Question 1

Find all specified roots.
-Ninth roots of 1 .

Accepted Answer

A)  $- 1 , \operatorname { cis } \frac { 2 \pi } { 9 } , \operatorname { cis } \frac { 4 \pi } { 9 } , \operatorname { cis } \frac { 2 \pi } { 3 } , \operatorname { cis } \frac { 8 \pi } { 9 } , \operatorname { cis } \frac { 10 \pi } { 9 } , \operatorname { cis } \frac { 4 \pi } { 3 } , \operatorname { cis } \frac { 14 \pi } { 9 } , \operatorname { cis } \frac { 16 \pi } { 9 }$ 
B)  $1 , \operatorname { cis } \frac { 2 \pi } { 9 } , \operatorname { cis } \frac { 4 \pi } { 9 } , \operatorname { cis } \frac { 2 \pi } { 3 } , \operatorname { cis } \frac { 8 \pi } { 9 } , \operatorname { cis } \frac { 10 \pi } { 9 } , \operatorname { cis } \frac { 4 \pi } { 3 } , \operatorname { cis } \frac { 14 \pi } { 9 } , \operatorname { cis } \frac { 16 \pi } { 9 }$ 
C)  $- 1 , \operatorname { cis } \frac { \pi } { 9 } , \operatorname { cis } \frac { 2 \pi } { 9 } , \operatorname { cis } \frac { \pi } { 3 } , \operatorname { cis } \frac { 4 \pi } { 9 } , \operatorname { cis } \frac { 5 \pi } { 9 } , \operatorname { cis } \frac { 2 \pi } { 3 } , \operatorname { cis } \frac { 7 \pi } { 9 } , \operatorname { cis } \frac { 8 \pi } { 9 }$ 
D)  $1 , \operatorname { cis } \frac { \pi } { 9 } , \operatorname { cis } \frac { 2 \pi } { 9 } , \operatorname { cis } \frac { \pi } { 3 } , \operatorname { cis } \frac { 4 \pi } { 9 } , \operatorname { cis } \frac { 5 \pi } { 9 } , \operatorname { cis } \frac { 2 \pi } { 3 } , \operatorname { cis } \frac { 7 \pi } { 9 } , \operatorname { cis } \frac { 8 \pi } { 9 }$ 
A)  $- 1 , \operatorname { cis } \frac { 2 \pi } { 9 } , \operatorname { cis } \frac { 4 \pi } { 9 } , \operatorname { cis } \frac { 2 \pi } { 3 } , \operatorname { cis } \frac { 8 \pi } { 9 } , \operatorname { cis } \frac { 10 \pi } { 9 } , \operatorname { cis } \frac { 4 \pi } { 3 } , \operatorname { cis } \frac { 14 \pi } { 9 } , \operatorname { cis } \frac { 16 \pi } { 9 }$ 
B)  $1 , \operatorname { cis } \frac { 2 \pi } { 9 } , \operatorname { cis } \frac { 4 \pi } { 9 } , \operatorname { cis } \frac { 2 \pi } { 3 } , \operatorname { cis } \frac { 8 \pi } { 9 } , \operatorname { cis } \frac { 10 \pi } { 9 } , \operatorname { cis } \frac { 4 \pi } { 3 } , \operatorname { cis } \frac { 14 \pi } { 9 } , \operatorname { cis } \frac { 16 \pi } { 9 }$ 
C)  $- 1 , \operatorname { cis } \frac { \pi } { 9 } , \operatorname { cis } \frac { 2 \pi } { 9 } , \operatorname { cis } \frac { \pi } { 3 } , \operatorname { cis } \frac { 4 \pi } { 9 } , \operatorname { cis } \frac { 5 \pi } { 9 } , \operatorname { cis } \frac { 2 \pi } { 3 } , \operatorname { cis } \frac { 7 \pi } { 9 } , \operatorname { cis } \frac { 8 \pi } { 9 }$ 
D)  $1 , \operatorname { cis } \frac { \pi } { 9 } , \operatorname { cis } \frac { 2 \pi } { 9 } , \operatorname { cis } \frac { \pi } { 3 } , \operatorname { cis } \frac { 4 \pi } { 9 } , \operatorname { cis } \frac { 5 \pi } { 9 } , \operatorname { cis } \frac { 2 \pi } { 3 } , \operatorname { cis } \frac { 7 \pi } { 9 } , \operatorname { cis } \frac { 8 \pi } { 9 }$

Question 2

Give the rectangular coordinates for the point.
-(10, $\left. 225 ^ { \circ } \right)$

