Question 1

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

Accepted Answer

A)  $x ^ { 2 } + 9 y ^ { 2 } = 1$ 
B)  $9 x ^ { 2 } + y ^ { 2 } = 9$ 
C)  $x ^ { 2 } + 9 y ^ { 2 } = 9$ 
D)  $9 x ^ { 2 } + y ^ { 2 } = 1$ 
A)  $x ^ { 2 } + 9 y ^ { 2 } = 1$ 
B)  $9 x ^ { 2 } + y ^ { 2 } = 9$ 
C)  $x ^ { 2 } + 9 y ^ { 2 } = 9$ 
D)  $9 x ^ { 2 } + y ^ { 2 } = 1$

Question 2

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

Accepted Answer

A)  $\mathrm { r } \sin \theta = 3$ 
B)  $r = 6 \sin \theta$ 
C)  $r = 3$ 
D)  $r = 6 \cos \theta$ 
A)  $\mathrm { r } \sin \theta = 3$ 
B)  $r = 6 \sin \theta$ 
C)  $r = 3$ 
D)  $r = 6 \cos \theta$

Question 3

Answer the question.
-If $\mathrm { n }$ is an odd number, then which of the following statements is true?

Accepted Answer

A)  The equation $x ^ { n } - 1 = 0$ has $n - 1$ roots. 
B)  The equation $x ^ { n } - 1 = 0$ has an odd number of nonreal roots. 
C)  The equation $x ^ { n } - 1 = 0$ has exactly one real root. 
D)  The equation $x ^ { n } - 1 = 0$ has an even number of real roots. 
A)  The equation $x ^ { n } - 1 = 0$ has $n - 1$ roots. 
B)  The equation $x ^ { n } - 1 = 0$ has an odd number of nonreal roots. 
C)  The equation $x ^ { n } - 1 = 0$ has exactly one real root. 
D)  The equation $x ^ { n } - 1 = 0$ has an even number of real roots.

Question 4

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)  $- \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } i$ 
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)  $- \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } i$

Question 5

Write the complex number in trigonometric form r(c $$r ( \cos \theta + i \sin \theta )$$ ), with $$\theta$$ in the interval [0 $$\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$$
-$2 \sqrt { 3 } - 2 i$

Accepted Answer

A)  $4 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
B)  $4 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
C)  $4 \left( \cos 300 ^ { \circ } + i \sin 300 ^ { \circ } \right)$ 
D)  $4 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$ 
A)  $4 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
B)  $4 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
C)  $4 \left( \cos 300 ^ { \circ } + i \sin 300 ^ { \circ } \right)$ 
D)  $4 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$

Question 6

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

Accepted Answer

A)  $60 ^ { \circ }$ 
B)  $45 ^ { \circ }$ 
C)  $30 ^ { \circ }$ 
D)  $75 ^ { \circ }$ 
A)  $60 ^ { \circ }$ 
B)  $45 ^ { \circ }$ 
C)  $30 ^ { \circ }$ 
D)  $75 ^ { \circ }$

Question 7

The rectangular coordinates of a point are given. Express the point in polar coordinates with $$r \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ } \text {. }$$
-$( 4 , - 4 )$

Accepted Answer

A)  $\left( 4 \sqrt { 2 } , 315 ^ { \circ } \right)$ 
B)  $\left( 4 \sqrt { 2 } , 225 ^ { \circ } \right)$ 
C)  $\left( 4 \sqrt { 2 } , 45 ^ { \circ } \right)$ 
D)  $\left( 4 \sqrt { 2 } , 135 ^ { \circ } \right)$ 
A)  $\left( 4 \sqrt { 2 } , 315 ^ { \circ } \right)$ 
B)  $\left( 4 \sqrt { 2 } , 225 ^ { \circ } \right)$ 
C)  $\left( 4 \sqrt { 2 } , 45 ^ { \circ } \right)$ 
D)  $\left( 4 \sqrt { 2 } , 135 ^ { \circ } \right)$

Question 8

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

Accepted Answer

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

Question 9

Find all specified roots.
-Sixth roots of i.

