Question 1

The rectangular coordinates of a point are given. Express the point in polar coordinates with r ≥ 0 and 0° ≤ θ < 360°.
-$$( \sqrt { 3 } , - \sqrt { 3 } )$$

Accepted Answer

A)  $\left( 6,225 ^ { \circ } \right)$ 
B)  $\left( \sqrt { 3 } , 135 ^ { \circ } \right)$ 
C)  $\left( \sqrt { 6 } , 225 ^ { \circ } \right)$ 
D)  $\left( \sqrt { 6 } , 315 ^ { \circ } \right)$ 
A)  $\left( 6,225 ^ { \circ } \right)$ 
B)  $\left( \sqrt { 3 } , 135 ^ { \circ } \right)$ 
C)  $\left( \sqrt { 6 } , 225 ^ { \circ } \right)$ 
D)  $\left( \sqrt { 6 } , 315 ^ { \circ } \right)$

Question 2

Find an equivalent equation in rectangular coordinates.
-$$r = \frac { 6 } { 2 \sin \theta + 3 \cos \theta }$$

Accepted Answer

A)  $2 y + 3 x = \frac { 6 } { \sqrt { x ^ { 2 } + y ^ { 2 } } }$ 
B)  $2 x + 3 y = 6$ 
C)  $2 y + 3 x = 6$ 
D)  $2 x - 3 y = 36$ 
A)  $2 y + 3 x = \frac { 6 } { \sqrt { x ^ { 2 } + y ^ { 2 } } }$ 
B)  $2 x + 3 y = 6$ 
C)  $2 y + 3 x = 6$ 
D)  $2 x - 3 y = 36$

Question 3

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

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 4

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal
places.
-$\left[ 5 \left( \cos 20 ^ { \circ } + i \sin 20 ^ { \circ } \right) \right] \left[ 4 \left( \cos 10 ^ { \circ } + i \sin 10 ^ { \circ } \right) \right]$

Accepted Answer

A)  $10.0000 + 17.3205 \mathrm { i }$ 
B)  $0.8660 + 0.5000 \mathrm { i }$ 
C)  $0.5000 + 0.8660 \mathrm { i }$ 
D)  $17.3205 + 10.0000 \mathrm { i }$ 
A)  $10.0000 + 17.3205 \mathrm { i }$ 
B)  $0.8660 + 0.5000 \mathrm { i }$ 
C)  $0.5000 + 0.8660 \mathrm { i }$ 
D)  $17.3205 + 10.0000 \mathrm { i }$

Question 5

Find all solutions of the equation. Leave answers in trigonometric form.
-$$x ^ { 2 } - 36 = 0$$

Accepted Answer

A)  $\left\{ 6 \sqrt { 2 } \operatorname { cis } 45 ^ { \circ } , 6 \sqrt { 2 } \operatorname { cis } 225 ^ { \circ } \right\}$ 
B)  $\left\{ 6 \sqrt { 2 } \operatorname { cis } 235 ^ { \circ } , 6 \sqrt { 2 } \operatorname { cis } 135 ^ { \circ } \right\}$ 
C)  $\left\{ 6 \operatorname { cis } 180 ^ { \circ } , 6 \operatorname { cis } 0 ^ { \circ } \right\}$ 
D)  $\left\{ 6 \operatorname { cis } 90 ^ { \circ } , 6 \operatorname { cis } 270 ^ { \circ } \right\}$ 
A)  $\left\{ 6 \sqrt { 2 } \operatorname { cis } 45 ^ { \circ } , 6 \sqrt { 2 } \operatorname { cis } 225 ^ { \circ } \right\}$ 
B)  $\left\{ 6 \sqrt { 2 } \operatorname { cis } 235 ^ { \circ } , 6 \sqrt { 2 } \operatorname { cis } 135 ^ { \circ } \right\}$ 
C)  $\left\{ 6 \operatorname { cis } 180 ^ { \circ } , 6 \operatorname { cis } 0 ^ { \circ } \right\}$ 
D)  $\left\{ 6 \operatorname { cis } 90 ^ { \circ } , 6 \operatorname { cis } 270 ^ { \circ } \right\}$

Question 6

Draw a sketch to represent the vector. Refer to the vectors pictured here.  
-$$a + d$$

Accepted Answer

A)   
B)   
A)   
B)

Question 7

Find the indicated vector.
-Let $\mathbf { a } = 3 \mathbf { i } , \mathbf { b } = \mathbf { i } + \mathbf { j }$. Find $3 \mathbf { a } + \mathbf { b }$.

