Question 1

Let $\mathbf { u } = \langle 6,5 \rangle$ and $\mathbf { v } = \langle 12,7 \rangle$. Find $\mathbf { u } - \mathbf { v }$.

Accepted Answer

A)  $\langle 13,17 \rangle$ 
B)  $\langle - 6 , - 2 \rangle$ 
C)  $\langle 18,12 \rangle$ 
D)  $\langle - 1 , - 7 \rangle$ 
E)  $\langle 17,13 \rangle$ 
A)  $\langle 13,17 \rangle$ 
B)  $\langle - 6 , - 2 \rangle$ 
C)  $\langle 18,12 \rangle$ 
D)  $\langle - 1 , - 7 \rangle$ 
E)  $\langle 17,13 \rangle$

Question 2

A 400 -pound trailer is sitting on an exit ramp inclined at 31° on Highway 35. How much force is required to keep the trailer from rolling back down the exit ramp? Round
Answer to two decimal places.

Accepted Answer

A)  206.02 pounds 
B)  342.87 pounds 
C)  274.44 pounds 
D)  308.65 pounds 
E)  325.76 pounds 
A)  206.02 pounds 
B)  342.87 pounds 
C)  274.44 pounds 
D)  308.65 pounds 
E)  325.76 pounds

Question 3

Given $a = 6 , b = 9$, and $c = 7$, use the Law of Cosines to solve the triangle for the value of $A$. Round answer to two decimal places.

Figure not drawn to scale

Accepted Answer

A)  $60.33^{\circ}$ 
B)  $50.98^{\circ}$ 
C)  $80.44^{\circ}$ 
D)  $87.27^{\circ}$ 
E)  $41.75^{\circ}$ 
A)  $60.33^{\circ}$ 
B)  $50.98^{\circ}$ 
C)  $80.44^{\circ}$ 
D)  $87.27^{\circ}$ 
E)  $41.75^{\circ}$

Question 4

Which of the following is a fourth root of 1 ?
I. $\cos \left( \frac { 3 \pi } { 2 } \right) + i \sin \left( \frac { 3 \pi } { 2 } \right)$
II. $\frac { 1 } { 4 } ( \cos ( 0 ) + i \sin ( 0 ) )$
III. $\cos \left( \frac { \pi } { 4 } \right) + i \sin \left( \frac { \pi } { 4 } \right)$

Accepted Answer

A)  None are roots. 
B)  II and III are the only roots. 
C)  I and II are the only roots. 
D)  I, II, and III are roots. 
E)  I is the only root. 
A)  None are roots. 
B)  II and III are the only roots. 
C)  I and II are the only roots. 
D)  I, II, and III are roots. 
E)  I is the only root.

Question 5

An airplane is flying at a bearing of $143 ^ { \circ }$ with an airspeed of 335 kilometers per hour. Because of the wind, its ground speed and direction are, respectively, 325 kilometers per hour and $138 ^ { \circ }$. Find the direction and speed of the wind. Round your answer to two decimals.

Accepted Answer

A)  $31.36 ^ { \circ } , 30.47$ kilometers per hour 
B)  $121.36 ^ { \circ } , 25.71$ kilometers per hour 
C)  $- 37.01 ^ { \circ } , 30.47$ kilometers per hour 
D)  $31.36 ^ { \circ } , 25.71$ kilometers per hour 
E)  $121.36 ^ { \circ } , 25.69$ kilometers per hour 
A)  $31.36 ^ { \circ } , 30.47$ kilometers per hour 
B)  $121.36 ^ { \circ } , 25.71$ kilometers per hour 
C)  $- 37.01 ^ { \circ } , 30.47$ kilometers per hour 
D)  $31.36 ^ { \circ } , 25.71$ kilometers per hour 
E)  $121.36 ^ { \circ } , 25.69$ kilometers per hour

Question 6

Given $A = 51 ^ { \circ } , b = 6$, and $c = 9$, use the Law of Cosines to solve the triangle for the value of $a$. Round answer to two decimal places.
 