Accepted Answer

A)  $( - 5 \sqrt { 2 } , - 5 \sqrt { 2 } )$ 
B)  $( 5 \sqrt { 2 } , 5 )$ 
C)  $( 5,5 \sqrt { 2 } )$ 
D)  $( - 5 \sqrt { 2 } , - 5 )$ 
A)  $( - 5 \sqrt { 2 } , - 5 \sqrt { 2 } )$ 
B)  $( 5 \sqrt { 2 } , 5 )$ 
C)  $( 5,5 \sqrt { 2 } )$ 
D)  $( - 5 \sqrt { 2 } , - 5 )$

Question 3

Find the area of triangle ABC with the given parts. Round to the nearest tenth when necessary.
-$$\begin{array} { l } 
\mathrm { a } = 42 \mathrm { ft } \
\mathrm { b } = 50 \mathrm { ft } \
\mathrm { c } = 64 \mathrm { ft }
\end{array}$$

Accepted Answer

A)  $1311 \mathrm { ft } ^ { 2 }$ 
B)  $1049 \mathrm { ft } ^ { 2 }$ 
C)  $119 \mathrm { ft } ^ { 2 }$ 
D)  $1574 \mathrm { ft } ^ { 2 }$ 
A)  $1311 \mathrm { ft } ^ { 2 }$ 
B)  $1049 \mathrm { ft } ^ { 2 }$ 
C)  $119 \mathrm { ft } ^ { 2 }$ 
D)  $1574 \mathrm { ft } ^ { 2 }$

Question 4

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal
places.
-$\left[ 4 \operatorname { cis } \frac { 5 \pi } { 9 } \right] ^ { 2 }$

Accepted Answer

A)  $- 3.7588 - 1.3681 \mathrm { i }$ 
B)  $- 15.0351 - 5.4723 \mathrm { i }$ 
C)  $- 2.7784 + 15.7569 \mathrm { i }$ 
D)  $- 5.4723 - 15.0351 \mathrm { i }$ 
A)  $- 3.7588 - 1.3681 \mathrm { i }$ 
B)  $- 15.0351 - 5.4723 \mathrm { i }$ 
C)  $- 2.7784 + 15.7569 \mathrm { i }$ 
D)  $- 5.4723 - 15.0351 \mathrm { i }$

Question 5

Write the complex number in trigonometric form $$r ( \cos \theta + i \sin \theta )$$ with θ in the interval [0°, 360°).
-$3 \sqrt { 3 } - 3 i$

Accepted Answer

A)  $6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
B)  $6 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$ 
C)  $6 \left( \cos 300 ^ { \circ } + i \sin 300 ^ { \circ } \right)$ 
D)  $6 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
A)  $6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
B)  $6 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$ 
C)  $6 \left( \cos 300 ^ { \circ } + i \sin 300 ^ { \circ } \right)$ 
D)  $6 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$

Question 6

Find the product. Write the product in rectangular form, using exact values.
-$\left[ 4 \left( \cos 315 ^ { \circ } + \mathrm { i } \sin 315 ^ { \circ } \right) \right] \left[ 6 \left( \cos 225 ^ { \circ } + \mathrm { i } \sin 225 ^ { \circ } \right) \right]$

Accepted Answer

A)  $- 24$ 
B)  $24 \mathrm { i }$ 
C)  24 
D)  $- 24 \mathrm { i }$ 
A)  $- 24$ 
B)  $24 \mathrm { i }$ 
C)  24 
D)  $- 24 \mathrm { i }$

Question 7

Write the complex number in trigonometric form $$r ( \cos \theta + i \sin \theta )$$ with θ in the interval [0°, 360°).
-$4 \sqrt { 3 } + 4 i$

Accepted Answer

A)  $8 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
B)  $8 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
C)  $8 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$ 
D)  $8 \left( \cos 150 ^ { \circ } + \mathrm { i } \sin 150 ^ { \circ } \right)$ 
A)  $8 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
B)  $8 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
C)  $8 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$ 
D)  $8 \left( \cos 150 ^ { \circ } + \mathrm { i } \sin 150 ^ { \circ } \right)$

Question 8

Answer the question.
-In which quadrants do the nonreal cube roots of $- 1$ lie?

