Question 1

Find all specified roots.
-Fifth roots of $1 .$

Accepted Answer

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

Question 2

Find the product. Write the product in rectangular form, using exact values.
-$\left[ 4 \operatorname { cis } 45 ^ { \circ } \right] \left[ 6 \operatorname { cis } 225 ^ { \circ } \right]$

Accepted Answer

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

Question 3

Find the missing parts of the triangle.
-$$\begin{array} { l } 
\mathrm { B } = 24.4 ^ { \circ } \
\mathrm { b } = 5.19 \
\mathrm { a } = 6.28
\end{array}$$
If necessary, round angles to the nearest tenth and side lengths to the nearest hundredth.

Accepted Answer

A)  $\mathrm { A } _ { 1 } = 29.99 ^ { \circ } , \mathrm { C } _ { 1 } = 125.61 ^ { \circ } , \mathrm { c } 1 = 10.21$; 
B)  $\mathrm { A } = 29.99 ^ { \circ } , \mathrm { C } = 125.61 ^ { \circ } , \mathrm { c } = 10.21$
$\mathrm { A } _ { 2 } = 150.01 ^ { \circ } , \mathrm { C } _ { 2 } = 5.59 ^ { \circ } , \mathrm { c } _ { 2 } = 1.22$ 
C)  no such triangle 
D)  $\mathrm { A } = 150.01 ^ { \circ } , \mathrm { C } = 5.59 ^ { \circ } , \mathrm { C } = 1.22$ 
A)  $\mathrm { A } _ { 1 } = 29.99 ^ { \circ } , \mathrm { C } _ { 1 } = 125.61 ^ { \circ } , \mathrm { c } 1 = 10.21$; 
B)  $\mathrm { A } = 29.99 ^ { \circ } , \mathrm { C } = 125.61 ^ { \circ } , \mathrm { c } = 10.21$
$\mathrm { A } _ { 2 } = 150.01 ^ { \circ } , \mathrm { C } _ { 2 } = 5.59 ^ { \circ } , \mathrm { c } _ { 2 } = 1.22$ 
C)  no such triangle 
D)  $\mathrm { A } = 150.01 ^ { \circ } , \mathrm { C } = 5.59 ^ { \circ } , \mathrm { C } = 1.22$

Question 4

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in [0,360°].
-$$\langle 2 \sqrt { 2 } , - 2 \sqrt { 2 } \rangle$$

Accepted Answer

A)  $4 \sqrt { 2 } ; 45 ^ { \circ }$ 
B)  $8 ; 315 ^ { \circ }$ 
C)  $4 ; 315 ^ { \circ }$ 
D)  $4 ; 135 ^ { \circ }$ 
A)  $4 \sqrt { 2 } ; 45 ^ { \circ }$ 
B)  $8 ; 315 ^ { \circ }$ 
C)  $4 ; 315 ^ { \circ }$ 
D)  $4 ; 135 ^ { \circ }$

Question 5

Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as
-$$\begin{array} { l } 
\mathrm { B } = 63 ^ { \circ } 30 ^ { \prime } \
\mathrm { a } = 12.2 \mathrm { ft } \
\mathrm { c } = 7.8 \mathrm { ft }
\end{array}$$

Accepted Answer

A)  No triangle satisfies the given conditions. 
B)  $\mathrm { b } = 12.2 \mathrm { ft } , \mathrm { A } = 75 ^ { \circ } 07 ^ { \prime } , \mathrm { C } = 41 ^ { \circ } 23 ^ { \prime }$ 
C)  $\mathrm { b } = 11.2 \mathrm { ft } , \mathrm { A } = 77 ^ { \circ } 49 ^ { \prime } , \mathrm { C } = 38 ^ { \circ } 41 ^ { \prime }$ 
D)  $\mathrm { b } = 13.2 \mathrm { ft } , \mathrm { A } = 73 ^ { \circ } 07 ^ { \prime } , \mathrm { C } = 37 ^ { \circ } 23 ^ { \prime }$ 
A)  No triangle satisfies the given conditions. 
B)  $\mathrm { b } = 12.2 \mathrm { ft } , \mathrm { A } = 75 ^ { \circ } 07 ^ { \prime } , \mathrm { C } = 41 ^ { \circ } 23 ^ { \prime }$ 
C)  $\mathrm { b } = 11.2 \mathrm { ft } , \mathrm { A } = 77 ^ { \circ } 49 ^ { \prime } , \mathrm { C } = 38 ^ { \circ } 41 ^ { \prime }$ 
D)  $\mathrm { b } = 13.2 \mathrm { ft } , \mathrm { A } = 73 ^ { \circ } 07 ^ { \prime } , \mathrm { C } = 37 ^ { \circ } 23 ^ { \prime }$

