Question 1

Find the component form of the vector v.
-

Accepted Answer

A)  $\approx \langle 0.64,0.77 \rangle$ 
B)  $\approx \langle 37.28,44.43 \rangle$ 
C)  $\approx ( 44.43,37.28 )$ 
D)  $\approx \langle 41.01,41.01 \rangle$ 
A)  $\approx \langle 0.64,0.77 \rangle$ 
B)  $\approx \langle 37.28,44.43 \rangle$ 
C)  $\approx ( 44.43,37.28 )$ 
D)  $\approx \langle 41.01,41.01 \rangle$

Question 2

Find the rectangular coordinates of the point with the given polar coordinates.
-$( 5,4 \pi / 3 )$

Accepted Answer

A)  $\left( \frac { - 5 \sqrt { 3 } } { 2 } , \frac { - 5 } { 2 } \right)$ 
B)  $\left( \frac { 5 } { 2 } , \frac { 5 } { 2 } \right)$ 
C)  $\left( \frac { - 5 } { 2 } , \frac { - 5 \sqrt { 3 } } { 2 } \right)$ 
D)  $\left( \frac { - 5 } { 2 } , \frac { - 5 } { 2 } \right)$ 
A)  $\left( \frac { - 5 \sqrt { 3 } } { 2 } , \frac { - 5 } { 2 } \right)$ 
B)  $\left( \frac { 5 } { 2 } , \frac { 5 } { 2 } \right)$ 
C)  $\left( \frac { - 5 } { 2 } , \frac { - 5 \sqrt { 3 } } { 2 } \right)$ 
D)  $\left( \frac { - 5 } { 2 } , \frac { - 5 } { 2 } \right)$

Question 3

Find the product or quotient. Write the answer in standard form.
-$( 9 - 5 i ) ( 6 - 4 i )$

Accepted Answer

A)  $74 + 6 \mathrm { i }$ 
B)  $34 - 66 i$ 
C)  $20 \mathrm { i } ^ { 2 } - 66 \mathrm { i } + 54$ 
D)  $34 + 66 \mathrm { i }$ 
A)  $74 + 6 \mathrm { i }$ 
B)  $34 - 66 i$ 
C)  $20 \mathrm { i } ^ { 2 } - 66 \mathrm { i } + 54$ 
D)  $34 + 66 \mathrm { i }$

Question 4

Solve the problem.
-The locations, given in polar coordinates, of two ships are $\left( 4 \mathrm { mi } , 59 ^ { \circ } \right)$ and $\left( 7 \mathrm { mi } , 119 ^ { \circ } \right)$. Find the distance between the two ships.

Accepted Answer

A)  $\approx 3609.00 \mathrm { mi }$ 
B)  $\approx 8.06 \mathrm { mi }$ 
C)  $\approx 4.06 \mathrm { mi }$ 
D)  $\sqrt { 37 } \approx 6.08 \mathrm { mi }$ 
A)  $\approx 3609.00 \mathrm { mi }$ 
B)  $\approx 8.06 \mathrm { mi }$ 
C)  $\approx 4.06 \mathrm { mi }$ 
D)  $\sqrt { 37 } \approx 6.08 \mathrm { mi }$

Question 5

The polar coordinates of point P are given. Find all of its polar coordinates.
-$$\mathrm { P } = \left( 8,79 ^ { \circ } \right)$$

