Question 1

Use DeMoivre's Theorem to find the indicated power of the following complex number.
$( 3 + 3 \sqrt { 3 } i ) ^ { 6 }$

Accepted Answer

A)  19,683 
B)  6,561 
C)  2,187 
D)  46,656 
E)  243 
A)  19,683 
B)  6,561 
C)  2,187 
D)  46,656 
E)  243

Question 2

Find the trigonometric form of the complex number shown below. $- 6 i$

Accepted Answer

A)  $6 ( \cos 0 + i \sin 0 )$ 
B)  $6 ( \cos \pi + i \sin \pi )$ 
C)  $6 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
D)  $6 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$ 
E)  $6 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
A)  $6 ( \cos 0 + i \sin 0 )$ 
B)  $6 ( \cos \pi + i \sin \pi )$ 
C)  $6 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
D)  $6 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$ 
E)  $6 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$

Question 3

Find the trigonometric form of the complex number shown below. $- 7 i$

Accepted Answer

A)  $7 ( \cos 0 + i \sin 0 )$ 
B)  $7 ( \cos \pi + i \sin \pi )$ 
C)  $7 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
D)  $7 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$ 
E)  $7 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
A)  $7 ( \cos 0 + i \sin 0 )$ 
B)  $7 ( \cos \pi + i \sin \pi )$ 
C)  $7 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
D)  $7 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$ 
E)  $7 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$

Question 4

If $\| \mathbf { u } \| = 5$ and $\| \mathbf { v } \| = 4$, and the vectors make angles of $60 ^ { \circ }$ and $80 ^ { \circ }$ with the $x$-axis respectively, find the component form of the sum of $\mathbf { u }$ and $\mathbf { v }$. Round answers to two decimal places.

Accepted Answer

A)  $\langle 3.19,8.27 \rangle$ 
B)  $\langle 1.81,0.39 \rangle$ 
C)  $\langle 5.02,6.44 \rangle$ 
D)  $\langle 3.64 , - 1.44 \rangle$ 
E)  $\langle 2.87,8.39 \rangle$ 
A)  $\langle 3.19,8.27 \rangle$ 
B)  $\langle 1.81,0.39 \rangle$ 
C)  $\langle 5.02,6.44 \rangle$ 
D)  $\langle 3.64 , - 1.44 \rangle$ 
E)  $\langle 2.87,8.39 \rangle$

Question 5

Find the absolute value of the complex number $- 4 + 6 i$.

Accepted Answer

A)  $\sqrt { 10 }$ 
B)  $3 \sqrt { 10 }$ 
C)  52 
D)  $5 \sqrt { 13 }$ 
E)  $2 \sqrt { 13 }$ 
A)  $\sqrt { 10 }$ 
B)  $3 \sqrt { 10 }$ 
C)  52 
D)  $5 \sqrt { 13 }$ 
E)  $2 \sqrt { 13 }$

Question 6

Use Heron's area formula to find the area of the triangle pictured below, if $a = 10$ inches, $\mathrm { b } = 12$ inches, and $\mathrm { c } = 4$ inches.

Accepted Answer

A)  $3 \sqrt { 3 }$ 
B)  20 
C)  60 
D)  $8 \sqrt { 77 }$ 
E)  $3 \sqrt { 39 }$ 
A)  $3 \sqrt { 3 }$ 
B)  20 
C)  60 
D)  $8 \sqrt { 77 }$ 
E)  $3 \sqrt { 39 }$

Question 7

A straight road makes an angle, $A$, of $16 ^ { \circ }$ with the horizontal. When the angle of elevation, $B$, of the sun is $58 ^ { \circ }$, a vertical pole beside the road casts a shadow 6 feet long parallel to the road. Approximate the length of the pole. Round answer to two decimal places.

Accepted Answer

A)  3.12 feet 
B)  4.75 feet 
C)  7.58 feet 
D)  9.60 feet 
E)  4.73 feet 
A)  3.12 feet 
B)  4.75 feet 
C)  7.58 feet 
D)  9.60 feet 
E)  4.73 feet

Question 8

Given a = 5 , b = 9 , and c =12 , use Heron's Area Formula to find the area of triangle ABC . Round answer to two decimal places.

