Question 1

Sketch the vectors u and w with angle θ between them and sketch the resultant.
-$$| \mathbf { u } | = 5 , | \mathbf { w } | = 1 , \theta = 60 ^ { \circ }$$

Accepted Answer

A) 
  
B) 
  
A) 
  
B)

Question 2

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form.
-$\frac { 25 \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right) } { 5 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right) }$

Accepted Answer

A)  $- 5 i$ 
B)  $- 5 i \sqrt { 2 }$ 
C)  $5 \sqrt { 2 } - 5 i \sqrt { 2 }$ 
D)  $5 \mathrm { i }$ 
A)  $- 5 i$ 
B)  $- 5 i \sqrt { 2 }$ 
C)  $5 \sqrt { 2 } - 5 i \sqrt { 2 }$ 
D)  $5 \mathrm { i }$

Question 3

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

Accepted Answer

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

Question 4

Find the dot product for the pair of vectors.
-$\langle 4,8 \rangle , \langle - 7 , - 14 \rangle$

Accepted Answer

A)  $- 28$ 
B)  84 
C)  $- 112$ 
D)  $- 140$ 
A)  $- 28$ 
B)  84 
C)  $- 112$ 
D)  $- 140$

Question 5

Find sum of the pair of complex numbers.
-$9 - 2 i , - 4 i$

Accepted Answer

A)  $9 - 6 \mathrm { i }$ 
B)  $9 + 2 \mathrm { i }$ 
C)  $5 - 2 \mathrm { i }$ 
D)  $3 \mathrm { i }$ 
A)  $9 - 6 \mathrm { i }$ 
B)  $9 + 2 \mathrm { i }$ 
C)  $5 - 2 \mathrm { i }$ 
D)  $3 \mathrm { i }$

Question 6

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

Accepted Answer

A)   
B)  $\longrightarrow$ 
A)   
B)  $\longrightarrow$

Question 7

Graph the polar equation for θ in [0°, 360°).
-$$r = 3 - \cos \theta$$

Accepted Answer

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

Question 8

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

Accepted Answer

A)  $\langle 20 , - 18 \rangle$ 
B)  $\langle 13.38 , - 14.86 \rangle$ 
C)  $\langle 0.74 , - 0.67 \rangle$ 
D)  $\langle 14.86 , - 13.38 \rangle$ 
A)  $\langle 20 , - 18 \rangle$ 
B)  $\langle 13.38 , - 14.86 \rangle$ 
C)  $\langle 0.74 , - 0.67 \rangle$ 
D)  $\langle 14.86 , - 13.38 \rangle$

Question 9

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 $z = 0.6 i$ belong to the Mandelbrot set?

Accepted Answer

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

Question 10

Find all cube roots of the complex number. Leave answers in trigonometric form.
-$$2 + 2 i \sqrt { 3 }$$

Accepted Answer

A)  $\sqrt [ 3 ] { 4 } \operatorname { cis } 30 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 150 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 270 ^ { \circ }$ 
B)  $\sqrt [ 3 ] { 4 } \operatorname { cis } 20 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 140 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 260 ^ { \circ }$ 
C)  $\sqrt { 2 } \operatorname { cis } 20 ^ { \circ } , \sqrt { 2 } \operatorname { cis } 140 ^ { \circ } , \sqrt { 2 } \operatorname { cis } 260 ^ { \circ }$ 
D)  $\sqrt { 4 } \operatorname { cis } 20 ^ { \circ } , \sqrt { 4 } \operatorname { cis } 140 ^ { \circ } , \sqrt { 4 } \operatorname { cis } 260 ^ { \circ }$ 
A)  $\sqrt [ 3 ] { 4 } \operatorname { cis } 30 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 150 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 270 ^ { \circ }$ 
B)  $\sqrt [ 3 ] { 4 } \operatorname { cis } 20 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 140 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 260 ^ { \circ }$ 
C)  $\sqrt { 2 } \operatorname { cis } 20 ^ { \circ } , \sqrt { 2 } \operatorname { cis } 140 ^ { \circ } , \sqrt { 2 } \operatorname { cis } 260 ^ { \circ }$ 
D)  $\sqrt { 4 } \operatorname { cis } 20 ^ { \circ } , \sqrt { 4 } \operatorname { cis } 140 ^ { \circ } , \sqrt { 4 } \operatorname { cis } 260 ^ { \circ }$

