Question 1

Write the complex number in rectangular form.
-$$3 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 6 } + \frac { \sqrt { 3 } } { 6 } \mathrm { i }$ 
B)  $\sqrt { 3 } + \mathrm { i }$ 
C)  $\frac { 3 } { 2 } + \frac { 3 \sqrt { 3 } } { 2 } \mathrm { i }$ 
D)  $\frac { \sqrt { 3 } } { 2 } + \frac { 3 \sqrt { 3 } } { 2 } \mathrm { i }$ 
A)  $\frac { \sqrt { 3 } } { 6 } + \frac { \sqrt { 3 } } { 6 } \mathrm { i }$ 
B)  $\sqrt { 3 } + \mathrm { i }$ 
C)  $\frac { 3 } { 2 } + \frac { 3 \sqrt { 3 } } { 2 } \mathrm { i }$ 
D)  $\frac { \sqrt { 3 } } { 2 } + \frac { 3 \sqrt { 3 } } { 2 } \mathrm { i }$

Question 2

Sketch the vectors u and w with angle $\theta$ between them and sketch the resultant.
-$$| \mathbf { u } | = 45 , | \mathbf { w } | = 25 , \theta = 30 ^ { \circ }$$

Accepted Answer

A) 
  
B) 
  
A) 
  
B)

Question 3

Write the complex number in rectangular form.
-$4 \left( \cos 15 ^ { \circ } + i \sin 15 ^ { \circ } \right)$

Accepted Answer

A)  $3.9 + 1 \mathrm { i }$ 
B)  $1.9 + 0.5 \mathrm { i }$ 
C)  $- 1.9 - 0.5 \mathrm { i }$ 
D)  $1 + 3.9 \mathrm { i }$ 
A)  $3.9 + 1 \mathrm { i }$ 
B)  $1.9 + 0.5 \mathrm { i }$ 
C)  $- 1.9 - 0.5 \mathrm { i }$ 
D)  $1 + 3.9 \mathrm { i }$

Question 4

Solve the problem.
-A rocket tracking station has two telescopes A and B placed 2.1 miles apart. The telescopes lock onto a rocket and transmit their angles of elevation to a computer after a rocket launch. What is the
Distance to the rocket from telescope B at the moment when both tracking stations are directly east
Of the rocket telescope A reports an angle of elevation of 20° and telescope B reports an angle of
Elevation of 52°? Round to the nearest hundredth of a mile.

Accepted Answer

A)  0.91 mi 
B)  1.36 mi 
C)  4.84 mi 
D)  3.12 mi 
A)  0.91 mi 
B)  1.36 mi 
C)  4.84 mi 
D)  3.12 mi

Question 5

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

Accepted Answer

A) 
  
B)

Question 6

Solve the problem.
-If u =-5, 7 , v =2, 6, and w =-11, 2 , evaluate u · (v + w).

Accepted Answer

A)  101 
B)  109 
C)  82 
D)  91 
A)  101 
B)  109 
C)  82 
D)  91

Question 7

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

Accepted Answer

A)  $\langle - 0.98,0.1 \rangle \rangle$ 
B)  $\langle - 8.86,1.56 \rangle$ 
C)  $\langle - 9,1.58 \rangle$ 
D)  $\langle - 8.46,3.08 \rangle$ 
A)  $\langle - 0.98,0.1 \rangle \rangle$ 
B)  $\langle - 8.86,1.56 \rangle$ 
C)  $\langle - 9,1.58 \rangle$ 
D)  $\langle - 8.46,3.08 \rangle$

Question 8

Write the complex number in trigonometric form r(c $$r ( \cos \theta + i \sin \theta )$$ ), with $$\theta$$ in the interval [0 $$\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$$
-$- \sqrt { 7 } - i \sqrt { 7 }$