Accepted Answer

A)  20.40 sq. units 
B)  16.97 sq. units 
C)  19.60 sq. units 
D)  12.65 sq. units 
E)  17.40 sq. units 
A)  20.40 sq. units 
B)  16.97 sq. units 
C)  19.60 sq. units 
D)  12.65 sq. units 
E)  17.40 sq. units

Question 9

Given vectors $\mathbf { u } = \left\langle u _ { 1 } , u _ { 2 } \right\rangle , \mathbf { v } = \left\langle v _ { 1 } , v _ { 2 } \right\rangle$, and $\mathbf { w } = \left\langle w _ { 1 } , w _ { 2 } \right\rangle$, determine whether the result of the following expression is a vector or a scalar.
$$\mathbf { u } \cdot \mathbf { w } + \mathbf { v } \cdot \mathbf { w }$$

Accepted Answer

A)  vector 
B)  scalar 
A)  vector 
B)  scalar

Question 10

$\text { Find the fourth roots of }-\frac{1}{2}-\frac{\sqrt{3}}{2} i \text {. Write the roots in trigonometric form. }$

Accepted Answer

A) 
$\begin{array}{l}
w_{1}=\cos \left(50^{\circ}\right)+i \sin \left(50^{\circ}\right) \
w_{2}=\cos \left(140^{\circ}\right)+i \sin \left(140^{\circ}\right) \
w_{3}=\cos \left(230^{\circ}\right)+i \sin \left(230^{\circ}\right) \
w_{4}=\cos \left(320^{\circ}\right)+i \sin \left(320^{\circ}\right)
\end{array}$ 
B) 
$\begin{array}{l}
w_{1}=\cos \left(60^{\circ}\right)+i \sin \left(60^{\circ}\right) \
w_{2}=\cos \left(150^{\circ}\right)+i \sin \left(150^{\circ}\right) \
w_{3}=\cos \left(240^{\circ}\right)+i \sin \left(240^{\circ}\right) \
w_{4}=\cos \left(330^{\circ}\right)+i \sin \left(330^{\circ}\right)
\end{array}$ 
C) 
$\begin{array}{l}
w_{1}=\cos \left(65^{\circ}\right)+i \sin \left(65^{\circ}\right) \
w_{2}=\cos \left(155^{\circ}\right)+i \sin \left(155^{\circ}\right) \
w_{3}=\cos \left(245^{\circ}\right)+i \sin \left(245^{\circ}\right) \
w_{4}=\cos \left(335^{\circ}\right)+i \sin \left(335^{\circ}\right)
\end{array}$ 
D) 
$\begin{array}{l}
w_{1}=\cos \left(55^{\circ}\right)+i \sin \left(55^{\circ}\right) \
w_{2}=\cos \left(145^{\circ}\right)+i \sin \left(145^{\circ}\right) \
w_{3}=\cos \left(235^{\circ}\right)+i \sin \left(235^{\circ}\right) \
w_{4}=\cos \left(325^{\circ}\right)+i \sin \left(325^{\circ}\right)
\end{array}$
 
E) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 70 ^ { \circ } \right) + i \sin \left( 70 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 160 ^ { \circ } \right) + i \sin \left( 160 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 250 ^ { \circ } \right) + i \sin \left( 250 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 340 ^ { \circ } \right) + i \sin \left( 340 ^ { \circ } \right)
\end{array}$ 
A) 
$\begin{array}{l}
w_{1}=\cos \left(50^{\circ}\right)+i \sin \left(50^{\circ}\right) \
w_{2}=\cos \left(140^{\circ}\right)+i \sin \left(140^{\circ}\right) \
w_{3}=\cos \left(230^{\circ}\right)+i \sin \left(230^{\circ}\right) \
w_{4}=\cos \left(320^{\circ}\right)+i \sin \left(320^{\circ}\right)
\end{array}$ 
B) 
$\begin{array}{l}
w_{1}=\cos \left(60^{\circ}\right)+i \sin \left(60^{\circ}\right) \
w_{2}=\cos \left(150^{\circ}\right)+i \sin \left(150^{\circ}\right) \
w_{3}=\cos \left(240^{\circ}\right)+i \sin \left(240^{\circ}\right) \
w_{4}=\cos \left(330^{\circ}\right)+i \sin \left(330^{\circ}\right)
\end{array}$ 
C) 
$\begin{array}{l}
w_{1}=\cos \left(65^{\circ}\right)+i \sin \left(65^{\circ}\right) \
w_{2}=\cos \left(155^{\circ}\right)+i \sin \left(155^{\circ}\right) \
w_{3}=\cos \left(245^{\circ}\right)+i \sin \left(245^{\circ}\right) \
w_{4}=\cos \left(335^{\circ}\right)+i \sin \left(335^{\circ}\right)
\end{array}$ 
D) 
$\begin{array}{l}
w_{1}=\cos \left(55^{\circ}\right)+i \sin \left(55^{\circ}\right) \
w_{2}=\cos \left(145^{\circ}\right)+i \sin \left(145^{\circ}\right) \
w_{3}=\cos \left(235^{\circ}\right)+i \sin \left(235^{\circ}\right) \
w_{4}=\cos \left(325^{\circ}\right)+i \sin \left(325^{\circ}\right)
\end{array}$
 