Question 6

Use a table of values to graph the plane curve defined by the following parametric equations. Find a rectangular
equation for the curve.
-$x = t ^ { 3 } + 1 , y = t ^ { 3 } - 10$, for $t$ in $[ - 2,2 ]$

Accepted Answer

A) 
 
$y = x ^ { 3 }$, for $x$ in $[ - 3,1 ]$ 
B) 
 
$y = x - 11$, for $x$ in $[ - 7,9 ]$ 
C) 
 
$y = - x ^ { 2 }$, for $x$ in $[ - 4,4 ]$ 
D) 
 
$y = - x - 11$, for $x$ in $[ - 7,9 ]$ 
A) 
 
$y = x ^ { 3 }$, for $x$ in $[ - 3,1 ]$ 
B) 
 
$y = x - 11$, for $x$ in $[ - 7,9 ]$ 
C) 
 
$y = - x ^ { 2 }$, for $x$ in $[ - 4,4 ]$ 
D) 
 
$y = - x - 11$, for $x$ in $[ - 7,9 ]$

Question 7

Solve the problem.
-A projectile is fired with an initial velocity of 500 feet per second at an angle of $$70 ^ { \circ }$$ with the horizontal. To the nearest foot, find the maximum altitude of the projectile.

Accepted Answer

A)  3432 ft 
B)  3449 ft 
C)  3441 ft 
D)  3465 ft 
A)  3432 ft 
B)  3449 ft 
C)  3441 ft 
D)  3465 ft

Question 8

Find the given power. Write answer in rectangular form.
-$\left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) 12$

Accepted Answer

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

Question 9

Find the indicated vector.
-Let $\mathbf { a } = \langle - 4,1 \rangle , \mathbf { b } = \langle - 9 , - 5 \rangle$. Find $\mathbf { a } - \mathbf { b }$.

Accepted Answer

A)  $\langle 5,6 \rangle$ 
B)  $\langle - 5 , - 4 \rangle$ 
C)  $\langle - 13 , - 4 \rangle$ 
D)  $\langle 1,10 \rangle$ 
A)  $\langle 5,6 \rangle$ 
B)  $\langle - 5 , - 4 \rangle$ 
C)  $\langle - 13 , - 4 \rangle$ 
D)  $\langle 1,10 \rangle$

Question 10

Find the indicated vector.
-Let $a = \langle - 4 , - 7 \rangle$. Find $- 5 a$.

Accepted Answer

A)  $\langle - 20 , - 35 \rangle$ 
B)  $\langle 20 , - 35 \rangle$ 
C)  $\langle - 20,35 \rangle$ 
D)  $\langle 20,35 \rangle$ 
A)  $\langle - 20 , - 35 \rangle$ 
B)  $\langle 20 , - 35 \rangle$ 
C)  $\langle - 20,35 \rangle$ 
D)  $\langle 20,35 \rangle$

Question 11

Determine whether the pair of vectors is orthogonal.
-$$\langle 3 , - 1 \rangle , \langle - 6 , - 18 \rangle$$

Accepted Answer

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

Question 12

Plot the point.
-$\left(-2,-135^{\circ}\right)$

Accepted Answer

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

Question 13

Write the complex number in trigonometric form $$r ( \cos \theta + i \sin \theta )$$ with θ in the interval [0°, 360°).
-$5 + 5 \mathrm { i }$