Accepted Answer

A)  $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 } , \operatorname { cis } \frac { 11 \pi } { 4 } , 1$ 
B)  $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 }$ 
C)  $\operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 } , \operatorname { cis } \frac { 11 \pi } { 4 } , \mathrm { i }$ 
D)  $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 5 \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 }$, cis $\frac { 7 \pi } { 4 } , \operatorname { cis } \frac { 13 \pi } { 12 } , \frac { 17 \pi } { 12 }$ 
A)  $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 } , \operatorname { cis } \frac { 11 \pi } { 4 } , 1$ 
B)  $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 }$ 
C)  $\operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 } , \operatorname { cis } \frac { 11 \pi } { 4 } , \mathrm { i }$ 
D)  $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 5 \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 }$, cis $\frac { 7 \pi } { 4 } , \operatorname { cis } \frac { 13 \pi } { 12 } , \frac { 17 \pi } { 12 }$

Question 10

Write the vector in the form <a, b>. If necessary, round values to the nearest hundredth.
-

Accepted Answer

A)  $\langle - 0.64 , - 0.77 \rangle$ 
B)  $\langle - 31.41 , - 26.35 \rangle$ 
C)  $\langle - 0.77 , - 0.64 \rangle$ 
D)  $\langle - 26.35 , - 31.41 \rangle$ 
A)  $\langle - 0.64 , - 0.77 \rangle$ 
B)  $\langle - 31.41 , - 26.35 \rangle$ 
C)  $\langle - 0.77 , - 0.64 \rangle$ 
D)  $\langle - 26.35 , - 31.41 \rangle$

Question 11

Find the indicated vector.
-Let $\mathbf { u } = \langle 1 , - 3 \rangle$. Find $2 \mathbf { u }$.

Accepted Answer

A)  $\langle - 2,6 \rangle$ 
B)  $\langle 2 , - 6 \rangle$ 
C)  $\langle - 2 , - 6 \rangle$ 
D)  $\langle 2,6 \rangle$ 
A)  $\langle - 2,6 \rangle$ 
B)  $\langle 2 , - 6 \rangle$ 
C)  $\langle - 2 , - 6 \rangle$ 
D)  $\langle 2,6 \rangle$

Question 12

Find sum of the pair of complex numbers.
-2 + 3i, 5

Accepted Answer

A)  10 
B)  2 + 8i 
C)  7 + 3i 
D)  2 - 2i 
A)  10 
B)  2 + 8i 
C)  7 + 3i 
D)  2 - 2i

Question 13

Find the product. Write the product in rectangular form, using exact values.
-$\left[ 4 \operatorname { cis } 30 ^ { \circ } \right] \left[ 5 \operatorname { cis } 120 ^ { \circ } \right]$

Accepted Answer

A)  $10 + 10 \sqrt { 3 } \mathrm { i }$ 
B)  $- 10 \sqrt { 3 } + 10 \mathrm { i }$ 
C)  $10 \sqrt { 3 } - 10 \mathrm { i }$ 
D)  $- 10 - 10 \sqrt { 3 } i$ 
A)  $10 + 10 \sqrt { 3 } \mathrm { i }$ 
B)  $- 10 \sqrt { 3 } + 10 \mathrm { i }$ 
C)  $10 \sqrt { 3 } - 10 \mathrm { i }$ 
D)  $- 10 - 10 \sqrt { 3 } i$

Question 14

Determine whether the pair of vectors is orthogonal.
-3, -6 , -6, -3

Accepted Answer

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

Question 15

Draw a sketch to represent the vector. Refer to the vectors pictured here.  
-b - c

Accepted Answer

A) 
  
B) 
  
A) 
  
B)

Question 16

Find the product. Write the product in rectangular form, using exact values.
-$\left[ 7 \left( \cos 120 ^ { \circ } + \mathrm { i } \sin 120 ^ { \circ } \right) \right] \left[ 2 \left( \cos 90 ^ { \circ } + \mathrm { i } \sin 90 ^ { \circ } \right) \right]$

Accepted Answer

A)  $5 \sqrt { 6 } - \sqrt { 2 } i$ 
B)  $7 \sqrt { 3 } - 28 \mathrm { i }$ 
C)  $14 - 14 \sqrt { 3 } i$ 
D)  $- 7 \sqrt { 3 } - 7 i$ 
A)  $5 \sqrt { 6 } - \sqrt { 2 } i$ 
B)  $7 \sqrt { 3 } - 28 \mathrm { i }$ 
C)  $14 - 14 \sqrt { 3 } i$ 
D)  $- 7 \sqrt { 3 } - 7 i$

Question 17

Provide an appropriate response.
-A projectile is launched from the ground with an initial velocity of $\mathrm { v } _ { 0 }$ feet per second at an angle of $\theta ^ { \circ }$ above the horizontal. Write parametric equations in terms of $\mathrm { v } _ { 0 }$ and $\theta$ that model the flight of the projectile. Explain how you would use these equations, for given values of $\mathrm { v } _ { 0 }$ and $\theta$, to determine the horizontal distance traveled by the projectile.