Accepted Answer

A)  The second and third quadrants 
B)  The second and fourth quadrants 
C)  The first and fourth quadrants 
D)  The first and third quadrants 
A)  The second and third quadrants 
B)  The second and fourth quadrants 
C)  The first and fourth quadrants 
D)  The first and third quadrants

Question 9

Give the rectangular coordinates for the point.
-$$\left( - 8,180 ^ { \circ } \right)$$

Accepted Answer

A)  $( 0,8 )$ 
B)  $( - 8,0 )$ 
C)  $( 0 , - 8 )$ 
D)  $( 8,0 )$ 
A)  $( 0,8 )$ 
B)  $( - 8,0 )$ 
C)  $( 0 , - 8 )$ 
D)  $( 8,0 )$

Question 10

Determine two pairs of polar coordinates for the point with 0° ≤ θ < 360°.
-$( 0 , \sqrt { 7 } )$

Accepted Answer

A)  $\left( 7,90 ^ { \circ } \right) , \left( - 7,270 ^ { \circ } \right)$ 
B)  $\left( \sqrt { 7 } , 270 ^ { \circ } \right) , \left( - \sqrt { 7 } , 90 ^ { \circ } \right)$ 
C)  $\left( 7,0 ^ { \circ } \right) , \left( - 7,180 ^ { \circ } \right)$ 
D)  $\left( \sqrt { 7 } , 90 ^ { \circ } \right) , \left( - \sqrt { 7 } , 270 ^ { \circ } \right)$ 
A)  $\left( 7,90 ^ { \circ } \right) , \left( - 7,270 ^ { \circ } \right)$ 
B)  $\left( \sqrt { 7 } , 270 ^ { \circ } \right) , \left( - \sqrt { 7 } , 90 ^ { \circ } \right)$ 
C)  $\left( 7,0 ^ { \circ } \right) , \left( - 7,180 ^ { \circ } \right)$ 
D)  $\left( \sqrt { 7 } , 90 ^ { \circ } \right) , \left( - \sqrt { 7 } , 270 ^ { \circ } \right)$

Question 11

Solve the problem.
-If $\mathbf { u } = \langle - 3,5 \rangle$ and $\mathbf { v } = \langle 4 , - 8 \rangle$, evaluate $\mathbf { u } \cdot ( 5 \mathbf { v } )$

Accepted Answer

A)  $- 265$ 
B)  $- 257$ 
C)  $- 260$ 
D)  $- 188$ 
A)  $- 265$ 
B)  $- 257$ 
C)  $- 260$ 
D)  $- 188$

Question 12

Write the complex number in trigonometric form $$r ( \cos \theta + i \sin \theta )$$ with θ in the interval [0°, 360°).
-$- 12 - 16 i$

Accepted Answer

A)  $20 \left( \cos 126.9 ^ { \circ } + i \sin 126.9 ^ { \circ } \right)$ 
B)  $20 \left( \cos 53.1 ^ { \circ } + i \sin 53.1 ^ { \circ } \right)$ 
C)  $20 \left( \cos 306.9 ^ { \circ } + i \sin 306.9 ^ { \circ } \right)$ 
D)  $20 \left( \cos 233.1 ^ { \circ } + i \sin 233.1 ^ { \circ } \right)$ 
A)  $20 \left( \cos 126.9 ^ { \circ } + i \sin 126.9 ^ { \circ } \right)$ 
B)  $20 \left( \cos 53.1 ^ { \circ } + i \sin 53.1 ^ { \circ } \right)$ 
C)  $20 \left( \cos 306.9 ^ { \circ } + i \sin 306.9 ^ { \circ } \right)$ 
D)  $20 \left( \cos 233.1 ^ { \circ } + i \sin 233.1 ^ { \circ } \right)$

Question 13

Solve the problem.
-Find the area of a triangular garden if the sides are approximately 10 feet, 8 feet, and 16 feet (to the nearest square foot).

Accepted Answer

A)  $8 \mathrm { ft } ^ { 2 }$ 
B)  $615 \mathrm { ft } ^ { 2 }$ 
C)  $33 \mathrm { ft } ^ { 2 }$ 
D)  $640 \mathrm { ft } ^ { 2 }$ 
A)  $8 \mathrm { ft } ^ { 2 }$ 
B)  $615 \mathrm { ft } ^ { 2 }$ 
C)  $33 \mathrm { ft } ^ { 2 }$ 
D)  $640 \mathrm { ft } ^ { 2 }$

Question 14

Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as
-$$\begin{array} { l } 
\mathrm { C } = 118.2 ^ { \circ } \
\mathrm { a } = 4.8 \mathrm {~m} \
\mathrm {~b} = 8.9 \mathrm {~m}
\end{array}$$