Figure not drawn to scale

Accepted Answer

A)  $7.00$ 
B)  $10.30$ 
C)  $7.83$ 
D)  $13.60$ 
E)  $8.65$ 
A)  $7.00$ 
B)  $10.30$ 
C)  $7.83$ 
D)  $13.60$ 
E)  $8.65$

Question 7

$$\text { In the figure below, } ( a , b ) = ( 4,6 ) \text { and } ( c , d ) = ( 6,5 ) . \text { Find } ( x , y ) \text { so that } \mathbf { u } = \mathbf { v }$$

Accepted Answer

A)  (2, -1) 
B)  (12,9) 
C)  (0,1) 
D)  (9,12) 
E)  (10,11) 
A)  (2, -1) 
B)  (12,9) 
C)  (0,1) 
D)  (9,12) 
E)  (10,11)

Question 8

Given $C = 130 ^ { \circ } , a = 19.9$, and $c = 15.3$, use the Law of Sines to solve the triangle (if possible) for the value of $b$. If two solutions exist, find both. Round answer to two decimal places.

Accepted Answer

A)  $b = 4.20$ 
B)  $b = 1.76$ and $6.08$ 
C)  $b = 7.45$ 
D)  $b = 0.83$ and $6.21$ 
E)  not possible 
A)  $b = 4.20$ 
B)  $b = 1.76$ and $6.08$ 
C)  $b = 7.45$ 
D)  $b = 0.83$ and $6.21$ 
E)  not possible

Question 9

A park ranger at point $A$ observes a fire in the direction $\mathrm { N } 29 ^ { \circ } 15 ^ { \prime } \mathrm { E }$. Another ranger at point $B , 9$ miles due east of $A$, sites the same fire at $\mathrm { N } 50 ^ { \circ } 39 ^ { \prime } \mathrm { W }$. Determine the distance from point $B$ to the fire. Round answer to two decimal places.

Accepted Answer

A)  4.47 miles 
B)  4.72 miles 
C)  7.98 miles 
D)  7.07 miles 
E)  7.47 miles 
A)  4.47 miles 
B)  4.72 miles 
C)  7.98 miles 
D)  7.07 miles 
E)  7.47 miles

Question 10

Determine whether u and v are orthogonal, parallel, or neither. $$\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 25,2 \rangle$$

Accepted Answer

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

Question 11

Perform the operation shown below and leave the result in trigonometric form.
$$\frac { 36 ( \cos 2.5 + i \sin 2.5 ) } { 6 ( \cos 1.7 + i \sin 1.7 ) }$$

Accepted Answer

A)  $6 ( \cos 0.8 + i \sin 0.8 )$ 
B)  $30 ( \cos 0.8 + i \sin 0.8 )$ 
C)  $30 ( \cos 1.5 + i \sin 1.5 )$ 
D)  $6 ( \cos 4.2 + i \sin 4.2 )$ 
E)  $6 ( \cos 4.25 + i \sin 4.25 )$ 
A)  $6 ( \cos 0.8 + i \sin 0.8 )$ 
B)  $30 ( \cos 0.8 + i \sin 0.8 )$ 
C)  $30 ( \cos 1.5 + i \sin 1.5 )$ 
D)  $6 ( \cos 4.2 + i \sin 4.2 )$ 
E)  $6 ( \cos 4.25 + i \sin 4.25 )$

Question 12

Which of the following is a fifth root of $2 ( - 1 + i ) ? \quad 0 \quad 2$
I. $\sqrt [ 10 ] { 8 } \left( \cos \left( \frac { 3 \pi } { 20 } \right) + i \sin \left( \frac { 3 \pi } { 20 } \right) \right)$
II. $\sqrt [ 10 ] { 8 } \left( \cos \left( \frac { 19 \pi } { 20 } \right) + i \sin \left( \frac { 19 \pi } { 20 } \right) \right)$
III. $\sqrt [ 5 ] { 2 } \left( \cos \left( \frac { 3 \pi } { 20 } \right) + i \sin \left( \frac { 3 \pi } { 20 } \right) \right)$

Accepted Answer

A)  I is the only root. 
B)  None are roots. 
C)  III is the only root. 
D)  I and II are the only roots. 
E)  I and III are the only roots. 
A)  I is the only root. 
B)  None are roots. 
C)  III is the only root. 
D)  I and II are the only roots. 
E)  I and III are the only roots.