Question 11

Find the angle between the pair of vectors to the nearest tenth of a degree.
-$$7 \mathbf { i } - 5 \mathbf { j } , 2 \mathbf { i } - 6 \mathbf { j }$$

Accepted Answer

A)  $36.0 ^ { \circ }$ 
B)  $44.4 ^ { \circ }$ 
C)  $117.2 ^ { \circ }$ 
D)  $104.7 ^ { \circ }$ 
A)  $36.0 ^ { \circ }$ 
B)  $44.4 ^ { \circ }$ 
C)  $117.2 ^ { \circ }$ 
D)  $104.7 ^ { \circ }$

Question 12

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form.
-$\frac { \sqrt { 3 } \left( \cos \frac { 7 \pi } { 4 } + i \sin \frac { 7 \pi } { 4 } \right) } { \sqrt { 6 } \left( \cos \frac { 9 \pi } { 4 } + i \sin \frac { 9 \pi } { 4 } \right) }$

Accepted Answer

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

Question 13

Find the missing parts of the triangle.
-$$\begin{array} { l } 
\mathrm { B } = 48 ^ { \circ } 30 ^ { \prime } \
\mathrm { b } = 6.15 \
\mathrm { a } = 27.75
\end{array}$$
If necessary, round side lengths to the nearest hundredth.

Accepted Answer

A)  $\mathrm { A } = 46 ^ { \circ } 15 ^ { \prime } , \mathrm { C } = 85 ^ { \circ } 15 ^ { \prime } , \mathrm { c } = 33.9$ 
B)  $\mathrm { A } = 47 ^ { \circ } 15 ^ { \prime } , \mathrm { C } = 86 ^ { \circ } 15 ^ { \prime } , \mathrm { c } = 35.4$ 
C)  $\mathrm { A } = 44 ^ { \circ } 15 ^ { \prime } , \mathrm { C } = 87 ^ { \circ } 15 ^ { \prime } , \mathrm { c } = 30.9$ 
D)  no such triangle 
A)  $\mathrm { A } = 46 ^ { \circ } 15 ^ { \prime } , \mathrm { C } = 85 ^ { \circ } 15 ^ { \prime } , \mathrm { c } = 33.9$ 
B)  $\mathrm { A } = 47 ^ { \circ } 15 ^ { \prime } , \mathrm { C } = 86 ^ { \circ } 15 ^ { \prime } , \mathrm { c } = 35.4$ 
C)  $\mathrm { A } = 44 ^ { \circ } 15 ^ { \prime } , \mathrm { C } = 87 ^ { \circ } 15 ^ { \prime } , \mathrm { c } = 30.9$ 
D)  no such triangle

Question 14

Find all solutions of the equation. Leave answers in trigonometric form.
-$$x ^ { 2 } + 1 - \sqrt { 3 } i = 0$$

Accepted Answer

A)  $\left\{ \sqrt { 2 } \operatorname { cis } 60 ^ { \circ } , \sqrt { 2 } \operatorname { cis } 240 ^ { \circ } \right\}$ 
B)  $\left\{ 2 \operatorname { cis } 90 ^ { \circ } , 2 \operatorname { cis } 270 ^ { \circ } \right\}$ 
C)  $\left\{ 2 \sqrt { 2 } \operatorname { cis } 90 ^ { \circ } , - 2 \sqrt { 2 } \operatorname { cis } 270 ^ { \circ } \right\}$ 
D)  $\left\{ 2 \operatorname { cis } 45 ^ { \circ } , 2 \operatorname { cis } 225 ^ { \circ } \right\}$ 
A)  $\left\{ \sqrt { 2 } \operatorname { cis } 60 ^ { \circ } , \sqrt { 2 } \operatorname { cis } 240 ^ { \circ } \right\}$ 
B)  $\left\{ 2 \operatorname { cis } 90 ^ { \circ } , 2 \operatorname { cis } 270 ^ { \circ } \right\}$ 
C)  $\left\{ 2 \sqrt { 2 } \operatorname { cis } 90 ^ { \circ } , - 2 \sqrt { 2 } \operatorname { cis } 270 ^ { \circ } \right\}$ 
D)  $\left\{ 2 \operatorname { cis } 45 ^ { \circ } , 2 \operatorname { cis } 225 ^ { \circ } \right\}$