Accepted Answer

A)  $\left( 8,79 ^ { \circ } + 360 n ^ { \circ } \right)$ or $\left( 8 , - 79 ^ { \circ } + ( 2 n + 1 ) 180 ^ { \circ } \right)$ 
B)  $\left( 8,79 ^ { \circ } + 180 n ^ { \circ } \right)$ or $\left( - 8,79 ^ { \circ } + 180 n ^ { \circ } \right)$ 
C)  $\left( 8,79 ^ { \circ } + 360 n ^ { \circ } \right)$ or $\left( - 8,79 ^ { \circ } + 360 n ^ { \circ } \right)$ 
D)  $\left( 8,79 ^ { \circ } + 360 n ^ { \circ } \right)$ or $\left( - 8,79 ^ { \circ } + ( 2 n + 1 ) 180 ^ { \circ } \right)$ 
A)  $\left( 8,79 ^ { \circ } + 360 n ^ { \circ } \right)$ or $\left( 8 , - 79 ^ { \circ } + ( 2 n + 1 ) 180 ^ { \circ } \right)$ 
B)  $\left( 8,79 ^ { \circ } + 180 n ^ { \circ } \right)$ or $\left( - 8,79 ^ { \circ } + 180 n ^ { \circ } \right)$ 
C)  $\left( 8,79 ^ { \circ } + 360 n ^ { \circ } \right)$ or $\left( - 8,79 ^ { \circ } + 360 n ^ { \circ } \right)$ 
D)  $\left( 8,79 ^ { \circ } + 360 n ^ { \circ } \right)$ or $\left( - 8,79 ^ { \circ } + ( 2 n + 1 ) 180 ^ { \circ } \right)$

Question 6

Find the indicated roots. Write the answer in polar form.
-Fifth roots of $243 \left( \cos 200 ^ { \circ } + i \sin 200 ^ { \circ } \right)$

Accepted Answer

A)  $3 \left( \cos 200 ^ { \circ } + i \sin 200 ^ { \circ } \right) , 3 \left( \cos 290 ^ { \circ } + i \sin 290 ^ { \circ } \right) , 3 \left( \cos 380 ^ { \circ } + i \sin 380 ^ { \circ } \right) , 3 \left( \cos 470 ^ { \circ } + i \sin 470 ^ { \circ } \right)$, $3 \left( \cos 515 ^ { \circ } + i \sin 515 ^ { \circ } \right)$ 
B)  $3 \left( \cos 200 ^ { \circ } + i \sin 200 ^ { \circ } \right) , 3 \left( \cos 272 ^ { \circ } + i \sin 272 ^ { \circ } \right) , 3 \left( \cos 344 ^ { \circ } + i \sin 344 ^ { \circ } \right) , 3 \left( \cos 416 ^ { \circ } + i \sin 416 ^ { \circ } \right)$, $3 \left( \cos 488 ^ { \circ } + i \sin 488 ^ { \circ } \right)$ 
C)  $3 \left( \cos 40 ^ { \circ } + i \sin 40 ^ { \circ } \right) , 3 \left( \cos 130 ^ { \circ } + i \sin 130 ^ { \circ } \right) , 3 \left( \cos 220 ^ { \circ } + i \sin 220 ^ { \circ } \right) , 3 \left( \cos 310 ^ { \circ } + i \sin 310 ^ { \circ } \right)$, $3 \left( \cos 355 ^ { \circ } + i \sin 355 ^ { \circ } \right)$ 
D)  $3 \left( \cos 40 ^ { \circ } + i \sin 40 ^ { \circ } \right) , 3 \left( \cos 112 ^ { \circ } + i \sin 112 ^ { \circ } \right) , 3 \left( \cos 184 ^ { \circ } + i \sin 184 ^ { \circ } \right) , 3 \left( \cos 256 ^ { \circ } + i \sin 256 ^ { \circ } \right)$, $3 \left( \cos 328 ^ { \circ } + i \sin 328 ^ { \circ } \right)$ 
A)  $3 \left( \cos 200 ^ { \circ } + i \sin 200 ^ { \circ } \right) , 3 \left( \cos 290 ^ { \circ } + i \sin 290 ^ { \circ } \right) , 3 \left( \cos 380 ^ { \circ } + i \sin 380 ^ { \circ } \right) , 3 \left( \cos 470 ^ { \circ } + i \sin 470 ^ { \circ } \right)$, $3 \left( \cos 515 ^ { \circ } + i \sin 515 ^ { \circ } \right)$ 
B)  $3 \left( \cos 200 ^ { \circ } + i \sin 200 ^ { \circ } \right) , 3 \left( \cos 272 ^ { \circ } + i \sin 272 ^ { \circ } \right) , 3 \left( \cos 344 ^ { \circ } + i \sin 344 ^ { \circ } \right) , 3 \left( \cos 416 ^ { \circ } + i \sin 416 ^ { \circ } \right)$, $3 \left( \cos 488 ^ { \circ } + i \sin 488 ^ { \circ } \right)$ 
C)  $3 \left( \cos 40 ^ { \circ } + i \sin 40 ^ { \circ } \right) , 3 \left( \cos 130 ^ { \circ } + i \sin 130 ^ { \circ } \right) , 3 \left( \cos 220 ^ { \circ } + i \sin 220 ^ { \circ } \right) , 3 \left( \cos 310 ^ { \circ } + i \sin 310 ^ { \circ } \right)$, $3 \left( \cos 355 ^ { \circ } + i \sin 355 ^ { \circ } \right)$ 
D)  $3 \left( \cos 40 ^ { \circ } + i \sin 40 ^ { \circ } \right) , 3 \left( \cos 112 ^ { \circ } + i \sin 112 ^ { \circ } \right) , 3 \left( \cos 184 ^ { \circ } + i \sin 184 ^ { \circ } \right) , 3 \left( \cos 256 ^ { \circ } + i \sin 256 ^ { \circ } \right)$, $3 \left( \cos 328 ^ { \circ } + i \sin 328 ^ { \circ } \right)$