Question 13

Use Heron's area formula to find the area of the triangle pictured below, if $a = 8$ inches, $\mathrm { b } = 14$ inches, and $\mathrm { c } = 10$ inches.

Accepted Answer

A)  $4 \sqrt { 6 }$ 
B)  40 
C)  56 
D)  $12 \sqrt { 66 }$ 
E)  $16 \sqrt { 6 }$ 
A)  $4 \sqrt { 6 }$ 
B)  40 
C)  56 
D)  $12 \sqrt { 66 }$ 
E)  $16 \sqrt { 6 }$

Question 14

A boat race runs along a triangular course marked by buoys A, B, and C. The race begins with the boats headed west for 3900 meters. The other two sides of the course
Lie to the north of the first side, and their lengths are 1900 meters and 2500 meters,
Respectively. Find the bearing for the last leg of the race. Round your answer to two
Decimals.

Accepted Answer

A)  -inf° 
B)  113.74° 
C)  180.01° 
D)  156.26° 
E)  90.01° 
A)  -inf° 
B)  113.74° 
C)  180.01° 
D)  156.26° 
E)  90.01°

Question 15

$\text { Find the fourth roots of } \frac{1}{2}-\frac{\sqrt{3}}{2} i \text {. Write the roots in trigonometric form. }$

Accepted Answer

A) 
$\begin{array}{l}
w_{1}=\cos \left(65^{\circ}\right)+i \sin \left(65^{\circ}\right) \
w_{2}=\cos \left(155^{\circ}\right)+i \sin \left(155^{\circ}\right) \
w_{3}=\cos \left(245^{\circ}\right)+i \sin \left(245^{\circ}\right) \
w_{4}=\cos \left(335^{\circ}\right)+i \sin \left(335^{\circ}\right)
\end{array}$ 
B)  
$\begin{array}{l}
w_{1}=\cos \left(75^{\circ}\right)+i \sin \left(75^{\circ}\right) \
w_{2}=\cos \left(165^{\circ}\right)+i \sin \left(165^{\circ}\right) \
w_{3}=\cos \left(255^{\circ}\right)+i \sin \left(255^{\circ}\right) \
w_{4}=\cos \left(345^{\circ}\right)+i \sin \left(345^{\circ}\right)
\end{array}$ 
C) 
$\begin{array}{l}
w_{1}=\cos \left(80^{\circ}\right)+i \sin \left(80^{\circ}\right) \
w_{2}=\cos \left(170^{\circ}\right)+i \sin \left(170^{\circ}\right) \
w_{3}=\cos \left(260^{\circ}\right)+i \sin \left(260^{\circ}\right) \
w_{4}=\cos \left(350^{\circ}\right)+i \sin \left(350^{\circ}\right)
\end{array}$ 
D) 
$\begin{array}{l}
w_{1}=\cos \left(70^{\circ}\right)+i \sin \left(70^{\circ}\right) \
w_{2}=\cos \left(160^{\circ}\right)+i \sin \left(160^{\circ}\right) \
w_{3}=\cos \left(250^{\circ}\right)+i \sin \left(250^{\circ}\right) \
w_{4}=\cos \left(340^{\circ}\right)+i \sin \left(340^{\circ}\right)
\end{array}$ 
E) 
$\begin{array}{l}
w_{1}=\cos \left(85^{\circ}\right)+i \sin \left(85^{\circ}\right) \
w_{2}=\cos \left(175^{\circ}\right)+i \sin \left(175^{\circ}\right) \
w_{3}=\cos \left(265^{\circ}\right)+i \sin \left(265^{\circ}\right) \
w_{4}=\cos \left(355^{\circ}\right)+i \sin \left(355^{\circ}\right)
\end{array}$ 
A) 
$\begin{array}{l}
w_{1}=\cos \left(65^{\circ}\right)+i \sin \left(65^{\circ}\right) \
w_{2}=\cos \left(155^{\circ}\right)+i \sin \left(155^{\circ}\right) \
w_{3}=\cos \left(245^{\circ}\right)+i \sin \left(245^{\circ}\right) \
w_{4}=\cos \left(335^{\circ}\right)+i \sin \left(335^{\circ}\right)
\end{array}$ 
B)  
$\begin{array}{l}
w_{1}=\cos \left(75^{\circ}\right)+i \sin \left(75^{\circ}\right) \
w_{2}=\cos \left(165^{\circ}\right)+i \sin \left(165^{\circ}\right) \
w_{3}=\cos \left(255^{\circ}\right)+i \sin \left(255^{\circ}\right) \
w_{4}=\cos \left(345^{\circ}\right)+i \sin \left(345^{\circ}\right)
\end{array}$ 
C) 
$\begin{array}{l}
w_{1}=\cos \left(80^{\circ}\right)+i \sin \left(80^{\circ}\right) \
w_{2}=\cos \left(170^{\circ}\right)+i \sin \left(170^{\circ}\right) \
w_{3}=\cos \left(260^{\circ}\right)+i \sin \left(260^{\circ}\right) \
w_{4}=\cos \left(350^{\circ}\right)+i \sin \left(350^{\circ}\right)
\end{array}$ 
D) 
$\begin{array}{l}
w_{1}=\cos \left(70^{\circ}\right)+i \sin \left(70^{\circ}\right) \
w_{2}=\cos \left(160^{\circ}\right)+i \sin \left(160^{\circ}\right) \
w_{3}=\cos \left(250^{\circ}\right)+i \sin \left(250^{\circ}\right) \
w_{4}=\cos \left(340^{\circ}\right)+i \sin \left(340^{\circ}\right)
\end{array}$ 
E) 
$\begin{array}{l}
w_{1}=\cos \left(85^{\circ}\right)+i \sin \left(85^{\circ}\right) \
w_{2}=\cos \left(175^{\circ}\right)+i \sin \left(175^{\circ}\right) \
w_{3}=\cos \left(265^{\circ}\right)+i \sin \left(265^{\circ}\right) \
w_{4}=\cos \left(355^{\circ}\right)+i \sin \left(355^{\circ}\right)
\end{array}$

