projectile is fired with an initial velocity of 400 feet per second at an angle of 70° with the horizontal. In how many seconds will the projectile reach its maximum altitude? (Round your
Answer to the nearest tenth of a second.)

Accepted Answer

A) 16.6 sec 
B) 8.4 sec 
C) 11.8 sec 
D) 19.9 sec 
A) 16.6 sec 
B) 8.4 sec 
C) 11.8 sec 
D) 19.9 sec

Question 14

Solve the quadratic equation and express all nonreal complex solutions in terms of i.
-$$\mathrm { x } ^ { 2 } + 35 = 5 \mathrm { x }$$

Accepted Answer

A)  $\{ 0,5 \}$ 
B)  $\left\{ \frac { 5 } { 2 } \pm \frac { 5 } { 2 } i \right\}$ 
C)  $\{ - 5,5 \}$ 
D)  $\left\{ \frac { 5 } { 2 } \pm \frac { \sqrt { 115 } } { 2 } \mathrm { i } \right\}$ 
A)  $\{ 0,5 \}$ 
B)  $\left\{ \frac { 5 } { 2 } \pm \frac { 5 } { 2 } i \right\}$ 
C)  $\{ - 5,5 \}$ 
D)  $\left\{ \frac { 5 } { 2 } \pm \frac { \sqrt { 115 } } { 2 } \mathrm { i } \right\}$

Question 15

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form.
-$\frac { 2 \sqrt { 3 } + 2 i } { \sqrt { 3 } - i }$

Accepted Answer

A)  $\sqrt { 3 } \mathrm { i }$ 
B)  $\sqrt { 3 }$ 
C)  $\sqrt { 3 } + \mathrm { i }$ 
D)  $1 + \sqrt { 3 } \mathrm { i }$ 
A)  $\sqrt { 3 } \mathrm { i }$ 
B)  $\sqrt { 3 }$ 
C)  $\sqrt { 3 } + \mathrm { i }$ 
D)  $1 + \sqrt { 3 } \mathrm { i }$

Question 16

Find the quotient. Write the answer in standard form.
-$$\frac { 5 - 6 i } { 4 + 2 i }$$

Accepted Answer

A)  $\frac { 8 } { 3 } + \frac { 17 } { 12 } \mathrm { i }$ 
B)  $\frac { 2 } { 5 } - \frac { 17 } { 10 } \mathrm { i }$ 
C)  $\frac { 16 } { 5 } + \frac { 7 } { 5 } \mathrm { i }$ 
D)  $\frac { 1 } { 3 } + \frac { 17 } { 12 } \mathrm { i }$ 
A)  $\frac { 8 } { 3 } + \frac { 17 } { 12 } \mathrm { i }$ 
B)  $\frac { 2 } { 5 } - \frac { 17 } { 10 } \mathrm { i }$ 
C)  $\frac { 16 } { 5 } + \frac { 7 } { 5 } \mathrm { i }$ 
D)  $\frac { 1 } { 3 } + \frac { 17 } { 12 } \mathrm { i }$

Question 17

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations.
-$$x = t , y = t ^ { 3 } ; t = - 2$$

Accepted Answer

A)  $( - 8 , - 2 )$ 
B)  $( 2,8 )$ 
C)  $( - 2 , - 8 )$ 
D)  $( - 2$, 
A)  $( - 8 , - 2 )$ 
B)  $( 2,8 )$ 
C)  $( - 2 , - 8 )$ 
D)  $( - 2$,

Question 18

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations.
-$x = - 4 t + 8 , y = 3 t - 3 ; t = - 7$

Accepted Answer

A)  $( - 3 , - 7 )$ 
B)  $( 36 , - 24 )$ 
C)  $( 20 , - 18 )$ 
D)  $( 4,0 )$ 
A)  $( - 3 , - 7 )$ 
B)  $( 36 , - 24 )$ 
C)  $( 20 , - 18 )$ 
D)  $( 4,0 )$

Question 19

Find the given power. Write answer in rectangular form.
-$\left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ }) \right.$$^{18}$

Accepted Answer

A)  $- 1$ 
B)  $\mathrm { i }$ 
C)  $- \mathrm { i }$ 
A)  $- 1$ 
B)  $\mathrm { i }$ 
C)  $- \mathrm { i }$

Question 20

Write the complex number in rectangular form.
-$9 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right)$