Question 7

Give a geometric interpretation of what happens to a complex number when it is squared. Describe what
happens to the argument and to the distance from the origin. Will the distance from the origin increase,
decrease, or remain the same?

Accepted Answer

The argument of the complex number will

Question 8

Find an equivalent equation in rectangular coordinates.
-$r ( \cos \theta - \sin \theta ) = 5$

Accepted Answer

A)  $x - y = 25$ 
B)  $x - y = 5$ 
C)  $\sqrt { x ^ { 2 } + y ^ { 2 } } ( x - y ) = 5$ 
D)  $x + y = 5$ 
A)  $x - y = 25$ 
B)  $x - y = 5$ 
C)  $\sqrt { x ^ { 2 } + y ^ { 2 } } ( x - y ) = 5$ 
D)  $x + y = 5$

Question 9

Find the component form and magnitude of the indicated vector.
-Given that $\mathrm { P } = ( - 4 , - 5 )$ and $\mathrm { Q } = ( - 2,4 )$, find the component form and magnitude of the vector $\overrightarrow { \mathrm { QP } }$.

Accepted Answer

A)  $\langle - 2 , - 9 \rangle , 85$ 
B)  $( - 2 , - 9 \rangle , \sqrt { 85 }$ 
C)  $( 2,9 ) , \sqrt { 85 }$ 
D)  $\langle - 6 , - 9 \rangle , \sqrt { 75 }$ 
A)  $\langle - 2 , - 9 \rangle , 85$ 
B)  $( - 2 , - 9 \rangle , \sqrt { 85 }$ 
C)  $( 2,9 ) , \sqrt { 85 }$ 
D)  $\langle - 6 , - 9 \rangle , \sqrt { 75 }$

Question 10

A directed line segment from $( 3,7 )$ to $( x , y )$ is equivalent to a directed line segment from $( - 4,6 )$ to the point $P$. Write an expression for the coordinates of the point $P$.

Accepted Answer

A)  $( x + 1 , y + 1 )$ 
B)  $( x + 1 , y - 1 )$ 
C)  $( x - 7 , y - 1 )$ 
D)  $( x - 7 , y + 1 )$ 
A)  $( x + 1 , y + 1 )$ 
B)  $( x + 1 , y - 1 )$ 
C)  $( x - 7 , y - 1 )$ 
D)  $( x - 7 , y + 1 )$

Question 11

Solve the problem.
-A rectangle has its center at the origin. It has two sides of length 4 a which are parallel to the horizontal axis and two sides of length 2 a parallel to the vertical axis. Find polar coordinates of the vertices.