Accepted Answer

A)  $\mathbf { i } + 10 \mathbf { j }$ 
B)  $10 \mathbf { i } + 3 \mathbf { j }$ 
C)  $11 \mathrm { i } + \mathbf { j }$ 
D)  $10 \mathbf { i } + \mathbf { j }$ 
A)  $\mathbf { i } + 10 \mathbf { j }$ 
B)  $10 \mathbf { i } + 3 \mathbf { j }$ 
C)  $11 \mathrm { i } + \mathbf { j }$ 
D)  $10 \mathbf { i } + \mathbf { j }$

Question 8

Give the rectangular coordinates for the point.
-$\left( - 9 , - \frac { \pi } { 3 } \right)$

Accepted Answer

A)  $\left( - \frac { 9 } { 2 } , \frac { - 9 \sqrt { 3 } } { 2 } \right)$ 
B)  $\left( \frac { 9 } { 2 } , \frac { 9 \sqrt { 3 } } { 2 } \right)$ 
C)  $\left( - \frac { 9 } { 2 } , \frac { 9 \sqrt { 3 } } { 2 } \right)$ 
D)  $\left( \frac { 9 } { 2 } , \frac { - 9 \sqrt { 3 } } { 2 } \right)$ 
A)  $\left( - \frac { 9 } { 2 } , \frac { - 9 \sqrt { 3 } } { 2 } \right)$ 
B)  $\left( \frac { 9 } { 2 } , \frac { 9 \sqrt { 3 } } { 2 } \right)$ 
C)  $\left( - \frac { 9 } { 2 } , \frac { 9 \sqrt { 3 } } { 2 } \right)$ 
D)  $\left( \frac { 9 } { 2 } , \frac { - 9 \sqrt { 3 } } { 2 } \right)$

Question 9

Find the product. Write the product in rectangular form, using exact values.
-[8( $\left. \left. \cos 210 ^ { \circ } + i \sin 210 ^ { \circ } \right) \right] \left[ 6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right) \right]$

Accepted Answer

A)  $- 12 + 12 \sqrt { 3 } \mathrm { i }$ 
B)  $- 48$ 
C)  $12 \sqrt { 3 } + 12 i$ 
D)  $24 \mathrm { i }$ 
A)  $- 12 + 12 \sqrt { 3 } \mathrm { i }$ 
B)  $- 48$ 
C)  $12 \sqrt { 3 } + 12 i$ 
D)  $24 \mathrm { i }$

Question 10

Solve the problem.
-Two airplanes leave an airport at the same time, one going northwest $$\text { (bearing } 135 ^ { \circ } \text { ) }$$ at 415 mph and the other going east at 329 mph. How far apart are the planes after 2 hours (to the nearest
Mile)?

Accepted Answer

A)  1228 mi 
B)  688 mi 
C)  1376 mi 
D)  1147 mi 
A)  1228 mi 
B)  688 mi 
C)  1376 mi 
D)  1147 mi

Question 11

Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as
-$$\begin{array} { l } 
A B = 329 y d \
A C = 187 y d \
B C = 163 y d
\end{array}$$