Question 16

Use DeMoivre's Theorem to find the indicated power of the following complex number. $$( 3 + 3 \sqrt { 3 } i ) ^ { 6 }$$

Accepted Answer

A)  19,683 
B)  6,561 
C)  2,187 
D)  46,656 
E)  243 
A)  19,683 
B)  6,561 
C)  2,187 
D)  46,656 
E)  243

Question 17

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form.
$$( - 4 - 4 i ) ( - 3 - 3 i )$$

Accepted Answer

A)  $- 12 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$ 
B)  $- 24 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
C)  $12 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
D)  $24 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
E)  $12 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$ 
A)  $- 12 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$ 
B)  $- 24 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
C)  $12 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
D)  $24 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
E)  $12 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$

Question 18

A plane flies 740 miles from City A to City B with a bearing of 25° (clockwise from north). Then it flies 614 miles from City B to City C with a bearing of 49°. Find the
Straight-line distance from City C to City A. Round your answer to two decimals.

Accepted Answer

A)  307.31 miles 
B)  1347.06 miles 
C)  1137.63 miles 
D)  911.87 miles 
E)  1324.67 miles 
A)  307.31 miles 
B)  1347.06 miles 
C)  1137.63 miles 
D)  911.87 miles 
E)  1324.67 miles

Question 19

Find the standard form of the complex number shown below.
$11 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)$