Accepted Answer

A)  $\sqrt { 14 } \left( \cos 225 ^ { \circ } + i \sin 225 ^ { \circ } \right)$ 
B)  $\sqrt { 14 } \left( \cos 315 ^ { \circ } + i \sin 315 ^ { \circ } \right)$ 
C)  $\sqrt { 14 } \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
D)  $\sqrt { 14 } \left( \cos 45 ^ { \circ } + \mathrm { i } \sin 45 ^ { \circ } \right)$ 
A)  $\sqrt { 14 } \left( \cos 225 ^ { \circ } + i \sin 225 ^ { \circ } \right)$ 
B)  $\sqrt { 14 } \left( \cos 315 ^ { \circ } + i \sin 315 ^ { \circ } \right)$ 
C)  $\sqrt { 14 } \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
D)  $\sqrt { 14 } \left( \cos 45 ^ { \circ } + \mathrm { i } \sin 45 ^ { \circ } \right)$

Question 9

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

Accepted Answer

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

Question 10

Plot the point.
-$$\left( 4 , \frac { - 3 \pi } { 4 } \right)$$

Accepted Answer

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

Question 11

Find the polar coordinates of the point(s) of intersection of the given curves for 0 $$0 \leq \theta < 2 \pi$$
-$$r = \frac { a \left( 1 - e ^ { 2 } \right) } { 1 + e \cos \theta }$$ Given that the polar equation r models the orbits of the planets, estimate the smallest possible distance between Earth, for which a = 1.00 and e = 0.017, and Venus, for which a = 0.78 and
E = 0.007.

Accepted Answer

A)  About 0.2 astronomical units 
B)  Zero astronomical units 
C)  About 2 astronomical units 
D)  About 0.4 astronomical units 
A)  About 0.2 astronomical units 
B)  Zero astronomical units 
C)  About 2 astronomical units 
D)  About 0.4 astronomical units

Question 12

Find the dot product for the pair of vectors.
-6i - 2j, 8i + j

Accepted Answer

A)  0 
B)  -10 
C)  46 
D)  50 
A)  0 
B)  -10 
C)  46 
D)  50

Question 13

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides,
by the law of sines.
-b, A, and B

Accepted Answer

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

Question 14

Find the angle between the pair of vectors to the nearest tenth of a degree.
-3, -9 , 4, 8

Accepted Answer

A)  57.6° 
B)  135.1° 
C)  145.1° 
D)  67.6° 
A)  57.6° 
B)  135.1° 
C)  145.1° 
D)  67.6°

Question 15

Find all cube roots of the complex number. Leave answers in trigonometric form.
-$$4 - 4 i \sqrt { 3 }$$

Accepted Answer

A)  $\sqrt [ 3 ] { 8 } \operatorname { cis } 100 ^ { \circ } , \sqrt [ 3 ] { 8 } \operatorname { cis } 220 ^ { \circ } , \sqrt [ 3 ] { 8 } \operatorname { cis } 340 ^ { \circ }$ 
B)  $\sqrt { 4 } \operatorname { cis } 60 ^ { \circ } , \sqrt { 4 } \operatorname { cis } 180 ^ { \circ } , \sqrt { 4 } \operatorname { cis } 300 ^ { \circ }$ 
C)  $\sqrt { 8 } \operatorname { cis } 100 ^ { \circ } , \sqrt { 8 } \operatorname { cis } 220 ^ { \circ } , \sqrt { 8 } \operatorname { cis } 340 ^ { \circ }$ 
D)  $\sqrt [ 3 ] { 4 } \operatorname { cis } 80 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 200 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 320 ^ { \circ }$ 
A)  $\sqrt [ 3 ] { 8 } \operatorname { cis } 100 ^ { \circ } , \sqrt [ 3 ] { 8 } \operatorname { cis } 220 ^ { \circ } , \sqrt [ 3 ] { 8 } \operatorname { cis } 340 ^ { \circ }$ 
B)  $\sqrt { 4 } \operatorname { cis } 60 ^ { \circ } , \sqrt { 4 } \operatorname { cis } 180 ^ { \circ } , \sqrt { 4 } \operatorname { cis } 300 ^ { \circ }$ 
C)  $\sqrt { 8 } \operatorname { cis } 100 ^ { \circ } , \sqrt { 8 } \operatorname { cis } 220 ^ { \circ } , \sqrt { 8 } \operatorname { cis } 340 ^ { \circ }$ 
D)  $\sqrt [ 3 ] { 4 } \operatorname { cis } 80 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 200 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 320 ^ { \circ }$

Question 16

Find the component form of the indicated vector.
-Let $\mathbf { u } = \langle 6 , - 7 \rangle$. Find $6 \mathbf { u }$.