Accepted Answer

A)  $\approx \left( 2 a \sqrt { 5 } , 63.4 ^ { \circ } \right) , \left( 2 a \sqrt { 5 } , 116.6 ^ { \circ } \right) , \left( 2 a \sqrt { 5 } , 243.4 ^ { \circ } \right) , \left( 2 a \sqrt { 5 } , 296.6 ^ { \circ } \right)$ 
B)  $\approx \left( a \sqrt { 5 } , 30 ^ { \circ } \right) , \left( a \sqrt { 5 } , 150 ^ { \circ } \right) , \left( a \sqrt { 5 } , 210 ^ { \circ } \right) , \left( a \sqrt { 5 } , 330 ^ { \circ } \right)$ 
C)  $\approx \left( a \sqrt { 5 } , 26.6 ^ { \circ } \right) , \left( a \sqrt { 5 } , 153.4 ^ { \circ } \right) , \left( a \sqrt { 5 } , 206.6 ^ { \circ } \right) , \left( a \sqrt { 5 } , 333.4 ^ { \circ } \right)$ 
D)  $\approx \left( 2 \mathrm { a } \sqrt { 5 } , 26.6 ^ { \circ } \right) , \left( 2 \mathrm { a } \sqrt { 5 } , 153.4 ^ { \circ } \right) , \left( 2 \mathrm { a } \sqrt { 5 } , 206.6 ^ { \circ } \right) , \left( 2 \mathrm { a } \sqrt { 5 } , 333.4 ^ { \circ } \right)$ 
A)  $\approx \left( 2 a \sqrt { 5 } , 63.4 ^ { \circ } \right) , \left( 2 a \sqrt { 5 } , 116.6 ^ { \circ } \right) , \left( 2 a \sqrt { 5 } , 243.4 ^ { \circ } \right) , \left( 2 a \sqrt { 5 } , 296.6 ^ { \circ } \right)$ 
B)  $\approx \left( a \sqrt { 5 } , 30 ^ { \circ } \right) , \left( a \sqrt { 5 } , 150 ^ { \circ } \right) , \left( a \sqrt { 5 } , 210 ^ { \circ } \right) , \left( a \sqrt { 5 } , 330 ^ { \circ } \right)$ 
C)  $\approx \left( a \sqrt { 5 } , 26.6 ^ { \circ } \right) , \left( a \sqrt { 5 } , 153.4 ^ { \circ } \right) , \left( a \sqrt { 5 } , 206.6 ^ { \circ } \right) , \left( a \sqrt { 5 } , 333.4 ^ { \circ } \right)$ 
D)  $\approx \left( 2 \mathrm { a } \sqrt { 5 } , 26.6 ^ { \circ } \right) , \left( 2 \mathrm { a } \sqrt { 5 } , 153.4 ^ { \circ } \right) , \left( 2 \mathrm { a } \sqrt { 5 } , 206.6 ^ { \circ } \right) , \left( 2 \mathrm { a } \sqrt { 5 } , 333.4 ^ { \circ } \right)$

Question 12

Determine if the graph is symmetric about the x-axis, the y-axis, or the origin.
-$$\mathrm { r } = - 3 + 4 \cos \theta$$

Accepted Answer

A)  $y$-axis only 
B)  Origin only 
C)  $x$-axis only 
D)  No symmetry 
A)  $y$-axis only 
B)  Origin only 
C)  $x$-axis only 
D)  No symmetry

Question 13

Determine two pairs of polar coordinates for the point with 0° ≤θ < 360°.
-$( - 4,0 )$

Accepted Answer

A)  $\left( 4,270 ^ { \circ } \right) , \left( - 4,90 ^ { \circ } \right)$ 
B)  $\left( 4,90 ^ { \circ } \right) , \left( - 4,270 ^ { \circ } \right)$ 
C)  $\left( 4,180 ^ { \circ } \right) , \left( - 4,0 ^ { \circ } \right)$ 
D)  $\left( 4,0 ^ { \circ } \right) , \left( - 4,180 ^ { \circ } \right)$ 
A)  $\left( 4,270 ^ { \circ } \right) , \left( - 4,90 ^ { \circ } \right)$ 
B)  $\left( 4,90 ^ { \circ } \right) , \left( - 4,270 ^ { \circ } \right)$ 
C)  $\left( 4,180 ^ { \circ } \right) , \left( - 4,0 ^ { \circ } \right)$ 
D)  $\left( 4,0 ^ { \circ } \right) , \left( - 4,180 ^ { \circ } \right)$