Question 15

Sketch the vectors u and w with angle θ between them and sketch the resultant.
-$$| \mathbf { u } | = 3 , | \mathbf { w } | = 12 , \theta = 40 ^ { \circ }$$

Accepted Answer

A) 
  
B) 
  
A) 
  
B)

Question 16

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

Accepted Answer

A)  $r = 8 \sin \theta$ 
B)  $r = 4 + \sin \theta$ 
C)  $\mathrm { r } = 4 + \cos \theta$ 
D)  $r = 8 \cos \theta$ 
A)  $r = 8 \sin \theta$ 
B)  $r = 4 + \sin \theta$ 
C)  $\mathrm { r } = 4 + \cos \theta$ 
D)  $r = 8 \cos \theta$

Question 17

Find all cube roots of the complex number. Leave answers in trigonometric form.
-$- 6 \sqrt { 3 } - 6 i$

Accepted Answer

A)  $\sqrt [ 3 ] { 12 } \operatorname { cis } 80 ^ { \circ } , \sqrt [ 3 ] { 6 } \operatorname { cis } 200 ^ { \circ } , \sqrt [ 3 ] { 12 }$ cis $320 ^ { \circ }$ 
B)  $\sqrt [ 3 ] { 6 } \operatorname { cis } 70 ^ { \circ } , \sqrt [ 3 ] { 6 } \operatorname { cis } 190 ^ { \circ } , \sqrt [ 3 ] { 6 } \operatorname { cis } 310 ^ { \circ }$ 
C)  $\sqrt { 6 } \operatorname { cis } 70 ^ { \circ } , \sqrt { 6 } \operatorname { cis } 190 ^ { \circ } , \sqrt { 6 } \operatorname { cis } 310 ^ { \circ }$ 
D)  $\sqrt [ 3 ] { 12 } \operatorname { cis } 70 ^ { \circ } , \sqrt [ 3 ] { 12 } \operatorname { cis } 190 ^ { \circ } , \sqrt [ 3 ] { 12 } \operatorname { cis } 310 ^ { \circ }$ 
A)  $\sqrt [ 3 ] { 12 } \operatorname { cis } 80 ^ { \circ } , \sqrt [ 3 ] { 6 } \operatorname { cis } 200 ^ { \circ } , \sqrt [ 3 ] { 12 }$ cis $320 ^ { \circ }$ 
B)  $\sqrt [ 3 ] { 6 } \operatorname { cis } 70 ^ { \circ } , \sqrt [ 3 ] { 6 } \operatorname { cis } 190 ^ { \circ } , \sqrt [ 3 ] { 6 } \operatorname { cis } 310 ^ { \circ }$ 
C)  $\sqrt { 6 } \operatorname { cis } 70 ^ { \circ } , \sqrt { 6 } \operatorname { cis } 190 ^ { \circ } , \sqrt { 6 } \operatorname { cis } 310 ^ { \circ }$ 
D)  $\sqrt [ 3 ] { 12 } \operatorname { cis } 70 ^ { \circ } , \sqrt [ 3 ] { 12 } \operatorname { cis } 190 ^ { \circ } , \sqrt [ 3 ] { 12 } \operatorname { cis } 310 ^ { \circ }$