Accepted Answer

A)  $\langle 36 , - 42 \rangle$ 
B)  $\langle 36,42 \rangle$ 
C)  $\langle - 36,42 \rangle$ 
D)  $\langle - 36 , - 42 \rangle$ 
A)  $\langle 36 , - 42 \rangle$ 
B)  $\langle 36,42 \rangle$ 
C)  $\langle - 36,42 \rangle$ 
D)  $\langle - 36 , - 42 \rangle$

Question 17

Give the rectangular coordinates for the point.
-$$\left( - 5,120 ^ { \circ } \right)$$

Accepted Answer

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

Question 18

The rectangular coordinates of a point are given. Express the point in polar coordinates with $$r \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ } \text {. }$$
-$( \sqrt { 5 } , - \sqrt { 5 } )$

Accepted Answer

A)  $\left( 10,225 ^ { \circ } \right)$ 
B)  $\left( \sqrt { 10 } , 225 ^ { \circ } \right)$ 
C)  $\left( \sqrt { 10 } , 315 ^ { \circ } \right)$ 
D)  $\left( \sqrt { 5 } , 135 ^ { \circ } \right)$ 
A)  $\left( 10,225 ^ { \circ } \right)$ 
B)  $\left( \sqrt { 10 } , 225 ^ { \circ } \right)$ 
C)  $\left( \sqrt { 10 } , 315 ^ { \circ } \right)$ 
D)  $\left( \sqrt { 5 } , 135 ^ { \circ } \right)$

Question 19

Find sum of the pair of complex numbers.
--8 + 9i, 12

Accepted Answer

A)  -20 + 21i 
B)  4 + 9i 
C)  4 + 21i 
D)  -20 + 9i 
A)  -20 + 21i 
B)  4 + 9i 
C)  4 + 21i 
D)  -20 + 9i

Question 20

Find the component form of the indicated vector.
-Let $\mathbf { u } = \langle - 5,4 \rangle , \mathbf { v } = \langle - 3,1 \rangle$. Find $\mathbf { u } + \mathbf { v }$.

Accepted Answer

A)  $\langle - 8,5 \rangle$ 
B)  $\langle - 1 , - 2 \rangle$ 
C)  $\langle - 2,3 \rangle$ 
D)  $\langle - 4,1 \rangle$ 
A)  $\langle - 8,5 \rangle$ 
B)  $\langle - 1 , - 2 \rangle$ 
C)  $\langle - 2,3 \rangle$ 
D)  $\langle - 4,1 \rangle$

Write the complex number in rectangular form. - $3 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$

Sketch the vectors u and w with angle $\theta$ between them and sketch the resultant. - $| \mathbf { u } | = 45 , | \mathbf { w } | = 25 , \theta = 30 ^ { \circ }$

Write the complex number in rectangular form. - $4 \left( \cos 15 ^ { \circ } + i \sin 15 ^ { \circ } \right)$

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

Solve the problem. -If u =-5, 7 , v =2, 6, and w =-11, 2 , evaluate u · (v + w).

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

Write the complex number in trigonometric form r(c $r ( \cos \theta + i \sin \theta )$ ), with $\theta$ in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ - $- \sqrt { 7 } - i \sqrt { 7 }$

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

Plot the point. - $\left( 4 , \frac { - 3 \pi } { 4 } \right)$ $Plot the point. - \left( 4 , \frac { - 3 \pi } { 4 } \right)$

Find the dot product for the pair of vectors. -6i - 2j, 8i + j

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides, by the law of sines. -b, A, and B

Find the angle between the pair of vectors to the nearest tenth of a degree. -3, -9 , 4, 8

Find all cube roots of the complex number. Leave answers in trigonometric form. - $4 - 4 i \sqrt { 3 }$

Find the component form of the indicated vector. -Let $\mathbf { u } = \langle 6 , - 7 \rangle$ . Find $6 \mathbf { u }$ .