E) 
$\begin{array} { l } 
w _ { 1 } = \cos \left( 70 ^ { \circ } \right) + i \sin \left( 70 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 160 ^ { \circ } \right) + i \sin \left( 160 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 250 ^ { \circ } \right) + i \sin \left( 250 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 340 ^ { \circ } \right) + i \sin \left( 340 ^ { \circ } \right)
\end{array}$

Question 11

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

Accepted Answer

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

Question 12

Given a =13, b = 11, and c = 6 , use Heron's Area Formula to find the area of triangle ABC . Round answer to two decimal places.

Accepted Answer

A)  32.86 sq. units 
B)  28.14 sq. units 
C)  20.78 sq. units 
D)  30.59 sq. units 
E)  28.10 sq. units 
A)  32.86 sq. units 
B)  28.14 sq. units 
C)  20.78 sq. units 
D)  30.59 sq. units 
E)  28.10 sq. units

Question 13

Given $C = 126 ^ { \circ } , a = 10.10$, and $c = 18$, 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)  $b = 3.39$ 
B)  $b = 0.58$ and $2.73$ 
C)  $b = 10.10$ 
D)  $b = 11.89$ and $4.03$ 
E)  not possible 
A)  $b = 3.39$ 
B)  $b = 0.58$ and $2.73$ 
C)  $b = 10.10$ 
D)  $b = 11.89$ and $4.03$ 
E)  not possible

Question 14

In the figure below, $a = 8 , b = 12$, and $c = 13$. Use this information to solve the parallelogram for $\theta$. The diagonals of the parallelogram are represented by $c$ and $d$. Round answer to two decimal places.

Accepted Answer

A)  101.72° 
B)  95.86 
C)  90.00° 
D)  91.46 
E)  78.28° 
A)  101.72° 
B)  95.86 
C)  90.00° 
D)  91.46 
E)  78.28°

Question 15

In the following triangle, $A = 58 ^ { \circ } , b = 105$, and $a = 126$. Use the Law of Sines to find the measure of angle $C$, in degrees. Round your answer to two decimals.

Accepted Answer

A)  $121.03 ^ { \circ }$ 
B)  $135.03 ^ { \circ }$ 
C)  $45.03 ^ { \circ }$ 
D)  $77.03 ^ { \circ }$ 
E)  $44.97 ^ { \circ }$ 
A)  $121.03 ^ { \circ }$ 
B)  $135.03 ^ { \circ }$ 
C)  $45.03 ^ { \circ }$ 
D)  $77.03 ^ { \circ }$ 
E)  $44.97 ^ { \circ }$

Question 16

Find the projection of $\mathbf { u }$ onto $\mathbf { v }$ if $\mathbf { u } = \langle 5 , - 4 \rangle , \mathbf { v } = \langle - 2,3 \rangle$.

Accepted Answer

A)  $\left\langle \frac { 44 } { \sqrt { 13 } } , \frac { 88 } { \sqrt { 13 } } \right\rangle$ 
B)  $\left\langle - \frac { 110 } { \sqrt { 13 } } , \frac { 44 } { \sqrt { 13 } } \right\rangle$ 
C)  $\left\langle - \frac { 110 } { \sqrt { 13 } } , \frac { 88 } { \sqrt { 13 } } \right\rangle$ 
D)  $\left\langle - \frac { 110 } { \sqrt { 13 } } , - \frac { 66 } { \sqrt { 13 } } \right\rangle$ 
E)  $\left\langle \frac { 44 } { \sqrt { 13 } } , - \frac { 66 } { \sqrt { 13 } } \right\rangle$ 
A)  $\left\langle \frac { 44 } { \sqrt { 13 } } , \frac { 88 } { \sqrt { 13 } } \right\rangle$ 
B)  $\left\langle - \frac { 110 } { \sqrt { 13 } } , \frac { 44 } { \sqrt { 13 } } \right\rangle$ 
C)  $\left\langle - \frac { 110 } { \sqrt { 13 } } , \frac { 88 } { \sqrt { 13 } } \right\rangle$ 
D)  $\left\langle - \frac { 110 } { \sqrt { 13 } } , - \frac { 66 } { \sqrt { 13 } } \right\rangle$ 
E)  $\left\langle \frac { 44 } { \sqrt { 13 } } , - \frac { 66 } { \sqrt { 13 } } \right\rangle$