Accepted Answer

A)  $\mathrm { c } = 11.9 \mathrm {~m} , \mathrm {~A} = 21 ^ { \circ } , \mathrm { B } = 40.8 ^ { \circ }$ 
B)  $\mathrm { c } = 17.7 \mathrm {~m} , \mathrm {~A} = 19 ^ { \circ } , \mathrm { B } = 42.8 ^ { \circ }$ 
C)  No triangle satisfies the given conditions. 
D)  $\mathrm { c } = 14.8 \mathrm {~m} , \mathrm {~A} = 23 ^ { \circ } , \mathrm { B } = 38.8 ^ { \circ }$ 
A)  $\mathrm { c } = 11.9 \mathrm {~m} , \mathrm {~A} = 21 ^ { \circ } , \mathrm { B } = 40.8 ^ { \circ }$ 
B)  $\mathrm { c } = 17.7 \mathrm {~m} , \mathrm {~A} = 19 ^ { \circ } , \mathrm { B } = 42.8 ^ { \circ }$ 
C)  No triangle satisfies the given conditions. 
D)  $\mathrm { c } = 14.8 \mathrm {~m} , \mathrm {~A} = 23 ^ { \circ } , \mathrm { B } = 38.8 ^ { \circ }$

Question 15

Solve the problem.
-A plane is heading due south with an airspeed of $276 \mathrm { mph }$. A wind from a direction of $52 ^ { \circ }$ is blowing at $17 \mathrm { mph }$. Find the bearing of the plane.

Accepted Answer

A)  $87 ^ { \circ }$ 
B)  $178 ^ { \circ }$ 
C)  $183 ^ { \circ }$ 
D)  $92 ^ { \circ }$ 
A)  $87 ^ { \circ }$ 
B)  $178 ^ { \circ }$ 
C)  $183 ^ { \circ }$ 
D)  $92 ^ { \circ }$

Question 16

Is the given number z in the Julia set?
-$$\mathrm { z } = - 8 + 1 \mathrm { i }$$

Accepted Answer

A)  Yes 
B)  $\mathrm { No }$ 
A)  Yes 
B)  $\mathrm { No }$

Question 17

Find the dot product for the pair of vectors.
-$$\langle 1,17 \rangle , \langle 3,1 \rangle$$

Accepted Answer

A)  20 
B)  53 
C)  14 
D)  22 
A)  20 
B)  53 
C)  14 
D)  22

Question 18

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations.
-$$x = \sin t , y = \cos t ; t = \frac { \pi } { 6 }$$

Accepted Answer

A)  $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right)$ 
B)  $\left( - \frac { \sqrt { 3 } } { 2 } , - \frac { 1 } { 2 } \right)$ 
C)  $\left( \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)$ 
D)  $\left( - \frac { 1 } { 2 } , - \frac { \sqrt { 3 } } { 2 } \right)$ 
A)  $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right)$ 
B)  $\left( - \frac { \sqrt { 3 } } { 2 } , - \frac { 1 } { 2 } \right)$ 
C)  $\left( \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)$ 
D)  $\left( - \frac { 1 } { 2 } , - \frac { \sqrt { 3 } } { 2 } \right)$

Question 19

Use the figure to find the specified vector.
-Find $\mathbf { a } - \mathbf { b }$.

Accepted Answer

A)  $\langle 11,3 \rangle$ 
B)  $\langle - 5,11 \rangle$ 
C)  $\langle - 11 , - 3 \rangle$ 
D)  $\langle - 5,3 \rangle$ 
A)  $\langle 11,3 \rangle$ 
B)  $\langle - 5,11 \rangle$ 
C)  $\langle - 11 , - 3 \rangle$ 
D)  $\langle - 5,3 \rangle$

Question 20

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form.
-$\frac { 8 ( \cos 90 + i \sin 90 ) } { 3 ( \cos 30 + i \sin 30 ) }$

Accepted Answer

A)  $8 + 8 \sqrt { 3 } \mathrm { i }$ 
B)  $\frac { 4 } { 3 } + \frac { 4 \sqrt { 3 } } { 3 } \mathrm { i }$ 
C)  $\frac { 5 } { 2 } + \frac { 5 \sqrt { 3 } } { 2 } \mathrm { i }$ 
D)  $1 + \sqrt { 3 } \mathrm { i }$ 
A)  $8 + 8 \sqrt { 3 } \mathrm { i }$ 
B)  $\frac { 4 } { 3 } + \frac { 4 \sqrt { 3 } } { 3 } \mathrm { i }$ 
C)  $\frac { 5 } { 2 } + \frac { 5 \sqrt { 3 } } { 2 } \mathrm { i }$ 
D)  $1 + \sqrt { 3 } \mathrm { i }$

Find all specified roots. -Ninth roots of 1 .