Question 14

Find the rectangular coordinates of the point with the given polar coordinates.
-

Accepted Answer

A)  $( \sqrt { 3 } , - 1 )$ 
B)  $( - \sqrt { 3 } , 1 )$ 
C)  $( 3 , - 2 )$ 
D)  $( \sqrt { 3 } / 2 , - 0.5 )$ 
A)  $( \sqrt { 3 } , - 1 )$ 
B)  $( - \sqrt { 3 } , 1 )$ 
C)  $( 3 , - 2 )$ 
D)  $( \sqrt { 3 } / 2 , - 0.5 )$

Question 15

Express the indicated roots of unity in standard form a + bi.
-Fourth roots of unity

Accepted Answer

A)  $ 1+0 \mathrm{i} $
$$
\begin{array}{l}
\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i \
1-i \
0-i
\end{array}
$$ 
B)  $ 1+0 i $
$ \cos \frac{\pi}{2}+i \sin \frac{\pi}{2}=0+i $
$ \cos \pi+i \sin \pi=-1+0 i $
$ \cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}=0-i $ 
C)  $ 1+0 i $
$ \cos \frac{\pi}{4}+i \sin \frac{\pi}{4}=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i $
$ \cos \frac{\pi}{2}+i \sin \frac{\pi}{2}=0+i $
$ \cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}=-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i $ 
D)  $ 1+0 \mathrm{i} $
$$
\begin{array}{l}
\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}=0+i \
\cos \pi+i \sin \pi=-1+0 i \
1-i
\end{array}
$$ 
A)  $ 1+0 \mathrm{i} $
$$
\begin{array}{l}
\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i \
1-i \
0-i
\end{array}
$$ 
B)  $ 1+0 i $
$ \cos \frac{\pi}{2}+i \sin \frac{\pi}{2}=0+i $
$ \cos \pi+i \sin \pi=-1+0 i $
$ \cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}=0-i $ 
C)  $ 1+0 i $
$ \cos \frac{\pi}{4}+i \sin \frac{\pi}{4}=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i $
$ \cos \frac{\pi}{2}+i \sin \frac{\pi}{2}=0+i $
$ \cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}=-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i $ 
D)  $ 1+0 \mathrm{i} $
$$
\begin{array}{l}
\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}=0+i \
\cos \pi+i \sin \pi=-1+0 i \
1-i
\end{array}
$$

Question 16

Find an equivalent equation in polar coordinates.
-$$x ^ { 2 } - y ^ { 2 } = 4$$

Accepted Answer

A)  $r \cos 2 \theta = 4$ 
B)  $\cos ^ { 2 } \theta - \sin ^ { 2 } \theta = 4$ 
C)  $\cos ^ { 2 } \theta - \sin ^ { 2 } \theta = 4 r$ 
D)  $r ^ { 2 } \cos 2 \theta = 4$ 
A)  $r \cos 2 \theta = 4$ 
B)  $\cos ^ { 2 } \theta - \sin ^ { 2 } \theta = 4$ 
C)  $\cos ^ { 2 } \theta - \sin ^ { 2 } \theta = 4 r$ 
D)  $r ^ { 2 } \cos 2 \theta = 4$

Question 17

Determine whether the vectors u and v are parallel, orthogonal, or neither.
-$$\mathbf { u } = \langle 9,7 \rangle , \mathbf { v } = \langle - 9,8 \rangle$$

Accepted Answer

A)  Orthogonal 
B)  Neither 
C)  Parallel 
A)  Orthogonal 
B)  Neither 
C)  Parallel

Question 18

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

Accepted Answer

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

Question 19

Find the component form of the vector v.
-

Accepted Answer

A)  $\approx ( - 7.78,7.78 )$ 
B)  $\approx \langle - 8.43,7.07 \rangle$ 
C)  $\approx ( - 0.77,0.64 )$ 
D)  $\approx ( - 7.07,8.43 )$ 
A)  $\approx ( - 7.78,7.78 )$ 
B)  $\approx \langle - 8.43,7.07 \rangle$ 
C)  $\approx ( - 0.77,0.64 )$ 
D)  $\approx ( - 7.07,8.43 )$