Accepted Answer

A)  $5 \sqrt { 2 } \left( \cos 315 ^ { \circ } + i \sin 315 ^ { \circ } \right)$ 
B)  $5 \sqrt { 2 } \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
C)  $5 \sqrt { 2 } \left( \cos 225 ^ { \circ } + i \sin 225 ^ { \circ } \right)$ 
D)  $5 \sqrt { 2 } \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right)$ 
A)  $5 \sqrt { 2 } \left( \cos 315 ^ { \circ } + i \sin 315 ^ { \circ } \right)$ 
B)  $5 \sqrt { 2 } \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
C)  $5 \sqrt { 2 } \left( \cos 225 ^ { \circ } + i \sin 225 ^ { \circ } \right)$ 
D)  $5 \sqrt { 2 } \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right)$

Question 14

Find the component form of the indicated vector.
-Let $\mathbf { u } = \langle 2 , - 8 \rangle$. Find $7 \mathbf { u }$.

Accepted Answer

A)  $\langle 14,56 \rangle$ 
B)  $\langle 14 , - 56 \rangle$ 
C)  $\langle - 14,56 \rangle$ 
D)  $\langle - 14 , - 56 \rangle$ 
A)  $\langle 14,56 \rangle$ 
B)  $\langle 14 , - 56 \rangle$ 
C)  $\langle - 14,56 \rangle$ 
D)  $\langle - 14 , - 56 \rangle$

Question 15

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides,
by the law of sines.
-A, c, and a

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 16

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides,
by the law of sines.
-b, A, and B

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 17

Find the missing parts of the triangle.
-$$\begin{array} { l } 
\mathrm { A } = 118.1 ^ { \circ } \
\mathrm { a } = 1229 \mathrm {~cm} \
\mathrm {~b} = 1329 \mathrm {~cm}
\end{array}$$
If necessary, round angles to the nearest tenth and side lengths to the nearest $\mathrm { cm }$.

Accepted Answer

A)  $\mathrm { B } = 36.7 ^ { \circ } , \mathrm { C } = 25.2 ^ { \circ } , \mathrm { c } = 1018 \mathrm {~cm}$ 
B)  $\mathrm { B } = 32.8 ^ { \circ } , \mathrm { C } = 29.1 ^ { \circ } , \mathrm { c } = 1018 \mathrm {~cm}$ 
C)  no such triangle 
D)  $B = 25.2 ^ { \circ } , C = 36.7 ^ { \circ } , c = 1018 \mathrm {~cm}$ 
A)  $\mathrm { B } = 36.7 ^ { \circ } , \mathrm { C } = 25.2 ^ { \circ } , \mathrm { c } = 1018 \mathrm {~cm}$ 
B)  $\mathrm { B } = 32.8 ^ { \circ } , \mathrm { C } = 29.1 ^ { \circ } , \mathrm { c } = 1018 \mathrm {~cm}$ 
C)  no such triangle 
D)  $B = 25.2 ^ { \circ } , C = 36.7 ^ { \circ } , c = 1018 \mathrm {~cm}$

Question 18

Find the component form of the indicated vector.
-Let $\mathbf { u } = \langle 9,2 \rangle , \mathbf { v } = \langle - 5,1 \rangle$. Find $\mathbf { u } - \mathbf { v }$.

Accepted Answer

A)  $\langle 8,7 \rangle$ 
B)  $\langle 7 , - 6 \rangle$ 
C)  $\langle 14,1 \rangle$ 
D)  $\langle 4,3 \rangle$ 
A)  $\langle 8,7 \rangle$ 
B)  $\langle 7 , - 6 \rangle$ 
C)  $\langle 14,1 \rangle$ 
D)  $\langle 4,3 \rangle$

Question 19

Answer the question.
-In which quadrants do the fourth roots of a complex number lie?

