Question 1

Find an equivalent equation in rectangular coordinates.
-$$r ( 1 - 2 \cos \theta ) = 1$$

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } = 2 + x$ 
B)  $\sqrt { x ^ { 2 } + y ^ { 2 } } = 2 + x$ 
C)  $\sqrt { x ^ { 2 } + y ^ { 2 } } = 1 + 2 x$ 
D)  $x ^ { 2 } + y ^ { 2 } = 1 + 2 x$ 
A)  $x ^ { 2 } + y ^ { 2 } = 2 + x$ 
B)  $\sqrt { x ^ { 2 } + y ^ { 2 } } = 2 + x$ 
C)  $\sqrt { x ^ { 2 } + y ^ { 2 } } = 1 + 2 x$ 
D)  $x ^ { 2 } + y ^ { 2 } = 1 + 2 x$

Question 2

For the given rectangular equation, give its equivalent polar equation.
-$$x - y = 5$$

Accepted Answer

A)  $r = \frac { 5 } { \sin \theta - \cos \theta }$ 
B)  $r = \frac { 1 } { 5 \cos \theta - 5 \sin \theta }$ 
C)  $r = \frac { 5 } { \cos \theta - \sin \theta }$ 
D)  $r = \frac { 5 } { \cos \theta + \sin \theta }$ 
A)  $r = \frac { 5 } { \sin \theta - \cos \theta }$ 
B)  $r = \frac { 1 } { 5 \cos \theta - 5 \sin \theta }$ 
C)  $r = \frac { 5 } { \cos \theta - \sin \theta }$ 
D)  $r = \frac { 5 } { \cos \theta + \sin \theta }$

Question 3

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations.
-$$x = \sin t , y = \cos t ; t = \frac { 2 \pi } { 3 }$$

Accepted Answer

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

Question 4

Find the missing parts of the triangle.
-$$\begin{array} { l } 
\mathrm { A } = 72 ^ { \circ } \
\mathrm { a } = 36 \mathrm { yd } \
\mathrm { b } = 73 \mathrm { yd }
\end{array}$$
If necessary, round angles to the nearest degree and side lengths to the nearest yard.

Accepted Answer

A)  $B = 80 ^ { \circ } , C = 28 ^ { \circ } , c = 79 \mathrm { yd }$ 
B)  $B = 28 ^ { \circ } , C = 80 ^ { \circ } , \mathrm { c } = 75 \mathrm { yd }$ 
C)  no such triangle 
D)  $B = 28 ^ { \circ } , C = 80 ^ { \circ } , \mathrm { c } = 81 \mathrm { yd }$ 
A)  $B = 80 ^ { \circ } , C = 28 ^ { \circ } , c = 79 \mathrm { yd }$ 
B)  $B = 28 ^ { \circ } , C = 80 ^ { \circ } , \mathrm { c } = 75 \mathrm { yd }$ 
C)  no such triangle 
D)  $B = 28 ^ { \circ } , C = 80 ^ { \circ } , \mathrm { c } = 81 \mathrm { yd }$

Question 5

Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. Round to
one decimal place.
-Suppose you would like to cross a 211-foot wide river in a boat. Assume that the boat can travel 34 mph relative to the water and that the current is flowing west at the rate of $5 \mathrm { mph }$. If the bearing is chosen so that the boat will land at a point exactly across from its starting point, how long will it take for the boat to make the crossing? Give your answer to the nearest second.

Accepted Answer

A)  $9 \mathrm { sec }$ 
B)  $4 \mathrm { sec }$ 
C)  $2 \mathrm { sec }$ 
D)  $17 \mathrm { sec }$ 
A)  $9 \mathrm { sec }$ 
B)  $4 \mathrm { sec }$ 
C)  $2 \mathrm { sec }$ 
D)  $17 \mathrm { sec }$

Question 6

Solve.
-A complex number $\mathrm { z }$ does not belong to the Mandelbrot set if any of the complex numbers in the sequence $z , z ^ { 2 } + z , \left( z ^ { 2 } + z \right) ^ { 2 } + z , \left[ \left( z ^ { 2 } + z \right) ^ { 2 } + z \right] ^ { 2 } + z , \ldots$ has modulus exceeding 2 . Does $z = 0.8$ belong to the Mandelbrot set?