Question 20

Analyze the graph of the given polar curve. Include the following information: If possible, describe the shape of the
graph (circle, rose curve, limacon, etc.), and state the domain, range, and maximum r-value of the graph. State whether the
graph is continuous and whether it is bounded. Describe any symmetry that the graph has. Give the equations of any
asymptotes or state that the graph has no asymptotes.
-$$r = 6 - 6 \cos \theta$$

Accepted Answer

Limacon curve with cardioid sh

Find the component form of the vector v. -

Find the rectangular coordinates of the point with the given polar coordinates. - $( 5,4 \pi / 3 )$

Find the product or quotient. Write the answer in standard form. - $( 9 - 5 i ) ( 6 - 4 i )$

Solve the problem. -The locations, given in polar coordinates, of two ships are $\left( 4 \mathrm { mi } , 59 ^ { \circ } \right)$ and $\left( 7 \mathrm { mi } , 119 ^ { \circ } \right)$ . Find the distance between the two ships.

The polar coordinates of point P are given. Find all of its polar coordinates. - $\mathrm { P } = \left( 8,79 ^ { \circ } \right)$

Find the indicated roots. Write the answer in polar form. -Fifth roots of $243 \left( \cos 200 ^ { \circ } + i \sin 200 ^ { \circ } \right)$

Give a geometric interpretation of what happens to a complex number when it is squared. Describe what happens to the argument and to the distance from the origin. Will the distance from the origin increase, decrease, or remain the same?

Find an equivalent equation in rectangular coordinates. - $r ( \cos \theta - \sin \theta ) = 5$

Find the component form and magnitude of the indicated vector. -Given that $\mathrm { P } = ( - 4 , - 5 )$ and $\mathrm { Q } = ( - 2,4 )$ , find the component form and magnitude of the vector $\overrightarrow { \mathrm { QP } }$ .

A directed line segment from $( 3,7 )$ to $( x , y )$ is equivalent to a directed line segment from $( - 4,6 )$ to the point $P$ . Write an expression for the coordinates of the point $P$ .

Solve the problem. -A rectangle has its center at the origin. It has two sides of length 4 a which are parallel to the horizontal axis and two sides of length 2 a parallel to the vertical axis. Find polar coordinates of the vertices.

Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. - $\mathrm { r } = - 3 + 4 \cos \theta$

Determine two pairs of polar coordinates for the point with 0° ≤θ < 360°. - $( - 4,0 )$

Find the rectangular coordinates of the point with the given polar coordinates. -

Express the indicated roots of unity in standard form a + bi. -Fourth roots of unity

Find an equivalent equation in polar coordinates. - $x ^ { 2 } - y ^ { 2 } = 4$

Determine whether the vectors u and v are parallel, orthogonal, or neither. - $\mathbf { u } = \langle 9,7 \rangle , \mathbf { v } = \langle - 9,8 \rangle$

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

Find the component form of the vector v. -

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An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

Filters

Exam 6: Applications of Trigonometry

Find the component form of the vector v. -

Find the rectangular coordinates of the point with the given polar coordinates. - (5,4π/3)( 5,4 \pi / 3 )(5,4π/3)

Find the product or quotient. Write the answer in standard form. - (9−5i)(6−4i)( 9 - 5 i ) ( 6 - 4 i )(9−5i)(6−4i)

Solve the problem. -The locations, given in polar coordinates, of two ships are (4mi,59∘)\left( 4 \mathrm { mi } , 59 ^ { \circ } \right)(4mi,59∘) and (7mi,119∘)\left( 7 \mathrm { mi } , 119 ^ { \circ } \right)(7mi,119∘) . Find the distance between the two ships.

The polar coordinates of point P are given. Find all of its polar coordinates. - P=(8,79∘)\mathrm { P } = \left( 8,79 ^ { \circ } \right)P=(8,79∘)

Find the indicated roots. Write the answer in polar form. -Fifth roots of 243(cos⁡200∘+isin⁡200∘)243 \left( \cos 200 ^ { \circ } + i \sin 200 ^ { \circ } \right)243(cos200∘+isin200∘)

Give a geometric interpretation of what happens to a complex number when it is squared. Describe what happens to the argument and to the distance from the origin. Will the distance from the origin increase, decrease, or remain the same?

