Question 1

Use the dot product to find the magnitude of u.  $\mathbf { u } = 35 \mathbf { i } + 40 \mathbf { j }$

Accepted Answer

A) 40 
B)  $\sqrt { 2829 }$ 
C)  $\sqrt { 2823 }$ 
D)  $5 \sqrt { 113 }$ 
E) 35 
A) 40 
B)  $\sqrt { 2829 }$ 
C)  $\sqrt { 2823 }$ 
D)  $5 \sqrt { 113 }$ 
E) 35

Question 2

Use DeMoivre's Theorem to find the indicated power of the complex number.Write the result in standard form. $\left[ - 5 \left( \cos \left( 210 ^ { \circ } \right) + i \sin \left( 210 ^ { \circ } \right) \right) \right] ^ { 10 }$

Accepted Answer

A)  $- \frac { 9765625 } { 2 } + \frac { 9765625 \sqrt { 3 } } { 2 } i$ 
B)  $\frac { 9765625 } { 2 } + \frac { 9765625 \sqrt { 3 } } { 2 } i$ 
C)  $\frac { 9765625 } { 2 } - \frac { 9765625 \sqrt { 3 } } { 2 } i$ 
D)  $\frac { 9765625 \sqrt { 3 } } { 2 } - \frac { 9765625 } { 2 } i$ 
E)  $- \frac { 9765625 \sqrt { 3 } } { 2 } + \frac { 9765625 } { 2 } i$ 
A)  $- \frac { 9765625 } { 2 } + \frac { 9765625 \sqrt { 3 } } { 2 } i$ 
B)  $\frac { 9765625 } { 2 } + \frac { 9765625 \sqrt { 3 } } { 2 } i$ 
C)  $\frac { 9765625 } { 2 } - \frac { 9765625 \sqrt { 3 } } { 2 } i$ 
D)  $\frac { 9765625 \sqrt { 3 } } { 2 } - \frac { 9765625 } { 2 } i$ 
E)  $- \frac { 9765625 \sqrt { 3 } } { 2 } + \frac { 9765625 } { 2 } i$

Question 3

Represent the complex number graphically.

Accepted Answer

A)    $Z = 2 ( \cos 0 - i \sin 0 )$ 
B)    $Z = 2 ( \cos 0 + i \sin 0 )$ 
C)     $Z = 2 ( \cos 0 - i \sin 0 )$ 
D)     $Z = 2 ( \cos 0 + i \sin 0 )$ 
E)     $Z = 2 ( \cos 0 - i \sin 0 )$ 
A)    $Z = 2 ( \cos 0 - i \sin 0 )$ 
B)    $Z = 2 ( \cos 0 + i \sin 0 )$ 
C)     $Z = 2 ( \cos 0 - i \sin 0 )$ 
D)     $Z = 2 ( \cos 0 + i \sin 0 )$ 
E)     $Z = 2 ( \cos 0 - i \sin 0 )$

Question 4

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. C = 73 $^\circ$ , a = 12.5 and b = 11.5

Accepted Answer

A) 68.96 sq. units 
B) 62.06 sq. units 
C) 55.17 sq. units 
D) 82.75 sq. units 
E) 75.85 sq. units 
A) 68.96 sq. units 
B) 62.06 sq. units 
C) 55.17 sq. units 
D) 82.75 sq. units 
E) 75.85 sq. units

Question 5

Select the graph of all complex numbers z satisfying the given condition.   $| z | = 4$

Accepted Answer

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

Question 6

Find the projection of u onto v if $\mathbf { u } = \langle 4 , - 5 \rangle$ , $\mathbf { v } = \langle 3 , - 1 \rangle$