Give the rectangular coordinates for the point. - $\left( - 5,120 ^ { \circ } \right)$

The rectangular coordinates of a point are given. Express the point in polar coordinates with $r \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ } \text {. }$ - $( \sqrt { 5 } , - \sqrt { 5 } )$

Find sum of the pair of complex numbers. --8 + 9i, 12

Find the component form of the indicated vector. -Let $\mathbf { u } = \langle - 5,4 \rangle , \mathbf { v } = \langle - 3,1 \rangle$ . Find $\mathbf { u } + \mathbf { v }$ .

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

Write the complex number in rectangular form. - 3(cos⁡π3+isin⁡π3)3 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)3(cos3π​+isin3π​)

Sketch the vectors u and w with angle θ\thetaθ between them and sketch the resultant. - ∣u∣=45,∣w∣=25,θ=30∘| \mathbf { u } | = 45 , | \mathbf { w } | = 25 , \theta = 30 ^ { \circ }∣u∣=45,∣w∣=25,θ=30∘

Write the complex number in rectangular form. - 4(cos⁡15∘+isin⁡15∘)4 \left( \cos 15 ^ { \circ } + i \sin 15 ^ { \circ } \right)4(cos15∘+isin15∘)

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

Solve the problem. -If u =-5, 7 , v =2, 6, and w =-11, 2 , evaluate u · (v + w).

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

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∘) - −7−i7- \sqrt { 7 } - i \sqrt { 7 }−7​−i7​

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

Plot the point. - (4,−3π4)\left( 4 , \frac { - 3 \pi } { 4 } \right)(4,4−3π​)

Find the dot product for the pair of vectors. -6i - 2j, 8i + j

Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides, by the law of sines. -b, A, and B

Find the angle between the pair of vectors to the nearest tenth of a degree. -3, -9 , 4, 8

Find all cube roots of the complex number. Leave answers in trigonometric form. - 4−4i34 - 4 i \sqrt { 3 }4−4i3​

Find the component form of the indicated vector. -Let u=⟨6,−7⟩\mathbf { u } = \langle 6 , - 7 \rangleu=⟨6,−7⟩ . Find 6u6 \mathbf { u }6u .

Give the rectangular coordinates for the point. - (−5,120∘)\left( - 5,120 ^ { \circ } \right)(−5,120∘)

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∘. - (5,−5)( \sqrt { 5 } , - \sqrt { 5 } )(5​,−5​)

Find sum of the pair of complex numbers. --8 + 9i, 12

Find the component form of the indicated vector. -Let u=⟨−5,4⟩,v=⟨−3,1⟩\mathbf { u } = \langle - 5,4 \rangle , \mathbf { v } = \langle - 3,1 \rangleu=⟨−5,4⟩,v=⟨−3,1⟩ . Find u+v\mathbf { u } + \mathbf { v }u+v .

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

Write the complex number in rectangular form. - $3 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$

Sketch the vectors u and w with angle $\theta$ between them and sketch the resultant. - $| \mathbf { u } | = 45 , | \mathbf { w } | = 25 , \theta = 30 ^ { \circ }$

Write the complex number in rectangular form. - $4 \left( \cos 15 ^ { \circ } + i \sin 15 ^ { \circ } \right)$

Write the complex number in trigonometric form r(c $r ( \cos \theta + i \sin \theta )$ ), with $\theta$ in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ - $- \sqrt { 7 } - i \sqrt { 7 }$

Plot the point. - $\left( 4 , \frac { - 3 \pi } { 4 } \right)$ $Plot the point. - \left( 4 , \frac { - 3 \pi } { 4 } \right)$

Find all cube roots of the complex number. Leave answers in trigonometric form. - $4 - 4 i \sqrt { 3 }$

Find the component form of the indicated vector. -Let $\mathbf { u } = \langle 6 , - 7 \rangle$ . Find $6 \mathbf { u }$ .

Give the rectangular coordinates for the point. - $\left( - 5,120 ^ { \circ } \right)$

The rectangular coordinates of a point are given. Express the point in polar coordinates with $r \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ } \text {. }$ - $( \sqrt { 5 } , - \sqrt { 5 } )$

Find the component form of the indicated vector. -Let $\mathbf { u } = \langle - 5,4 \rangle , \mathbf { v } = \langle - 3,1 \rangle$ . Find $\mathbf { u } + \mathbf { v }$ .