Find an equivalent equation in rectangular coordinates. - r(cos⁡θ−sin⁡θ)=5r ( \cos \theta - \sin \theta ) = 5r(cosθ−sinθ)=5

Find the component form and magnitude of the indicated vector. -Given that P=(−4,−5)\mathrm { P } = ( - 4 , - 5 )P=(−4,−5) and Q=(−2,4)\mathrm { Q } = ( - 2,4 )Q=(−2,4) , find the component form and magnitude of the vector QP→\overrightarrow { \mathrm { QP } }QP​ .

A directed line segment from (3,7)( 3,7 )(3,7) to (x,y)( x , y )(x,y) is equivalent to a directed line segment from (−4,6)( - 4,6 )(−4,6) to the point PPP . Write an expression for the coordinates of the point PPP .

Solve the problem. -A rectangle has its center at the origin. It has two sides of length 4 a which are parallel to the horizontal axis and two sides of length 2 a parallel to the vertical axis. Find polar coordinates of the vertices.

Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. - r=−3+4cos⁡θ\mathrm { r } = - 3 + 4 \cos \thetar=−3+4cosθ

Determine two pairs of polar coordinates for the point with 0° ≤θ < 360°. - (−4,0)( - 4,0 )(−4,0)

Find the rectangular coordinates of the point with the given polar coordinates. -

Express the indicated roots of unity in standard form a + bi. -Fourth roots of unity

Find an equivalent equation in polar coordinates. - x2−y2=4x ^ { 2 } - y ^ { 2 } = 4x2−y2=4

Determine whether the vectors u and v are parallel, orthogonal, or neither. - u=⟨9,7⟩,v=⟨−9,8⟩\mathbf { u } = \langle 9,7 \rangle , \mathbf { v } = \langle - 9,8 \rangleu=⟨9,7⟩,v=⟨−9,8⟩

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

Find the component form of the vector v. -

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Systems and Matrices

Analytic Geometry in Two and Three Dimensions

Discrete Mathematics

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

Filters

Find the rectangular coordinates of the point with the given polar coordinates. - $( 5,4 \pi / 3 )$

Find the product or quotient. Write the answer in standard form. - $( 9 - 5 i ) ( 6 - 4 i )$

Solve the problem. -The locations, given in polar coordinates, of two ships are $\left( 4 \mathrm { mi } , 59 ^ { \circ } \right)$ and $\left( 7 \mathrm { mi } , 119 ^ { \circ } \right)$ . Find the distance between the two ships.

The polar coordinates of point P are given. Find all of its polar coordinates. - $\mathrm { P } = \left( 8,79 ^ { \circ } \right)$

Find the indicated roots. Write the answer in polar form. -Fifth roots of $243 \left( \cos 200 ^ { \circ } + i \sin 200 ^ { \circ } \right)$

Find an equivalent equation in rectangular coordinates. - $r ( \cos \theta - \sin \theta ) = 5$

Find the component form and magnitude of the indicated vector. -Given that $\mathrm { P } = ( - 4 , - 5 )$ and $\mathrm { Q } = ( - 2,4 )$ , find the component form and magnitude of the vector $\overrightarrow { \mathrm { QP } }$ .

A directed line segment from $( 3,7 )$ to $( x , y )$ is equivalent to a directed line segment from $( - 4,6 )$ to the point $P$ . Write an expression for the coordinates of the point $P$ .

Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. - $\mathrm { r } = - 3 + 4 \cos \theta$

Determine two pairs of polar coordinates for the point with 0° ≤θ < 360°. - $( - 4,0 )$

Find an equivalent equation in polar coordinates. - $x ^ { 2 } - y ^ { 2 } = 4$

Determine whether the vectors u and v are parallel, orthogonal, or neither. - $\mathbf { u } = \langle 9,7 \rangle , \mathbf { v } = \langle - 9,8 \rangle$

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