Accepted Answer

A)  $- \frac { 11 } { 2 } + \frac { 11 \sqrt { 3 } } { 2 } i$ 
B)  $\frac { 11 \sqrt { 3 } } { 4 } - \frac { 11 } { 4 } i$ 
C)  $- \frac { 2 } { 11 } + \frac { 2 \sqrt { 3 } } { 11 } i$ 
D)  $- \frac { 11 \sqrt { 3 } } { 2 } + \frac { 11 } { 2 } i$ 
E)  $\frac { 4 \sqrt { 3 } } { 11 } - \frac { 4 } { 11 } i$ 
A)  $- \frac { 11 } { 2 } + \frac { 11 \sqrt { 3 } } { 2 } i$ 
B)  $\frac { 11 \sqrt { 3 } } { 4 } - \frac { 11 } { 4 } i$ 
C)  $- \frac { 2 } { 11 } + \frac { 2 \sqrt { 3 } } { 11 } i$ 
D)  $- \frac { 11 \sqrt { 3 } } { 2 } + \frac { 11 } { 2 } i$ 
E)  $\frac { 4 \sqrt { 3 } } { 11 } - \frac { 4 } { 11 } i$

Question 20

A park ranger at point A observes a fire in the direction $\mathrm { N } 27 ^ { \circ } 27 ^ { \prime } \mathrm { E }$ . Another ranger at point B, 9 miles due east of A, sites the same fire at $\mathrm { N } 54 ^ { \circ } 28 ^ { \prime } \mathrm { W }$ . Determine the Distance from point B to the fire. Round answer to two decimal places.

Accepted Answer

A)  4.19 miles 
B)  4.66 miles 
C)  8.07 miles 
D)  7.40 miles 
E)  8.22 miles 
A)  4.19 miles 
B)  4.66 miles 
C)  8.07 miles 
D)  7.40 miles 
E)  8.22 miles

Let $\mathbf { u } = \langle 6,5 \rangle$ and $\mathbf { v } = \langle 12,7 \rangle$ . Find $\mathbf { u } - \mathbf { v }$ .

A 400 -pound trailer is sitting on an exit ramp inclined at 31° on Highway 35. How much force is required to keep the trailer from rolling back down the exit ramp? Round Answer to two decimal places.

Given $a = 6 , b = 9$ , and $c = 7$ , use the Law of Cosines to solve the triangle for the value of $A$ . Round answer to two decimal places. Figure not drawn to scale

Which of the following is a fourth root of 1 ? I. $\cos \left( \frac { 3 \pi } { 2 } \right) + i \sin \left( \frac { 3 \pi } { 2 } \right)$ II. $\frac { 1 } { 4 } ( \cos ( 0 ) + i \sin ( 0 ) )$ III. $\cos \left( \frac { \pi } { 4 } \right) + i \sin \left( \frac { \pi } { 4 } \right)$

An airplane is flying at a bearing of $143 ^ { \circ }$ with an airspeed of 335 kilometers per hour. Because of the wind, its ground speed and direction are, respectively, 325 kilometers per hour and $138 ^ { \circ }$ . Find the direction and speed of the wind. Round your answer to two decimals.

Given $C = 130 ^ { \circ } , a = 19.9$ , and $c = 15.3$ , use the Law of Sines to solve the triangle (if possible) for the value of $b$ . If two solutions exist, find both. Round answer to two decimal places.

Determine whether u and v are orthogonal, parallel, or neither. $\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 25,2 \rangle$

Perform the operation shown below and leave the result in trigonometric form. $\frac { 36 ( \cos 2.5 + i \sin 2.5 ) } { 6 ( \cos 1.7 + i \sin 1.7 ) }$

$\text { Find the fourth roots of } \frac{1}{2}-\frac{\sqrt{3}}{2} i \text {. Write the roots in trigonometric form. }$

Use DeMoivre's Theorem to find the indicated power of the following complex number. $( 3 + 3 \sqrt { 3 } i ) ^ { 6 }$

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. $( - 4 - 4 i ) ( - 3 - 3 i )$

A plane flies 740 miles from City A to City B with a bearing of 25° (clockwise from north). Then it flies 614 miles from City B to City C with a bearing of 49°. Find the Straight-line distance from City C to City A. Round your answer to two decimals.

Find the standard form of the complex number shown below. $11 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Review of Graphs, Equations, and Inequalities

Filters

Exam 6: Additional Topics in Trigonometry

Let u=⟨6,5⟩\mathbf { u } = \langle 6,5 \rangleu=⟨6,5⟩ and v=⟨12,7⟩\mathbf { v } = \langle 12,7 \ranglev=⟨12,7⟩ . Find u−v\mathbf { u } - \mathbf { v }u−v .