Accepted Answer

A)   $\left\langle \frac { 51 } { \sqrt { 10 } } , \frac { - 17 } { \sqrt { 10 } } \right\rangle$ 
B)  $\left\langle \frac { - 51 } { \sqrt { 10 } } , \frac { - 17 } { \sqrt { 10 } } \right\rangle$ 
C)   $\left\langle \frac { 51 } { \sqrt { 10 } } , \frac { - 51 } { \sqrt { 10 } } \right\rangle$ 
D)   $\left\langle \frac { 17 } { \sqrt { 10 } } , \frac { - 17 } { \sqrt { 10 } } \right\rangle$ 
E)   $\left\langle \frac { 17 } { \sqrt { 10 } } , \frac { 17 } { \sqrt { 10 } } \right\rangle$ 
A)   $\left\langle \frac { 51 } { \sqrt { 10 } } , \frac { - 17 } { \sqrt { 10 } } \right\rangle$ 
B)  $\left\langle \frac { - 51 } { \sqrt { 10 } } , \frac { - 17 } { \sqrt { 10 } } \right\rangle$ 
C)   $\left\langle \frac { 51 } { \sqrt { 10 } } , \frac { - 51 } { \sqrt { 10 } } \right\rangle$ 
D)   $\left\langle \frac { 17 } { \sqrt { 10 } } , \frac { - 17 } { \sqrt { 10 } } \right\rangle$ 
E)   $\left\langle \frac { 17 } { \sqrt { 10 } } , \frac { 17 } { \sqrt { 10 } } \right\rangle$

Question 7

Find the component form of v if $\| \mathbf { v } \| = 6$ and the angle it makes with the x-axis is 150°.

Accepted Answer

A)   $\langle - 6 \sqrt { 3 } , - 6 \rangle$ 
B)   $\langle - 6,6 \sqrt { 3 } \rangle$ 
C)   $\langle - 3 \sqrt { 3 } , 3 \rangle$ 
D)   $\langle - 3 \sqrt { 2 } , 3 \sqrt { 2 } \rangle$ 
E)   $\langle - 3,3 \sqrt { 3 } \rangle$ 
A)   $\langle - 6 \sqrt { 3 } , - 6 \rangle$ 
B)   $\langle - 6,6 \sqrt { 3 } \rangle$ 
C)   $\langle - 3 \sqrt { 3 } , 3 \rangle$ 
D)   $\langle - 3 \sqrt { 2 } , 3 \sqrt { 2 } \rangle$ 
E)   $\langle - 3,3 \sqrt { 3 } \rangle$

Question 8

Use the vectors $\mathbf { u } = \langle 2,4 \rangle$ , $\mathbf { v } = \langle - 5,3 \rangle$ and $\mathbf { w } = \langle 3 , - 1 \rangle$ to find the indicated quantity.State whether the result is a vector or a scalar.
  $\left( \mathbf { v } ^ { * } \mathbf { u } \right) - \left( \mathbf { w } ^ { \cdot } \mathbf { v } \right)$

Accepted Answer

A) 16; scalar 
B)  $\langle 24,16 \rangle$ ; vector 
C)  $\langle 0,20 \rangle$ ; vector 
D) 22; scalar 
E) 20; scalar 
A) 16; scalar 
B)  $\langle 24,16 \rangle$ ; vector 
C)  $\langle 0,20 \rangle$ ; vector 
D) 22; scalar 
E) 20; scalar

Question 9

Use the dot product to find the magnitude of u if $\mathbf { u } = \langle 6 , - 2 \rangle$

Accepted Answer

A)  $\| \mathbf { u } \| = 4$ 
B)  $\| \mathbf { u } \| = 12$ 
C)  $\| \mathbf { u } \| = 20$ 
D)  $\| \mathbf { u } \| =$ $4 \sqrt { 10 }$ 
E)  $\| \mathbf { u } \| =$ $2 \sqrt { 10 }$ 
A)  $\| \mathbf { u } \| = 4$ 
B)  $\| \mathbf { u } \| = 12$ 
C)  $\| \mathbf { u } \| = 20$ 
D)  $\| \mathbf { u } \| =$ $4 \sqrt { 10 }$ 
E)  $\| \mathbf { u } \| =$ $2 \sqrt { 10 }$

Question 10

Determine whether u are v and orthogonal, parallel, or neither. $\mathbf { u } = \langle - 4 , - 7 \rangle , \mathbf { v } = \langle 14,24 \rangle$

Accepted Answer

A) orthogonal 
B) parallel 
C) neither 
A) orthogonal 
B) parallel 
C) neither

Question 11

Find the product $( - 5 + 5 i ) ( 7 - 7 i )$ using trigonometric forms.Leave the result in trigonometric form.