Accepted Answer

A)  $- 9$ 
B)  $9 \mathrm { i }$ 
C)  9 
D)  $- 9 \mathrm { i }$ 
A)  $- 9$ 
B)  $9 \mathrm { i }$ 
C)  9 
D)  $- 9 \mathrm { i }$

The graph of a polar equation is given. Select the polar equation for the graph. -

Perform the indicated operation. Write the result in standard form. - $( - 4 + 6 i ) - 5$

The rectangular coordinates of a point are given. Express the point in polar coordinates with $\mathrm { r } \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ }$ - $( 3,3 )$

Find the product. Write the answer in standard form. - $\mathrm { i } ( 6 - 9 \mathrm { i } ) ( 6 - 5 \mathrm { i } )$

Graph the cycloid for t in the indicated interval. - $x = 4 ( t - \sin t ) , y = 4 ( 1 - \cos t ) , 0 \leq t \leq 4 \pi$ $Graph the cycloid for t in the indicated interval. - x = 4 ( t - \sin t ) , y = 4 ( 1 - \cos t ) , 0 \leq t \leq 4 \pi$

For the given rectangular equation, give its equivalent polar equation. - $2 x - 9 y = - 10$

Find the polar coordinates of the point(s)of intersection of the given curves for 0 $0 \leq \theta < 2 \pi$ - $\mathrm { r } = 7 , \mathrm { r } = 7 + \sin \theta$

Find all specified roots. -Fifth roots of i.

Find the product. Write the answer in standard form. - $( 3 - 7 \mathrm { i } ) ( 2 - 5 \mathrm { i } )$

Find the given power. Write answer in rectangular form. - $\left( - \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } i \right) ^ { 10 }$

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations. -

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal places. - $\left( 12 \operatorname { cis } 13.5 ^ { \circ } \right) \left( 2 \operatorname { cis } 7.5 ^ { \circ } \right)$

Solve the problem. -A projectile is fired with an initial velocity of 400 feet per second at an angle of 70° with the horizontal. In how many seconds will the projectile reach its maximum altitude? (Round your Answer to the nearest tenth of a second.)

Solve the quadratic equation and express all nonreal complex solutions in terms of i. - $\mathrm { x } ^ { 2 } + 35 = 5 \mathrm { x }$

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 2 \sqrt { 3 } + 2 i } { \sqrt { 3 } - i }$

Find the quotient. Write the answer in standard form. - $\frac { 5 - 6 i } { 4 + 2 i }$

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations. - $x = t , y = t ^ { 3 } ; t = - 2$

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations. - $x = - 4 t + 8 , y = 3 t - 3 ; t = - 7$

Find the given power. Write answer in rectangular form. - $\left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ }) \right.$ $^{18}$

Write the complex number in rectangular form. - $9 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right)$

Trigonometric Functions

Acute Angles and Right Triangles

Radian Measure and the Unit Circle

Graphs of the Circular Functions

Trigonometric Identities

Inverse Circular Functions and Trigonometric Equations

Applications of Trigonometry and Vectors

Filters

Exam 8: Complex Numbers, Polar Equations, and Parametric Equations

The graph of a polar equation is given. Select the polar equation for the graph. -

Perform the indicated operation. Write the result in standard form. - (−4+6i)−5( - 4 + 6 i ) - 5(−4+6i)−5

The rectangular coordinates of a point are given. Express the point in polar coordinates with r≥0 and 0∘≤θ<360∘\mathrm { r } \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ }r≥0 and 0∘≤θ<360∘ - (3,3)( 3,3 )(3,3)

Find the product. Write the answer in standard form. - i(6−9i)(6−5i)\mathrm { i } ( 6 - 9 \mathrm { i } ) ( 6 - 5 \mathrm { i } )i(6−9i)(6−5i)

Graph the cycloid for t in the indicated interval. - x=4(t−sin⁡t),y=4(1−cos⁡t),0≤t≤4πx = 4 ( t - \sin t ) , y = 4 ( 1 - \cos t ) , 0 \leq t \leq 4 \pix=4(t−sint),y=4(1−cost),0≤t≤4π

For the given rectangular equation, give its equivalent polar equation. - 2x−9y=−102 x - 9 y = - 102x−9y=−10

Find the polar coordinates of the point(s)of intersection of the given curves for 0 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - r=7,r=7+sin⁡θ\mathrm { r } = 7 , \mathrm { r } = 7 + \sin \thetar=7,r=7+sinθ