Accepted Answer

@#LAT-DLM&
First, solve @#LAT-DLM& for @#LAT-DLM& to determi

Question 18

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

Accepted Answer

A) 
  
B) 
  
A) 
  
B)

Question 19

Find all specified roots.
-Seventh roots of 1.

Accepted Answer

A)  $1 , \operatorname { cis } \frac { \pi } { 7 } , \operatorname { cis } \frac { 2 \pi } { 7 } , \operatorname { cis } \frac { 3 \pi } { 7 } , \operatorname { cis } \frac { 4 \pi } { 7 } , \operatorname { cis } \frac { 5 \pi } { 7 } , \operatorname { cis } \frac { 6 \pi } { 7 }$ 
B)  $1 , \operatorname { cis } \frac { 2 \pi } { 7 } , \operatorname { cis } \frac { 4 \pi } { 7 } , \operatorname { cis } \frac { 6 \pi } { 7 } , \operatorname { cis } \frac { 8 \pi } { 7 } , \operatorname { cis } \frac { 10 \pi } { 7 } , \operatorname { cis } \frac { 12 \pi } { 7 } , - 1$ 
C)  $1 , \operatorname { cis } \frac { 2 \pi } { 7 } , \operatorname { cis } \frac { 4 \pi } { 7 } , \operatorname { cis } \frac { 6 \pi } { 7 } , \operatorname { cis } \frac { 8 \pi } { 7 } , \operatorname { cis } \frac { 10 \pi } { 7 } , \operatorname { cis } \frac { 12 \pi } { 7 }$ 
D)  $- 1 , \operatorname { cis } \frac { \pi } { 7 } , \operatorname { cis } \frac { 2 \pi } { 7 } , \operatorname { cis } \frac { 3 \pi } { 7 } , \operatorname { cis } \frac { 4 \pi } { 7 } , \operatorname { cis } \frac { 5 \pi } { 7 } , \operatorname { cis } \frac { 6 \pi } { 7 }$ 
A)  $1 , \operatorname { cis } \frac { \pi } { 7 } , \operatorname { cis } \frac { 2 \pi } { 7 } , \operatorname { cis } \frac { 3 \pi } { 7 } , \operatorname { cis } \frac { 4 \pi } { 7 } , \operatorname { cis } \frac { 5 \pi } { 7 } , \operatorname { cis } \frac { 6 \pi } { 7 }$ 
B)  $1 , \operatorname { cis } \frac { 2 \pi } { 7 } , \operatorname { cis } \frac { 4 \pi } { 7 } , \operatorname { cis } \frac { 6 \pi } { 7 } , \operatorname { cis } \frac { 8 \pi } { 7 } , \operatorname { cis } \frac { 10 \pi } { 7 } , \operatorname { cis } \frac { 12 \pi } { 7 } , - 1$ 
C)  $1 , \operatorname { cis } \frac { 2 \pi } { 7 } , \operatorname { cis } \frac { 4 \pi } { 7 } , \operatorname { cis } \frac { 6 \pi } { 7 } , \operatorname { cis } \frac { 8 \pi } { 7 } , \operatorname { cis } \frac { 10 \pi } { 7 } , \operatorname { cis } \frac { 12 \pi } { 7 }$ 
D)  $- 1 , \operatorname { cis } \frac { \pi } { 7 } , \operatorname { cis } \frac { 2 \pi } { 7 } , \operatorname { cis } \frac { 3 \pi } { 7 } , \operatorname { cis } \frac { 4 \pi } { 7 } , \operatorname { cis } \frac { 5 \pi } { 7 } , \operatorname { cis } \frac { 6 \pi } { 7 }$

Question 20

Answer the question.
-If $\mathrm { n }$ is an even number, then which of the following statements is true?

Accepted Answer

A)  The equation $x ^ { n } + 1 = 0$ has exactly one real root. 
B)  The equation $x ^ { n } + 1 = 0$ has $n + 1$ roots. 
C)  The equation $x ^ { n } + 1 = 0$ has $n$ nonreal roots. 
D)  The equation $x ^ { n } + 1 = 0$ has at least one real root. 
A)  The equation $x ^ { n } + 1 = 0$ has exactly one real root. 
B)  The equation $x ^ { n } + 1 = 0$ has $n + 1$ roots. 
C)  The equation $x ^ { n } + 1 = 0$ has $n$ nonreal roots. 
D)  The equation $x ^ { n } + 1 = 0$ has at least one real root.