Question 18

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

Accepted Answer

A)  $\langle 41.01,41.01 \rangle$ 
B)  $\langle 44.43,37.28 \rangle$ 
C)  $\langle 0.64,0.7 \rangle \rangle$ 
D)  $\langle 37.28,44.43 \rangle$ 
A)  $\langle 41.01,41.01 \rangle$ 
B)  $\langle 44.43,37.28 \rangle$ 
C)  $\langle 0.64,0.7 \rangle \rangle$ 
D)  $\langle 37.28,44.43 \rangle$

Question 19

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in [0,360°].
-$\langle - 6 \sqrt { 2 } , - 6 \sqrt { 2 } \rangle$

Accepted Answer

A)  $24 ; 45 ^ { \circ }$ 
B)  $12 \sqrt { 2 } ; 135 ^ { \circ }$ 
C)  $12 ; 225 ^ { \circ }$ 
D)  $12 ; 45 ^ { \circ }$ 
A)  $24 ; 45 ^ { \circ }$ 
B)  $12 \sqrt { 2 } ; 135 ^ { \circ }$ 
C)  $12 ; 225 ^ { \circ }$ 
D)  $12 ; 45 ^ { \circ }$

Question 20

Solve the problem.
-A pilot wants to fly on a bearing of $60.5 ^ { \circ }$. By flying due east, he finds that a 57 -mph wind, blowing from the south, puts him on course. Find the airspeed of the plane.

Accepted Answer

A)  $50 \mathrm { mph }$ 
B)  $101 \mathrm { mph }$ 
C)  $116 \mathrm { mph }$ 
D)  $151 \mathrm { mph }$ 
A)  $50 \mathrm { mph }$ 
B)  $101 \mathrm { mph }$ 
C)  $116 \mathrm { mph }$ 
D)  $151 \mathrm { mph }$

Sketch the vectors u and w with angle θ between them and sketch the resultant. - $| \mathbf { u } | = 5 , | \mathbf { w } | = 1 , \theta = 60 ^ { \circ }$

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 25 \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right) } { 5 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right) }$

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

Find the dot product for the pair of vectors. - $\langle 4,8 \rangle , \langle - 7 , - 14 \rangle$

Find sum of the pair of complex numbers. - $9 - 2 i , - 4 i$

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

Graph the polar equation for θ in [0°, 360°). - $r = 3 - \cos \theta$ $Graph the polar equation for θ in [0°, 360°). - r = 3 - \cos \theta$

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

Find all cube roots of the complex number. Leave answers in trigonometric form. - $2 + 2 i \sqrt { 3 }$

Find the angle between the pair of vectors to the nearest tenth of a degree. - $7 \mathbf { i } - 5 \mathbf { j } , 2 \mathbf { i } - 6 \mathbf { j }$

Find the missing parts of the triangle. - =43 =6.15 =27.75 If necessary, round side lengths to the nearest hundredth.

Find all solutions of the equation. Leave answers in trigonometric form. - $x ^ { 2 } + 1 - \sqrt { 3 } i = 0$

Sketch the vectors u and w with angle θ between them and sketch the resultant. - $| \mathbf { u } | = 3 , | \mathbf { w } | = 12 , \theta = 40 ^ { \circ }$

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

Find all cube roots of the complex number. Leave answers in trigonometric form. - $- 6 \sqrt { 3 } - 6 i$

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

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in [0,360°]. - $\langle - 6 \sqrt { 2 } , - 6 \sqrt { 2 } \rangle$

Solve the problem. -A pilot wants to fly on a bearing of $60.5 ^ { \circ }$ . By flying due east, he finds that a 57 -mph wind, blowing from the south, puts him on course. Find the airspeed of the plane.