Accepted Answer

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

Question 7

Find the product. Write the product in rectangular form, using exact values.
-$\left[ 4 \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right) \right] \left[ 6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right) \right]$

Accepted Answer

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

Question 8

Find the indicated vector.
-Let $\mathbf { u } = 3 i , \mathbf { v } = \mathbf { i } + \mathrm { j }$. Find $2 \mathbf { u } - \mathbf { v }$.

Accepted Answer

A)  $5 \mathbf { i } - 2 \mathbf { j }$ 
B)  $\mathbf { i } - 5 \mathbf { j }$ 
C)  $11 \mathbf { i } + \mathbf { j }$ 
D)  $5 \mathbf { i } - \mathbf { j }$ 
A)  $5 \mathbf { i } - 2 \mathbf { j }$ 
B)  $\mathbf { i } - 5 \mathbf { j }$ 
C)  $11 \mathbf { i } + \mathbf { j }$ 
D)  $5 \mathbf { i } - \mathbf { j }$

Question 9

Find a rectangular equation for the plane curve defined by the parametric equations.
-$$x = 3 t , y = t + 7$$

Accepted Answer

A)  $y = 3 x - 7$ 
B)  $y = \frac { x } { 3 } + 7$ 
C)  $y = 3 x + 7$ 
D)  $y = \frac { x } { 3 } - 7$ 
A)  $y = 3 x - 7$ 
B)  $y = \frac { x } { 3 } + 7$ 
C)  $y = 3 x + 7$ 
D)  $y = \frac { x } { 3 } - 7$

Question 10

Find the angle between the pair of vectors to the nearest tenth of a degree.
-6i - 4j, 2i - 6j

Accepted Answer

A)  23.6° 
B)  120.0° 
C)  107.5° 
D)  37.9° 
A)  23.6° 
B)  120.0° 
C)  107.5° 
D)  37.9°

Question 11

Solve.
-A complex number $z$ does not belong to the Mandelbrot set if any of the complex numbers in the sequence $z , z ^ { 2 } + z , \left( z ^ { 2 } + z \right) ^ { 2 } + z , \left[ \left( z ^ { 2 } + z \right) ^ { 2 } + z \right] ^ { 2 } + z , \ldots$ has modulus exceeding 2. Does $z = 3$ belong to the Mandelbrot set?

Accepted Answer

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

Question 12

Answer the question.
-In which quadrants do the square roots of $i$ lie?

Accepted Answer

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

Question 13

Solve.
-A complex number $\mathrm { z }$ does not belong to the Mandelbrot set if any of the complex numbers in the sequence $z , z ^ { 2 } + z , \left( z ^ { 2 } + z \right) ^ { 2 } + z , \left[ \left( z ^ { 2 } + z \right) ^ { 2 } + z \right] ^ { 2 } + z , \ldots$ has modulus exceeding 2 . Does $z = 0.4 i$ belong to the Mandelbrot set?

Accepted Answer

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

Question 14

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form.
-$$\frac { 2 \sqrt { 3 } + 2 i } { \sqrt { 3 } - i }$$

Accepted Answer

A)  $\sqrt { 3 }$ 
B)  $1 + \sqrt { 3 } \mathrm { i }$ 
C)  $\sqrt { 3 } i$ 
D)  $\sqrt { 3 } + i$ 
A)  $\sqrt { 3 }$ 
B)  $1 + \sqrt { 3 } \mathrm { i }$ 
C)  $\sqrt { 3 } i$ 
D)  $\sqrt { 3 } + i$

Question 15

Find the polar coordinates of the point(s) of intersection of the given curves for 0 $$0 \leq \theta < 2 \pi$$
-$$\mathrm { r } = \frac { \mathrm { a } \left( 1 - \mathrm { e } ^ { 2 } \right) } { 1 + \mathrm { e } \cos \theta }$$ Given that the polar equation r models the orbits of the planets about the sun, estimate the closest approach of a planet for which a = 21 and e = 0.79 to the sun.