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

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

Answer the question. -If $\mathrm { n }$ is an odd number, then which of the following statements is true?

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

Write the complex number in trigonometric form r(c $r ( \cos \theta + i \sin \theta )$ ), with $\theta$ in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ - $2 \sqrt { 3 } - 2 i$

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

The rectangular coordinates of a point are given. Express the point in polar coordinates with $r \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ } \text {. }$ - $( 4 , - 4 )$

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

Find all specified roots. -Sixth roots of i.

Write the vector in the form <a, b>. If necessary, round values to the nearest hundredth. -

Find the indicated vector. -Let $\mathbf { u } = \langle 1 , - 3 \rangle$ . Find $2 \mathbf { u }$ .

Find sum of the pair of complex numbers. -2 + 3i, 5

Find the product. Write the product in rectangular form, using exact values. - $\left[ 4 \operatorname { cis } 30 ^ { \circ } \right] \left[ 5 \operatorname { cis } 120 ^ { \circ } \right]$

Determine whether the pair of vectors is orthogonal. -3, -6 , -6, -3

Draw a sketch to represent the vector. Refer to the vectors pictured here. -b - c

Find the product. Write the product in rectangular form, using exact values. - $\left[ 7 \left( \cos 120 ^ { \circ } + \mathrm { i } \sin 120 ^ { \circ } \right) \right] \left[ 2 \left( \cos 90 ^ { \circ } + \mathrm { i } \sin 90 ^ { \circ } \right) \right]$

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

Find all specified roots. -Seventh roots of 1.

Answer the question. -If $\mathrm { n }$ is an even number, then which of the following statements is true?

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 a rectangular equation for the plane curve defined by the parametric equations. - x=sin⁡t,y=3cos⁡tx = \sin t , y = 3 \cos tx=sint,y=3cost

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

Answer the question. -If n\mathrm { n }n is an odd number, then which of the following statements is true?

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

Write the complex number in trigonometric form r(c r(cos⁡θ+isin⁡θ)r ( \cos \theta + i \sin \theta )r(cosθ+isinθ) ), with θ\thetaθ in the interval [0 [0∘,360∘)\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)[0∘,360∘) - 23−2i2 \sqrt { 3 } - 2 i23​−2i

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

The rectangular coordinates of a point are given. Express the point in polar coordinates with r≥0 and 0∘≤θ<360∘. r \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ } \text {. }r≥0 and 0∘≤θ<360∘. - (4,−4)( 4 , - 4 )(4,−4)

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

Find all specified roots. -Sixth roots of i.

Write the vector in the form <a, b>. If necessary, round values to the nearest hundredth. -

Find the indicated vector. -Let u=⟨1,−3⟩\mathbf { u } = \langle 1 , - 3 \rangleu=⟨1,−3⟩ . Find 2u2 \mathbf { u }2u .

Find sum of the pair of complex numbers. -2 + 3i, 5

Find the product. Write the product in rectangular form, using exact values. - [4cis⁡30∘][5cis⁡120∘]\left[ 4 \operatorname { cis } 30 ^ { \circ } \right] \left[ 5 \operatorname { cis } 120 ^ { \circ } \right][4cis30∘][5cis120∘]

Determine whether the pair of vectors is orthogonal. -3, -6 , -6, -3

Draw a sketch to represent the vector. Refer to the vectors pictured here. -b - c

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

Find all specified roots. -Seventh roots of 1.

Answer the question. -If n\mathrm { n }n is an even number, then which of the following statements is true?

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 a rectangular equation for the plane curve defined by the parametric equations. - $x = \sin t , y = 3 \cos t$

Answer the question. -If $\mathrm { n }$ is an odd number, then which of the following statements is true?

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

Write the complex number in trigonometric form r(c $r ( \cos \theta + i \sin \theta )$ ), with $\theta$ in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ - $2 \sqrt { 3 } - 2 i$

The rectangular coordinates of a point are given. Express the point in polar coordinates with $r \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ } \text {. }$ - $( 4 , - 4 )$

Find the indicated vector. -Let $\mathbf { u } = \langle 1 , - 3 \rangle$ . Find $2 \mathbf { u }$ .

Find the product. Write the product in rectangular form, using exact values. - $\left[ 4 \operatorname { cis } 30 ^ { \circ } \right] \left[ 5 \operatorname { cis } 120 ^ { \circ } \right]$

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

Answer the question. -If $\mathrm { n }$ is an even number, then which of the following statements is true?