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

Review of Basic Concepts

Filters

Exam 8: Applications of Trigonometry

Sketch the vectors u and w with angle θ between them and sketch the resultant. - ∣u∣=5,∣w∣=1,θ=60∘| \mathbf { u } | = 5 , | \mathbf { w } | = 1 , \theta = 60 ^ { \circ }∣u∣=5,∣w∣=1,θ=60∘

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=2π3\mathrm { x } = \sin \mathrm { t } , \mathrm { y } = \cos \mathrm { t } ; \mathrm { t } = \frac { 2 \pi } { 3 }x=sint,y=cost;t=32π​

Find the dot product for the pair of vectors. - ⟨4,8⟩,⟨−7,−14⟩\langle 4,8 \rangle , \langle - 7 , - 14 \rangle⟨4,8⟩,⟨−7,−14⟩

Find sum of the pair of complex numbers. - 9−2i,−4i9 - 2 i , - 4 i9−2i,−4i

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

Graph the polar equation for θ in [0°, 360°). - r=3−cos⁡θr = 3 - \cos \thetar=3−cosθ

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

Find all cube roots of the complex number. Leave answers in trigonometric form. - 2+2i32 + 2 i \sqrt { 3 }2+2i3​

Find the angle between the pair of vectors to the nearest tenth of a degree. - 7i−5j,2i−6j7 \mathbf { i } - 5 \mathbf { j } , 2 \mathbf { i } - 6 \mathbf { j }7i−5j,2i−6j

Find the missing parts of the triangle. - =43 =6.15 =27.75 If necessary, round side lengths to the nearest hundredth.

Find all solutions of the equation. Leave answers in trigonometric form. - x2+1−3i=0x ^ { 2 } + 1 - \sqrt { 3 } i = 0x2+1−3​i=0

Sketch the vectors u and w with angle θ between them and sketch the resultant. - ∣u∣=3,∣w∣=12,θ=40∘| \mathbf { u } | = 3 , | \mathbf { w } | = 12 , \theta = 40 ^ { \circ }∣u∣=3,∣w∣=12,θ=40∘

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

Find all cube roots of the complex number. Leave answers in trigonometric form. - −63−6i- 6 \sqrt { 3 } - 6 i−63​−6i

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

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in [0,360°]. - ⟨−62,−62⟩\langle - 6 \sqrt { 2 } , - 6 \sqrt { 2 } \rangle⟨−62​,−62​⟩

Solve the problem. -A pilot wants to fly on a bearing of 60.5∘60.5 ^ { \circ }60.5∘ . By flying due east, he finds that a 57 -mph wind, blowing from the south, puts him on course. Find the airspeed of the plane.

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

Review of Basic Concepts

Filters

Sketch the vectors u and w with angle θ between them and sketch the resultant. - $| \mathbf { u } | = 5 , | \mathbf { w } | = 1 , \theta = 60 ^ { \circ }$

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

Find the dot product for the pair of vectors. - $\langle 4,8 \rangle , \langle - 7 , - 14 \rangle$

Find sum of the pair of complex numbers. - $9 - 2 i , - 4 i$

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

Graph the polar equation for θ in [0°, 360°). - $r = 3 - \cos \theta$ $Graph the polar equation for θ in [0°, 360°). - r = 3 - \cos \theta$

Find all cube roots of the complex number. Leave answers in trigonometric form. - $2 + 2 i \sqrt { 3 }$

Find the angle between the pair of vectors to the nearest tenth of a degree. - $7 \mathbf { i } - 5 \mathbf { j } , 2 \mathbf { i } - 6 \mathbf { j }$

Find all solutions of the equation. Leave answers in trigonometric form. - $x ^ { 2 } + 1 - \sqrt { 3 } i = 0$

Sketch the vectors u and w with angle θ between them and sketch the resultant. - $| \mathbf { u } | = 3 , | \mathbf { w } | = 12 , \theta = 40 ^ { \circ }$

Find all cube roots of the complex number. Leave answers in trigonometric form. - $- 6 \sqrt { 3 } - 6 i$

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in [0,360°]. - $\langle - 6 \sqrt { 2 } , - 6 \sqrt { 2 } \rangle$

Solve the problem. -A pilot wants to fly on a bearing of $60.5 ^ { \circ }$ . By flying due east, he finds that a 57 -mph wind, blowing from the south, puts him on course. Find the airspeed of the plane.