Accepted Answer

A)  $35 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
B)   $70 \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right)$ 
C)   $35 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right)$ 
D)   $70 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
E)   $70 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right)$ 
A)  $35 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
B)   $70 \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right)$ 
C)   $35 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right)$ 
D)   $70 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
E)   $70 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right)$

Question 12

Find all solutions to the following equation. $x ^ { 3 } - 8 = 0$

Accepted Answer

A)  $x = 1,1 + i \sqrt { 3 } , 1 - i \sqrt { 3 }$ 
B)  $x = 2 , - 2$ 
C)  $x = - 2,2 - i \sqrt { 3 } , 2 + i \sqrt { 3 }$ 
D)  $x = 2$ 
E)  $x = 2 , - 1 + i \sqrt { 3 } , - 1 - i \sqrt { 3 }$ 
A)  $x = 1,1 + i \sqrt { 3 } , 1 - i \sqrt { 3 }$ 
B)  $x = 2 , - 2$ 
C)  $x = - 2,2 - i \sqrt { 3 } , 2 + i \sqrt { 3 }$ 
D)  $x = 2$ 
E)  $x = 2 , - 1 + i \sqrt { 3 } , - 1 - i \sqrt { 3 }$

Question 13

Given vectors $\mathbf { u } = \left\langle u _ { 1 } , u _ { 2 } \right\rangle$ , $\mathbf { v } = \left\langle v _ { 1 } , v _ { 2 } \right\rangle$ and $\mathbf { w } = \left\langle w _ { 1 } , w _ { 2 } \right\rangle$ determine whether the result of the following expression is a vector or a scalar. $6 w \cdot 3 w$

Accepted Answer

A) vector 
B) scalar 
A) vector 
B) scalar

Question 14

Perform the indicated operation using the standard form.   $( \sqrt { 7 } + i ) ( 1 + i )$

Accepted Answer

A)  $( \sqrt { 7 } + i ) ( 1 + i ) \approx 1 + i$ 
B)   $( \sqrt { 7 } + i ) ( 1 + i ) \approx 3.646 - 1.646 i$ 
C)   $( \sqrt { 7 } + i ) ( 1 + i ) \approx 1.646 - 3.646 i$ 
D)   $( \sqrt { 7 } + i ) ( 1 + i ) \approx 1.646 + 3.646 i$ 
E)   $( \sqrt { 7 } + i ) ( 1 + i ) \approx 3.646 + 1.646 i$ 
A)  $( \sqrt { 7 } + i ) ( 1 + i ) \approx 1 + i$ 
B)   $( \sqrt { 7 } + i ) ( 1 + i ) \approx 3.646 - 1.646 i$ 
C)   $( \sqrt { 7 } + i ) ( 1 + i ) \approx 1.646 - 3.646 i$ 
D)   $( \sqrt { 7 } + i ) ( 1 + i ) \approx 1.646 + 3.646 i$ 
E)   $( \sqrt { 7 } + i ) ( 1 + i ) \approx 3.646 + 1.646 i$

Question 15

Perform the operation and leave the result in trigonometric form.   $\frac { 2 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) } { 3 \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right) }$

Accepted Answer

A)  $\frac { 2 } { 3 } \left( \cos 270 ^ { \circ } + i \sin 270 ^ { \circ } \right)$ 
B)   $\frac { 2 } { 3 } \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
C)  $\frac { 2 } { 3 } \left( \cos 270 ^ { \circ } - i \sin 270 ^ { \circ } \right)$ 
D)   $\frac { 2 } { 3 } \left( \cos 30 ^ { \circ } - i \sin 30 ^ { \circ } \right)$ 
E)  $\frac { 2 } { 3 } \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right)$ 
A)  $\frac { 2 } { 3 } \left( \cos 270 ^ { \circ } + i \sin 270 ^ { \circ } \right)$ 
B)   $\frac { 2 } { 3 } \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
C)  $\frac { 2 } { 3 } \left( \cos 270 ^ { \circ } - i \sin 270 ^ { \circ } \right)$ 
D)   $\frac { 2 } { 3 } \left( \cos 30 ^ { \circ } - i \sin 30 ^ { \circ } \right)$ 
E)  $\frac { 2 } { 3 } \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right)$