Accepted Answer

A)  $\mathrm { A } = 18.6 ^ { \circ } , \mathrm { B } = 21.4 ^ { \circ } , \mathrm { C } = 140 ^ { \circ }$ 
B)  No triangle satisfies the given conditions 
C)  $\mathrm { A } = 18.6 ^ { \circ } , \mathrm { B } = 10.7 ^ { \circ } , \mathrm { C } = 150.7 ^ { \circ }$ 
D)  $\mathrm { A } = 21.4 ^ { \circ } , \mathrm { B } = 18.6 ^ { \circ } , \mathrm { C } = 140 ^ { \circ }$ 
A)  $\mathrm { A } = 18.6 ^ { \circ } , \mathrm { B } = 21.4 ^ { \circ } , \mathrm { C } = 140 ^ { \circ }$ 
B)  No triangle satisfies the given conditions 
C)  $\mathrm { A } = 18.6 ^ { \circ } , \mathrm { B } = 10.7 ^ { \circ } , \mathrm { C } = 150.7 ^ { \circ }$ 
D)  $\mathrm { A } = 21.4 ^ { \circ } , \mathrm { B } = 18.6 ^ { \circ } , \mathrm { C } = 140 ^ { \circ }$

Question 12

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

Accepted Answer

A)  $r \sin \theta = 1$ 
B)  $r = 2 \sin \theta$ 
C)  $r = 2 \cos \theta$ 
D)  $\mathrm { r } = 1$ 
A)  $r \sin \theta = 1$ 
B)  $r = 2 \sin \theta$ 
C)  $r = 2 \cos \theta$ 
D)  $\mathrm { r } = 1$

Question 13

Determine whether the pair of vectors is orthogonal.
-$$3 \mathbf { i } - 3 \mathbf { j } , - 8 \mathbf { i } - 8 \mathbf { j }$$

Accepted Answer

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

Question 14

The rectangular coordinates of a point are given. Express the point in polar coordinates with r ≥ 0 and 0° ≤ θ < 360°.
-$$\left( \frac { 1 } { 4 } , \frac { - \sqrt { 3 } } { 4 } \right)$$

Accepted Answer

A)  $\left( \frac { 1 } { 2 } , 30 ^ { \circ } \right)$ 
B)  $\left( \frac { 1 } { 4 } , 150 ^ { \circ } \right)$ 
C)  $\left( \frac { 1 } { 2 } , 60 ^ { \circ } \right)$ 
D)  $\left( \frac { 1 } { 2 } , 300 ^ { \circ } \right)$ 
A)  $\left( \frac { 1 } { 2 } , 30 ^ { \circ } \right)$ 
B)  $\left( \frac { 1 } { 4 } , 150 ^ { \circ } \right)$ 
C)  $\left( \frac { 1 } { 2 } , 60 ^ { \circ } \right)$ 
D)  $\left( \frac { 1 } { 2 } , 300 ^ { \circ } \right)$

Question 15

Solve the problem.
-Tia is going to stain her triangular concrete patio. The patio is approximately 15 feet by 18 feet by 10 feet. If one can of stain covers $$15 \mathrm { ft } ^ { 2 }$$ how many cans of stain will she need?

Accepted Answer

A)  66 cans 
B)  2 cans 
C)  90 cans 
D)  5 cans 
A)  66 cans 
B)  2 cans 
C)  90 cans 
D)  5 cans

Question 16

Sketch the vectors u and w with angle θ between them and sketch the resultant.
-$$| \mathbf { u } | = 20 , | \mathbf { w } | = 50 , \theta = 80 ^ { \circ }$$

Accepted Answer

A) 
  
B) 
  
A) 
  
B)

Question 17

For the given rectangular equation, give its equivalent polar equation.
-$$6 x - y = 2$$

Accepted Answer

A)  $r = \frac { 6 } { 2 \cos \theta - \sin \theta }$ 
B)  $r = \frac { 2 } { 6 \cos \theta + \sin \theta }$ 
C)  $r = \frac { 2 } { 6 \cos \theta - \sin \theta }$ 
D)  $r = \frac { 2 } { 6 \sin \theta - \cos \theta }$ 
A)  $r = \frac { 6 } { 2 \cos \theta - \sin \theta }$ 
B)  $r = \frac { 2 } { 6 \cos \theta + \sin \theta }$ 
C)  $r = \frac { 2 } { 6 \cos \theta - \sin \theta }$ 
D)  $r = \frac { 2 } { 6 \sin \theta - \cos \theta }$

Question 18

Solve the problem.
-A projectile is fired with an initial velocity of 600 feet per second at an angle of $45 ^ { \circ }$ with the horizontal. To the nearest 10 feet, find the horizontal distance covered by the projectile.