Give the rectangular coordinates for the point. -(10, $\left. 225 ^ { \circ } \right)$

Find the area of triangle ABC with the given parts. Round to the nearest tenth when necessary. - =42 =50 =64

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal places. - $\left[ 4 \operatorname { cis } \frac { 5 \pi } { 9 } \right] ^ { 2 }$

Write the complex number in trigonometric form $r ( \cos \theta + i \sin \theta )$ with θ in the interval [0°, 360°). - $3 \sqrt { 3 } - 3 i$

Find the product. Write the product in rectangular form, using exact values. - $\left[ 4 \left( \cos 315 ^ { \circ } + \mathrm { i } \sin 315 ^ { \circ } \right) \right] \left[ 6 \left( \cos 225 ^ { \circ } + \mathrm { i } \sin 225 ^ { \circ } \right) \right]$

Write the complex number in trigonometric form $r ( \cos \theta + i \sin \theta )$ with θ in the interval [0°, 360°). - $4 \sqrt { 3 } + 4 i$

Answer the question. -In which quadrants do the nonreal cube roots of $- 1$ lie?

Give the rectangular coordinates for the point. - $\left( - 8,180 ^ { \circ } \right)$

Determine two pairs of polar coordinates for the point with 0° ≤ θ < 360°. - $( 0 , \sqrt { 7 } )$

Solve the problem. -If $\mathbf { u } = \langle - 3,5 \rangle$ and $\mathbf { v } = \langle 4 , - 8 \rangle$ , evaluate $\mathbf { u } \cdot ( 5 \mathbf { v } )$

Write the complex number in trigonometric form $r ( \cos \theta + i \sin \theta )$ with θ in the interval [0°, 360°). - $- 12 - 16 i$

Solve the problem. -Find the area of a triangular garden if the sides are approximately 10 feet, 8 feet, and 16 feet (to the nearest square foot).

Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as - =118. =4.8 =8.9

Solve the problem. -A plane is heading due south with an airspeed of $276 \mathrm { mph }$ . A wind from a direction of $52 ^ { \circ }$ is blowing at $17 \mathrm { mph }$ . Find the bearing of the plane.

Is the given number z in the Julia set? - $\mathrm { z } = - 8 + 1 \mathrm { i }$

Find the dot product for the pair of vectors. - $\langle 1,17 \rangle , \langle 3,1 \rangle$

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations. - $x = \sin t , y = \cos t ; t = \frac { \pi } { 6 }$

Use the figure to find the specified vector. -Find $\mathbf { a } - \mathbf { b }$ . $Use the figure to find the specified vector. -Find \mathbf { a } - \mathbf { b } .$

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 8 ( \cos 90 + i \sin 90 ) } { 3 ( \cos 30 + i \sin 30 ) }$

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Review of Basic Concepts

Filters

Exam 8: Applications of Trigonometry

Find all specified roots. -Ninth roots of 1 .

Give the rectangular coordinates for the point. -(10, 225∘)\left. 225 ^ { \circ } \right)225∘)

Find the area of triangle ABC with the given parts. Round to the nearest tenth when necessary. - =42 =50 =64

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal places. - [4cis⁡5π9]2\left[ 4 \operatorname { cis } \frac { 5 \pi } { 9 } \right] ^ { 2 }[4cis95π​]2

Write the complex number in trigonometric form r(cos⁡θ+isin⁡θ)r ( \cos \theta + i \sin \theta )r(cosθ+isinθ) with θ in the interval [0°, 360°). - 33−3i3 \sqrt { 3 } - 3 i33​−3i

Write the complex number in trigonometric form r(cos⁡θ+isin⁡θ)r ( \cos \theta + i \sin \theta )r(cosθ+isinθ) with θ in the interval [0°, 360°). - 43+4i4 \sqrt { 3 } + 4 i43​+4i

Answer the question. -In which quadrants do the nonreal cube roots of −1- 1−1 lie?

Give the rectangular coordinates for the point. - (−8,180∘)\left( - 8,180 ^ { \circ } \right)(−8,180∘)

Determine two pairs of polar coordinates for the point with 0° ≤ θ < 360°. - (0,7)( 0 , \sqrt { 7 } )(0,7​)

Solve the problem. -If u=⟨−3,5⟩\mathbf { u } = \langle - 3,5 \rangleu=⟨−3,5⟩ and v=⟨4,−8⟩\mathbf { v } = \langle 4 , - 8 \ranglev=⟨4,−8⟩ , evaluate u⋅(5v)\mathbf { u } \cdot ( 5 \mathbf { v } )u⋅(5v)

Write the complex number in trigonometric form r(cos⁡θ+isin⁡θ)r ( \cos \theta + i \sin \theta )r(cosθ+isinθ) with θ in the interval [0°, 360°). - −12−16i- 12 - 16 i−12−16i