Question 16

Given a = 7, b = 10, and c = 11, use the Law of Cosines to solve the triangle for the value of С.Round your answer to two decimal places.   Figure not drawn to scale

Accepted Answer

A) 62.96° 
B) 80.44° 
C) 78.46° 
D) 38.57° 
E) 60.33° 
A) 62.96° 
B) 80.44° 
C) 78.46° 
D) 38.57° 
E) 60.33°

Question 17

Given a = 9, b = 13, and c = 8, use the Law of Cosines to solve the triangle for the value of A.Round your answer to two decimal places.   Figure not drawn to scale

Accepted Answer

A) 99.59° 
B) 80.44° 
C) 37.36° 
D) 43.05° 
E) 60.33° 
A) 99.59° 
B) 80.44° 
C) 37.36° 
D) 43.05° 
E) 60.33°

Question 18

Find values for b such that the triangle has no solutions.  
A = 54°, a = 19

Accepted Answer

A)  $b  \frac { 19 } { \sin 54 ^ { \circ } }$ 
A)  $b  \frac { 19 } { \sin 54 ^ { \circ } }$

Question 19

Given a = 10, b = 12, and c = 8, use the Law of Cosines to solve the triangle for the value of B.Round your answer to two decimal places. ​

Accepted Answer

A) 82.82° 
B) 55.77° 
C) 41.41° 
D) 80.44° 
E) 60.33° 
A) 82.82° 
B) 55.77° 
C) 41.41° 
D) 80.44° 
E) 60.33°

Question 20

A force of $y = 50$ pounds exerted at an angle of $30 ^ { \circ }$ above the horizontal is required to slide a table across a floor (see figure).The table is dragged $x = 12$ feet.Determine the work done in sliding the table.

Accepted Answer

A) 539.6 ft-lb 
B) 599.6 ft-lb 
C) 519.6 ft-lb 
D) 579.6 ft-lb 
E) 559.6 ft-lb 
A) 539.6 ft-lb 
B) 599.6 ft-lb 
C) 519.6 ft-lb 
D) 579.6 ft-lb 
E) 559.6 ft-lb

Use the dot product to find the magnitude of u. $\mathbf { u } = 35 \mathbf { i } + 40 \mathbf { j }$

Use DeMoivre's Theorem to find the indicated power of the complex number.Write the result in standard form. $\left[ - 5 \left( \cos \left( 210 ^ { \circ } \right) + i \sin \left( 210 ^ { \circ } \right) \right) \right] ^ { 10 }$

Represent the complex number graphically.

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. C = 73 $^\circ$ , a = 12.5 and b = 11.5

Select the graph of all complex numbers z satisfying the given condition. $| z | = 4$

Find the projection of u onto v if $\mathbf { u } = \langle 4 , - 5 \rangle$ , $\mathbf { v } = \langle 3 , - 1 \rangle$

Find the component form of v if $\| \mathbf { v } \| = 6$ and the angle it makes with the x-axis is 150°.

Use the dot product to find the magnitude of u if $\mathbf { u } = \langle 6 , - 2 \rangle$

Determine whether u are v and orthogonal, parallel, or neither. $\mathbf { u } = \langle - 4 , - 7 \rangle , \mathbf { v } = \langle 14,24 \rangle$

Find the product $( - 5 + 5 i ) ( 7 - 7 i )$ using trigonometric forms.Leave the result in trigonometric form.

Find all solutions to the following equation. $x ^ { 3 } - 8 = 0$

Perform the indicated operation using the standard form. $( \sqrt { 7 } + i ) ( 1 + i )$

Perform the operation and leave the result in trigonometric form. $\frac { 2 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) } { 3 \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right) }$

Given a = 7, b = 10, and c = 11, use the Law of Cosines to solve the triangle for the value of С.Round your answer to two decimal places. Figure not drawn to scale

Given a = 9, b = 13, and c = 8, use the Law of Cosines to solve the triangle for the value of A.Round your answer to two decimal places. Figure not drawn to scale

Find values for b such that the triangle has no solutions. A = 54°, a = 19

Given a = 10, b = 12, and c = 8, use the Law of Cosines to solve the triangle for the value of B.Round your answer to two decimal places.