Question 17

Find all solutions to the following equation. $$x ^ { 3 } - 216 = 0$$

Accepted Answer

A)  $x = 6,3 + 3 \sqrt { 3 } i , 3 - 3 \sqrt { 3 } i$ 
B)  $x = 6 , - 6$ 
C)  $x = - 6,3 - 3 \sqrt { 3 } i , \quad 3 + 3 \sqrt { 3 } i$ 
D)  $x = 6$ 
E)  $\quad x = 6 , - 3 + 3 \sqrt { 3 } i , \quad - 3 - 3 \sqrt { 3 } i$ 
A)  $x = 6,3 + 3 \sqrt { 3 } i , 3 - 3 \sqrt { 3 } i$ 
B)  $x = 6 , - 6$ 
C)  $x = - 6,3 - 3 \sqrt { 3 } i , \quad 3 + 3 \sqrt { 3 } i$ 
D)  $x = 6$ 
E)  $\quad x = 6 , - 3 + 3 \sqrt { 3 } i , \quad - 3 - 3 \sqrt { 3 } i$

Question 18

Given vectors $\mathbf { u } = \langle - 3,1 \rangle$ and $\mathbf { v } = \langle 2 , - 2 \rangle$, determine the quantity indicated below. $\mathbf { u } \cdot - 4 \mathbf { v }$

Accepted Answer

A)  20 
B)  - 16 
C)  2 
D)  32 
E)  14 
A)  20 
B)  - 16 
C)  2 
D)  32 
E)  14

Question 19

Find the standard form of the complex number shown below. Round answers to two decimal places.
$$2 \left( \cos 340 ^ { \circ } + i \sin 340 ^ { \circ } \right)$$

Accepted Answer

A)  $1.88 - 0.68 i$ 
B)  $2.11 - 1.07 i$ 
C)  $2.63 - 1.78 i$ 
D)  $3.12 - 2.03 i$ 
E)  $3.30 - 2.21 i$ 
A)  $1.88 - 0.68 i$ 
B)  $2.11 - 1.07 i$ 
C)  $2.63 - 1.78 i$ 
D)  $3.12 - 2.03 i$ 
E)  $3.30 - 2.21 i$

Question 20

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form.
$$( 8 + 8 i ) ( 2 + 2 i )$$

Accepted Answer

A)  $- 16 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$ 
B)  $- 32 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
C)  $16 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
D)  $32 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
E)  $16 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$ 
A)  $- 16 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$ 
B)  $- 32 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
C)  $16 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
D)  $32 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
E)  $16 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$

Use DeMoivre's Theorem to find the indicated power of the following complex number. $( 3 + 3 \sqrt { 3 } i ) ^ { 6 }$

Find the trigonometric form of the complex number shown below. $- 6 i$

Find the trigonometric form of the complex number shown below. $- 7 i$

If $\| \mathbf { u } \| = 5$ and $\| \mathbf { v } \| = 4$ , and the vectors make angles of $60 ^ { \circ }$ and $80 ^ { \circ }$ with the $x$ -axis respectively, find the component form of the sum of $\mathbf { u }$ and $\mathbf { v }$ . Round answers to two decimal places.

Find the absolute value of the complex number $- 4 + 6 i$ .

Given a = 5 , b = 9 , and c =12 , use Heron's Area Formula to find the area of triangle ABC . Round answer to two decimal places.

$\text { Find the fourth roots of }-\frac{1}{2}-\frac{\sqrt{3}}{2} i \text {. Write the roots in trigonometric form. }$

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

Given a =13, b = 11, and c = 6 , use Heron's Area Formula to find the area of triangle ABC . Round answer to two decimal places.

Given $C = 126 ^ { \circ } , a = 10.10$ , and $c = 18$ , 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 projection of $\mathbf { u }$ onto $\mathbf { v }$ if $\mathbf { u } = \langle 5 , - 4 \rangle , \mathbf { v } = \langle - 2,3 \rangle$ .

Find all solutions to the following equation. $x ^ { 3 } - 216 = 0$

Given vectors $\mathbf { u } = \langle - 3,1 \rangle$ and $\mathbf { v } = \langle 2 , - 2 \rangle$ , determine the quantity indicated below. $\mathbf { u } \cdot - 4 \mathbf { v }$

Find the standard form of the complex number shown below. Round answers to two decimal places. $2 \left( \cos 340 ^ { \circ } + i \sin 340 ^ { \circ } \right)$

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. $( 8 + 8 i ) ( 2 + 2 i )$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

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Limits and an Introduction to Calculus

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Exam 6: Additional Topics in Trigonometry

Use DeMoivre's Theorem to find the indicated power of the following complex number. (3+33i)6( 3 + 3 \sqrt { 3 } i ) ^ { 6 }(3+33​i)6

Find the trigonometric form of the complex number shown below. −6i- 6 i−6i

Find the trigonometric form of the complex number shown below. −7i- 7 i−7i

Find the absolute value of the complex number −4+6i- 4 + 6 i−4+6i .