Accepted Answer

A)  $11,990 \mathrm { ft }$ 
B)  $11,250 \mathrm { ft }$ 
C)  $11,760 \mathrm { ft }$ 
D)  $10,910 \mathrm { ft }$ 
A)  $11,990 \mathrm { ft }$ 
B)  $11,250 \mathrm { ft }$ 
C)  $11,760 \mathrm { ft }$ 
D)  $10,910 \mathrm { ft }$

Question 19

Find a rectangular equation for the plane curve defined by the parametric equations.
-$$x = \sqrt { t } , y = 2 t + 5$$

Accepted Answer

A)  $y = 2 x ^ { 2 } - 5$ 
B)  $y = 2 \sqrt { x } + 5 , x \geq 0$ 
C)  $y = 2 \sqrt { x } - 5 , x \geq 0$ 
D)  $y = 2 x ^ { 2 } + 5$ 
A)  $y = 2 x ^ { 2 } - 5$ 
B)  $y = 2 \sqrt { x } + 5 , x \geq 0$ 
C)  $y = 2 \sqrt { x } - 5 , x \geq 0$ 
D)  $y = 2 x ^ { 2 } + 5$

Question 20

Give two parametric representations for the equation of the parabola.
-$$y = ( x - 4 ) ^ { 2 } - 10$$

Accepted Answer

A)  $x = t , y = ( t - 4 ) ^ { 2 } - 10$ for $t$ in $( - \infty , \infty ) ; x = t - 4 , y = t ^ { 2 } - 10$ for $t$ in $( - \infty , \infty )$ 
B)  $x = t , y = ( t - 4 ) ^ { 2 } - 10$ for $t$ in $( - \infty , \infty ) ; x = t + 4 , y = t ^ { 2 } - 10$ for $t$ in $( - \infty , \infty )$ 
C)  $x = t , y = ( t + 4 ) ^ { 2 } - 10$ for $t$ in $( - \infty , \infty ) ; x = t + 4 , y = t ^ { 2 } - 10$ for $t$ in $( - \infty , \infty )$ 
D)  $x = t , y = ( t + 4 ) ^ { 2 } - 10$ for $t$ in $( - \infty , \infty ) ; x = t - 4 , y = t ^ { 2 } - 10$ for $t$ in $( - \infty , \infty )$ 
A)  $x = t , y = ( t - 4 ) ^ { 2 } - 10$ for $t$ in $( - \infty , \infty ) ; x = t - 4 , y = t ^ { 2 } - 10$ for $t$ in $( - \infty , \infty )$ 
B)  $x = t , y = ( t - 4 ) ^ { 2 } - 10$ for $t$ in $( - \infty , \infty ) ; x = t + 4 , y = t ^ { 2 } - 10$ for $t$ in $( - \infty , \infty )$ 
C)  $x = t , y = ( t + 4 ) ^ { 2 } - 10$ for $t$ in $( - \infty , \infty ) ; x = t + 4 , y = t ^ { 2 } - 10$ for $t$ in $( - \infty , \infty )$ 
D)  $x = t , y = ( t + 4 ) ^ { 2 } - 10$ for $t$ in $( - \infty , \infty ) ; x = t - 4 , y = t ^ { 2 } - 10$ for $t$ in $( - \infty , \infty )$

The rectangular coordinates of a point are given. Express the point in polar coordinates with r ≥ 0 and 0° ≤ θ < 360°. - $( \sqrt { 3 } , - \sqrt { 3 } )$

Find an equivalent equation in rectangular coordinates. - $r = \frac { 6 } { 2 \sin \theta + 3 \cos \theta }$

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

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal places. - $\left[ 5 \left( \cos 20 ^ { \circ } + i \sin 20 ^ { \circ } \right) \right] \left[ 4 \left( \cos 10 ^ { \circ } + i \sin 10 ^ { \circ } \right) \right]$

Find all solutions of the equation. Leave answers in trigonometric form. - $x ^ { 2 } - 36 = 0$

Draw a sketch to represent the vector. Refer to the vectors pictured here. - $a + d$

Find the indicated vector. -Let $\mathbf { a } = 3 \mathbf { i } , \mathbf { b } = \mathbf { i } + \mathbf { j }$ . Find $3 \mathbf { a } + \mathbf { b }$ .