Accepted Answer

A)  8 astronomical units 
B)  2 astronomical units 
C)  4 astronomical units 
D)  38 astronomical units 
A)  8 astronomical units 
B)  2 astronomical units 
C)  4 astronomical units 
D)  38 astronomical units

Question 16

Plot the point.
-$$\left( - 4,315 ^ { \circ } \right)$$

Accepted Answer

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

Question 17

Give the rectangular coordinates for the point.
-$$\left( - 3,0 ^ { \circ } \right)$$

Accepted Answer

A)  $( 3,0 )$ 
B)  $( 0,3 )$ 
C)  $( - 3,0 )$ 
D)  $( 0 , - 3 )$ 
A)  $( 3,0 )$ 
B)  $( 0,3 )$ 
C)  $( - 3,0 )$ 
D)  $( 0 , - 3 )$

Question 18

Find the product. Write the product in rectangular form, using exact values.
-$\left[ 8 \operatorname { cis } 240 ^ { \circ } \right] \left[ 6 \operatorname { cis } 330 ^ { \circ } \right]$

Accepted Answer

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

Question 19

Find the component form of the indicated vector.
-Let $\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 6 , - 3 \rangle$. Find $- \mathbf { u } - \mathbf { v }$

Accepted Answer

A)  $\langle 5 , - 2 \rangle$ 
B)  $\langle - 5,2 \rangle$ 
C)  $\langle 7,8 \rangle$ 
D)  $\langle - 7 , - 8 \rangle$ 
A)  $\langle 5 , - 2 \rangle$ 
B)  $\langle - 5,2 \rangle$ 
C)  $\langle 7,8 \rangle$ 
D)  $\langle - 7 , - 8 \rangle$

Question 20

Find the indicated angle or side. Give an exact answer.
-Find the exact length of side a.

Accepted Answer

A)  $4 \sqrt { 3 }$ 
B)  4 
C)  $2 \sqrt { 7 }$ 
D)  $2 \sqrt { 3 }$ 
A)  $4 \sqrt { 3 }$ 
B)  4 
C)  $2 \sqrt { 7 }$ 
D)  $2 \sqrt { 3 }$

Find an equivalent equation in rectangular coordinates. - $r ( 1 - 2 \cos \theta ) = 1$

For the given rectangular equation, give its equivalent polar equation. - $x - y = 5$

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations. - $x = \sin t , y = \cos t ; t = \frac { 2 \pi } { 3 }$

Find the missing parts of the triangle. - =7 =36 =73 If necessary, round angles to the nearest degree and side lengths to the nearest yard.

Find the product. Write the product in rectangular form, using exact values. - $\left[ 4 \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right) \right] \left[ 6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right) \right]$

Find the indicated vector. -Let $\mathbf { u } = 3 i , \mathbf { v } = \mathbf { i } + \mathrm { j }$ . Find $2 \mathbf { u } - \mathbf { v }$ .

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = 3 t , y = t + 7$

Find the angle between the pair of vectors to the nearest tenth of a degree. -6i - 4j, 2i - 6j

Answer the question. -In which quadrants do the square roots of $i$ lie?

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 2 \sqrt { 3 } + 2 i } { \sqrt { 3 } - i }$

Plot the point. - $\left( - 4,315 ^ { \circ } \right)$ $Plot the point. - \left( - 4,315 ^ { \circ } \right)$

Give the rectangular coordinates for the point. - $\left( - 3,0 ^ { \circ } \right)$

Find the product. Write the product in rectangular form, using exact values. - $\left[ 8 \operatorname { cis } 240 ^ { \circ } \right] \left[ 6 \operatorname { cis } 330 ^ { \circ } \right]$

Find the component form of the indicated vector. -Let $\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 6 , - 3 \rangle$ . Find $- \mathbf { u } - \mathbf { v }$

Find the indicated angle or side. Give an exact answer. -Find the exact length of side a.