Accepted Answer

A)  The first and third quadrants if the number is real, and the second and fourth quadrants if the number is not real 
B)  The first and third quadrants if the number is real and positive, and the second and fourth quadrants otherwise 
C)  The first and second quadrants if the number is real, and the third and fourth quadrants if the number is not real 
D)  All four quadrants 
A)  The first and third quadrants if the number is real, and the second and fourth quadrants if the number is not real 
B)  The first and third quadrants if the number is real and positive, and the second and fourth quadrants otherwise 
C)  The first and second quadrants if the number is real, and the third and fourth quadrants if the number is not real 
D)  All four quadrants

Question 20

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form.
-$\frac { - 6 i } { - 9 + 4 i }$

Accepted Answer

A)  $- \frac { 24 } { 97 } + \frac { 54 } { 97 } \mathrm { i }$ 
B)  $\frac { 24 } { 65 } + \frac { 54 } { 65 } \mathrm { i }$ 
C)  $- \frac { 24 } { 65 } + \frac { 54 } { 65 } \mathrm { i }$ 
D)  $\frac { 24 } { 97 } + \frac { 54 } { 97 } \mathrm { i }$ 
A)  $- \frac { 24 } { 97 } + \frac { 54 } { 97 } \mathrm { i }$ 
B)  $\frac { 24 } { 65 } + \frac { 54 } { 65 } \mathrm { i }$ 
C)  $- \frac { 24 } { 65 } + \frac { 54 } { 65 } \mathrm { i }$ 
D)  $\frac { 24 } { 97 } + \frac { 54 } { 97 } \mathrm { i }$

Find all specified roots. -Fifth roots of $1 .$

Find the product. Write the product in rectangular form, using exact values. - $\left[ 4 \operatorname { cis } 45 ^ { \circ } \right] \left[ 6 \operatorname { cis } 225 ^ { \circ } \right]$

Find the missing parts of the triangle. - =24. =5.19 =6.28 If necessary, round angles to the nearest tenth and side lengths to the nearest hundredth.

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in [0,360°]. - $\langle 2 \sqrt { 2 } , - 2 \sqrt { 2 } \rangle$

Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as - =63 =12.2 =7.8

Solve the problem. -A projectile is fired with an initial velocity of 500 feet per second at an angle of $70 ^ { \circ }$ with the horizontal. To the nearest foot, find the maximum altitude of the projectile.

Find the given power. Write answer in rectangular form. - $\left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) 12$

Find the indicated vector. -Let $\mathbf { a } = \langle - 4,1 \rangle , \mathbf { b } = \langle - 9 , - 5 \rangle$ . Find $\mathbf { a } - \mathbf { b }$ .

Find the indicated vector. -Let $a = \langle - 4 , - 7 \rangle$ . Find $- 5 a$ .

Determine whether the pair of vectors is orthogonal. - $\langle 3 , - 1 \rangle , \langle - 6 , - 18 \rangle$

Plot the point. - $\left(-2,-135^{\circ}\right)$ $Plot the point. - \left(-2,-135^{\circ}\right)$

Write the complex number in trigonometric form $r ( \cos \theta + i \sin \theta )$ with θ in the interval [0°, 360°). - $5 + 5 \mathrm { i }$

Find the component form of the indicated vector. -Let $\mathbf { u } = \langle 2 , - 8 \rangle$ . Find $7 \mathbf { u }$ .

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides, by the law of sines. -A, c, and a

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides, by the law of sines. -b, A, and B

Find the missing parts of the triangle. - =118. =1229 =1329 If necessary, round angles to the nearest tenth and side lengths to the nearest $\mathrm { cm }$ .

Find the component form of the indicated vector. -Let $\mathbf { u } = \langle 9,2 \rangle , \mathbf { v } = \langle - 5,1 \rangle$ . Find $\mathbf { u } - \mathbf { v }$ .

Answer the question. -In which quadrants do the fourth roots of a complex number lie?

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { - 6 i } { - 9 + 4 i }$

Equations and Inequalities

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Inverse, Exponential, and Logarithmic Functions

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Trigonometric Identities and Equations

Systems and Matrices

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Further Topics in Algebra

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Exam 8: Applications of Trigonometry

Find all specified roots. -Fifth roots of 1.1 .1.

Find the product. Write the product in rectangular form, using exact values. - [4cis⁡45∘][6cis⁡225∘]\left[ 4 \operatorname { cis } 45 ^ { \circ } \right] \left[ 6 \operatorname { cis } 225 ^ { \circ } \right][4cis45∘][6cis225∘]