Use Heron's area formula to find the area of the triangle pictured below, if a=10a = 10a=10 inches, b=12\mathrm { b } = 12b=12 inches, and c=4\mathrm { c } = 4c=4 inches.

Given a = 5 , b = 9 , and c =12 , use Heron's Area Formula to find the area of triangle ABC . Round answer to two decimal places.

Find the fourth roots of −12−32i. Write the roots in trigonometric form. \text { Find the fourth roots of }-\frac{1}{2}-\frac{\sqrt{3}}{2} i \text {. Write the roots in trigonometric form. } Find the fourth roots of −21​−23​​i. Write the roots in trigonometric form.

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

Given a =13, b = 11, and c = 6 , use Heron's Area Formula to find the area of triangle ABC . Round answer to two decimal places.

Given C=126∘,a=10.10C = 126 ^ { \circ } , a = 10.10C=126∘,a=10.10 , and c=18c = 18c=18 , 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.

In the figure below, a=8,b=12a = 8 , b = 12a=8,b=12 , and c=13c = 13c=13 . Use this information to solve the parallelogram for θ\thetaθ . The diagonals of the parallelogram are represented by ccc and ddd . Round answer to two decimal places.

In the following triangle, A=58∘,b=105A = 58 ^ { \circ } , b = 105A=58∘,b=105 , and a=126a = 126a=126 . Use the Law of Sines to find the measure of angle CCC , in degrees. Round your answer to two decimals.

Find the projection of u\mathbf { u }u onto v\mathbf { v }v if u=⟨5,−4⟩,v=⟨−2,3⟩\mathbf { u } = \langle 5 , - 4 \rangle , \mathbf { v } = \langle - 2,3 \rangleu=⟨5,−4⟩,v=⟨−2,3⟩ .

Find all solutions to the following equation. x3−216=0x ^ { 3 } - 216 = 0x3−216=0

Given vectors u=⟨−3,1⟩\mathbf { u } = \langle - 3,1 \rangleu=⟨−3,1⟩ and v=⟨2,−2⟩\mathbf { v } = \langle 2 , - 2 \ranglev=⟨2,−2⟩ , determine the quantity indicated below. u⋅−4v\mathbf { u } \cdot - 4 \mathbf { v }u⋅−4v

Find the standard form of the complex number shown below. Round answers to two decimal places. 2(cos⁡340∘+isin⁡340∘)2 \left( \cos 340 ^ { \circ } + i \sin 340 ^ { \circ } \right)2(cos340∘+isin340∘)

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. (8+8i)(2+2i)( 8 + 8 i ) ( 2 + 2 i )(8+8i)(2+2i)

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Review of Graphs, Equations, and Inequalities

Filters

Use DeMoivre's Theorem to find the indicated power of the following complex number. $( 3 + 3 \sqrt { 3 } i ) ^ { 6 }$

Find the trigonometric form of the complex number shown below. $- 6 i$

Find the trigonometric form of the complex number shown below. $- 7 i$

Find the absolute value of the complex number $- 4 + 6 i$ .

$\text { Find the fourth roots of }-\frac{1}{2}-\frac{\sqrt{3}}{2} i \text {. Write the roots in trigonometric form. }$

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

Given $C = 126 ^ { \circ } , a = 10.10$ , and $c = 18$ , 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 projection of $\mathbf { u }$ onto $\mathbf { v }$ if $\mathbf { u } = \langle 5 , - 4 \rangle , \mathbf { v } = \langle - 2,3 \rangle$ .

Find all solutions to the following equation. $x ^ { 3 } - 216 = 0$

Given vectors $\mathbf { u } = \langle - 3,1 \rangle$ and $\mathbf { v } = \langle 2 , - 2 \rangle$ , determine the quantity indicated below. $\mathbf { u } \cdot - 4 \mathbf { v }$

Find the standard form of the complex number shown below. Round answers to two decimal places. $2 \left( \cos 340 ^ { \circ } + i \sin 340 ^ { \circ } \right)$

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. $( 8 + 8 i ) ( 2 + 2 i )$