Review of Basic Concepts

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

Filters

Exam 9: Applications of Trigonometry

Find an equivalent equation in rectangular coordinates. - r(1−2cos⁡θ)=1r ( 1 - 2 \cos \theta ) = 1r(1−2cosθ)=1

For the given rectangular equation, give its equivalent polar equation. - x−y=5x - y = 5x−y=5

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=2π3x = \sin t , y = \cos t ; t = \frac { 2 \pi } { 3 }x=sint,y=cost;t=32π​

Find the missing parts of the triangle. - =7 =36 =73 If necessary, round angles to the nearest degree and side lengths to the nearest yard.

Find the indicated vector. -Let u=3i,v=i+j\mathbf { u } = 3 i , \mathbf { v } = \mathbf { i } + \mathrm { j }u=3i,v=i+j . Find 2u−v2 \mathbf { u } - \mathbf { v }2u−v .

Find a rectangular equation for the plane curve defined by the parametric equations. - x=3t,y=t+7x = 3 t , y = t + 7x=3t,y=t+7

Find the angle between the pair of vectors to the nearest tenth of a degree. -6i - 4j, 2i - 6j

Answer the question. -In which quadrants do the square roots of iii lie?

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - 23+2i3−i\frac { 2 \sqrt { 3 } + 2 i } { \sqrt { 3 } - i }3​−i23​+2i​

Plot the point. - (−4,315∘)\left( - 4,315 ^ { \circ } \right)(−4,315∘)

Give the rectangular coordinates for the point. - (−3,0∘)\left( - 3,0 ^ { \circ } \right)(−3,0∘)

Find the product. Write the product in rectangular form, using exact values. - [8cis⁡240∘][6cis⁡330∘]\left[ 8 \operatorname { cis } 240 ^ { \circ } \right] \left[ 6 \operatorname { cis } 330 ^ { \circ } \right][8cis240∘][6cis330∘]

Find the component form of the indicated vector. -Let u=⟨1,5⟩,v=⟨−6,−3⟩\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 6 , - 3 \rangleu=⟨1,5⟩,v=⟨−6,−3⟩ . Find −u−v- \mathbf { u } - \mathbf { v }−u−v

Find the indicated angle or side. Give an exact answer. -Find the exact length of side a.

Review of Basic Concepts

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

Filters

Find an equivalent equation in rectangular coordinates. - $r ( 1 - 2 \cos \theta ) = 1$

For the given rectangular equation, give its equivalent polar equation. - $x - y = 5$

Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations. - $x = \sin t , y = \cos t ; t = \frac { 2 \pi } { 3 }$

Find the indicated vector. -Let $\mathbf { u } = 3 i , \mathbf { v } = \mathbf { i } + \mathrm { j }$ . Find $2 \mathbf { u } - \mathbf { v }$ .

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = 3 t , y = t + 7$

Answer the question. -In which quadrants do the square roots of $i$ lie?

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 2 \sqrt { 3 } + 2 i } { \sqrt { 3 } - i }$

Plot the point. - $\left( - 4,315 ^ { \circ } \right)$ $Plot the point. - \left( - 4,315 ^ { \circ } \right)$

Give the rectangular coordinates for the point. - $\left( - 3,0 ^ { \circ } \right)$

Find the product. Write the product in rectangular form, using exact values. - $\left[ 8 \operatorname { cis } 240 ^ { \circ } \right] \left[ 6 \operatorname { cis } 330 ^ { \circ } \right]$

Find the component form of the indicated vector. -Let $\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 6 , - 3 \rangle$ . Find $- \mathbf { u } - \mathbf { v }$