Give the rectangular coordinates for the point. - $\left( - 9 , - \frac { \pi } { 3 } \right)$

Find the product. Write the product in rectangular form, using exact values. -[8( $\left. \left. \cos 210 ^ { \circ } + i \sin 210 ^ { \circ } \right) \right] \left[ 6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right) \right]$

Solve the problem. -Two airplanes leave an airport at the same time, one going northwest $\text { (bearing } 135 ^ { \circ } \text { ) }$ at 415 mph and the other going east at 329 mph. How far apart are the planes after 2 hours (to the nearest Mile)?

Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as - AB=329yd AC=187yd BC=163yd

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

Determine whether the pair of vectors is orthogonal. - $3 \mathbf { i } - 3 \mathbf { j } , - 8 \mathbf { i } - 8 \mathbf { j }$

The rectangular coordinates of a point are given. Express the point in polar coordinates with r ≥ 0 and 0° ≤ θ < 360°. - $\left( \frac { 1 } { 4 } , \frac { - \sqrt { 3 } } { 4 } \right)$

Solve the problem. -Tia is going to stain her triangular concrete patio. The patio is approximately 15 feet by 18 feet by 10 feet. If one can of stain covers $15 \mathrm { ft } ^ { 2 }$ how many cans of stain will she need?

Sketch the vectors u and w with angle θ between them and sketch the resultant. - $| \mathbf { u } | = 20 , | \mathbf { w } | = 50 , \theta = 80 ^ { \circ }$

For the given rectangular equation, give its equivalent polar equation. - $6 x - y = 2$

Solve the problem. -A projectile is fired with an initial velocity of 600 feet per second at an angle of $45 ^ { \circ }$ with the horizontal. To the nearest 10 feet, find the horizontal distance covered by the projectile.

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = \sqrt { t } , y = 2 t + 5$

Give two parametric representations for the equation of the parabola. - $y = ( x - 4 ) ^ { 2 } - 10$

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

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Exam 8: Applications of Trigonometry

The rectangular coordinates of a point are given. Express the point in polar coordinates with r ≥ 0 and 0° ≤ θ < 360°. - (3,−3)( \sqrt { 3 } , - \sqrt { 3 } )(3​,−3​)

Find an equivalent equation in rectangular coordinates. - r=62sin⁡θ+3cos⁡θr = \frac { 6 } { 2 \sin \theta + 3 \cos \theta }r=2sinθ+3cosθ6​

Find the given power. Write answer in rectangular form. - (−12−32i)10\left( - \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } \mathrm { i } \right) ^ { 10 }(−21​−23​​i)10

Find all solutions of the equation. Leave answers in trigonometric form. - x2−36=0x ^ { 2 } - 36 = 0x2−36=0

Draw a sketch to represent the vector. Refer to the vectors pictured here. - a+da + da+d

Find the indicated vector. -Let a=3i,b=i+j\mathbf { a } = 3 \mathbf { i } , \mathbf { b } = \mathbf { i } + \mathbf { j }a=3i,b=i+j . Find 3a+b3 \mathbf { a } + \mathbf { b }3a+b .

Give the rectangular coordinates for the point. - (−9,−π3)\left( - 9 , - \frac { \pi } { 3 } \right)(−9,−3π​)

Solve the problem. -Two airplanes leave an airport at the same time, one going northwest (bearing 135∘ ) \text { (bearing } 135 ^ { \circ } \text { ) } (bearing 135∘ ) at 415 mph and the other going east at 329 mph. How far apart are the planes after 2 hours (to the nearest Mile)?