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

Filters

Exam 6: Additional Topics In Trigonometery

Use the dot product to find the magnitude of u. u=35i+40j\mathbf { u } = 35 \mathbf { i } + 40 \mathbf { j }u=35i+40j

Represent the complex number graphically.

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. C = 73 ∘^\circ∘ , a = 12.5 and b = 11.5

Select the graph of all complex numbers z satisfying the given condition. ∣z∣=4| z | = 4∣z∣=4

Find the projection of u onto v if u=⟨4,−5⟩\mathbf { u } = \langle 4 , - 5 \rangleu=⟨4,−5⟩ , v=⟨3,−1⟩\mathbf { v } = \langle 3 , - 1 \ranglev=⟨3,−1⟩

Find the component form of v if ∥v∥=6\| \mathbf { v } \| = 6∥v∥=6 and the angle it makes with the x-axis is 150°.

Use the dot product to find the magnitude of u if u=⟨6,−2⟩\mathbf { u } = \langle 6 , - 2 \rangleu=⟨6,−2⟩

Determine whether u are v and orthogonal, parallel, or neither. u=⟨−4,−7⟩,v=⟨14,24⟩\mathbf { u } = \langle - 4 , - 7 \rangle , \mathbf { v } = \langle 14,24 \rangleu=⟨−4,−7⟩,v=⟨14,24⟩

Find the product (−5+5i)(7−7i)( - 5 + 5 i ) ( 7 - 7 i )(−5+5i)(7−7i) using trigonometric forms.Leave the result in trigonometric form.

Find all solutions to the following equation. x3−8=0x ^ { 3 } - 8 = 0x3−8=0

Perform the indicated operation using the standard form. (7+i)(1+i)( \sqrt { 7 } + i ) ( 1 + i )(7​+i)(1+i)

Given a = 7, b = 10, and c = 11, use the Law of Cosines to solve the triangle for the value of С.Round your answer to two decimal places. Figure not drawn to scale

Given a = 9, b = 13, and c = 8, use the Law of Cosines to solve the triangle for the value of A.Round your answer to two decimal places. Figure not drawn to scale

Find values for b such that the triangle has no solutions. A = 54°, a = 19

Given a = 10, b = 12, and c = 8, use the Law of Cosines to solve the triangle for the value of B.Round your answer to two decimal places. ​

A force of y=50y = 50y=50 pounds exerted at an angle of 30∘30 ^ { \circ }30∘ above the horizontal is required to slide a table across a floor (see figure).The table is dragged x=12x = 12x=12 feet.Determine the work done in sliding the table.

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

Filters

Use the dot product to find the magnitude of u. $\mathbf { u } = 35 \mathbf { i } + 40 \mathbf { j }$

Determine the area of a triangle having the following measurements.Round your answer to two decimal places. C = 73 $^\circ$ , a = 12.5 and b = 11.5

Select the graph of all complex numbers z satisfying the given condition. $| z | = 4$

Find the projection of u onto v if $\mathbf { u } = \langle 4 , - 5 \rangle$ , $\mathbf { v } = \langle 3 , - 1 \rangle$

Find the component form of v if $\| \mathbf { v } \| = 6$ and the angle it makes with the x-axis is 150°.

Use the dot product to find the magnitude of u if $\mathbf { u } = \langle 6 , - 2 \rangle$

Determine whether u are v and orthogonal, parallel, or neither. $\mathbf { u } = \langle - 4 , - 7 \rangle , \mathbf { v } = \langle 14,24 \rangle$

Find the product $( - 5 + 5 i ) ( 7 - 7 i )$ using trigonometric forms.Leave the result in trigonometric form.

Find all solutions to the following equation. $x ^ { 3 } - 8 = 0$

Perform the indicated operation using the standard form. $( \sqrt { 7 } + i ) ( 1 + i )$

Given a = 10, b = 12, and c = 8, use the Law of Cosines to solve the triangle for the value of B.Round your answer to two decimal places.