A 400 -pound trailer is sitting on an exit ramp inclined at 31° on Highway 35. How much force is required to keep the trailer from rolling back down the exit ramp? Round Answer to two decimal places.

Given a=6,b=9a = 6 , b = 9a=6,b=9 , and c=7c = 7c=7 , use the Law of Cosines to solve the triangle for the value of AAA . Round answer to two decimal places. Figure not drawn to scale

Given A=51∘,b=6A = 51 ^ { \circ } , b = 6A=51∘,b=6 , and c=9c = 9c=9 , use the Law of Cosines to solve the triangle for the value of aaa . Round answer to two decimal places. Figure not drawn to scale

Given C=130∘,a=19.9C = 130 ^ { \circ } , a = 19.9C=130∘,a=19.9 , and c=15.3c = 15.3c=15.3 , use the Law of Sines to solve the triangle (if possible) for the value of bbb . If two solutions exist, find both. Round answer to two decimal places.

Determine whether u and v are orthogonal, parallel, or neither. u=⟨1,5⟩,v=⟨−25,2⟩\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 25,2 \rangleu=⟨1,5⟩,v=⟨−25,2⟩

Perform the operation shown below and leave the result in trigonometric form. 36(cos⁡2.5+isin⁡2.5)6(cos⁡1.7+isin⁡1.7)\frac { 36 ( \cos 2.5 + i \sin 2.5 ) } { 6 ( \cos 1.7 + i \sin 1.7 ) }6(cos1.7+isin1.7)36(cos2.5+isin2.5)​

Use Heron's area formula to find the area of the triangle pictured below, if a=8a = 8a=8 inches, b=14\mathrm { b } = 14b=14 inches, and c=10\mathrm { c } = 10c=10 inches.

Find the fourth roots of 12−32i. Write the roots in trigonometric form. \text { Find the fourth roots of } \frac{1}{2}-\frac{\sqrt{3}}{2} i \text {. Write the roots in trigonometric form. } Find the fourth roots of 21​−23​​i. Write the roots in trigonometric form.

Use DeMoivre's Theorem to find the indicated power of the following complex number. (3+33i)6( 3 + 3 \sqrt { 3 } i ) ^ { 6 }(3+33​i)6

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. (−4−4i)(−3−3i)( - 4 - 4 i ) ( - 3 - 3 i )(−4−4i)(−3−3i)

A plane flies 740 miles from City A to City B with a bearing of 25° (clockwise from north). Then it flies 614 miles from City B to City C with a bearing of 49°. Find the Straight-line distance from City C to City A. Round your answer to two decimals.

Find the standard form of the complex number shown below. 11(cos⁡5π6+isin⁡5π6)11 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)11(cos65π​+isin65π​)

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Review of Graphs, Equations, and Inequalities

Filters

Let $\mathbf { u } = \langle 6,5 \rangle$ and $\mathbf { v } = \langle 12,7 \rangle$ . Find $\mathbf { u } - \mathbf { v }$ .

Given $a = 6 , b = 9$ , and $c = 7$ , use the Law of Cosines to solve the triangle for the value of $A$ . Round answer to two decimal places. Figure not drawn to scale

Given $C = 130 ^ { \circ } , a = 19.9$ , and $c = 15.3$ , use the Law of Sines to solve the triangle (if possible) for the value of $b$ . If two solutions exist, find both. Round answer to two decimal places.

Determine whether u and v are orthogonal, parallel, or neither. $\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 25,2 \rangle$

Perform the operation shown below and leave the result in trigonometric form. $\frac { 36 ( \cos 2.5 + i \sin 2.5 ) } { 6 ( \cos 1.7 + i \sin 1.7 ) }$

$\text { Find the fourth roots of } \frac{1}{2}-\frac{\sqrt{3}}{2} i \text {. Write the roots in trigonometric form. }$

Use DeMoivre's Theorem to find the indicated power of the following complex number. $( 3 + 3 \sqrt { 3 } i ) ^ { 6 }$

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. $( - 4 - 4 i ) ( - 3 - 3 i )$

Find the standard form of the complex number shown below. $11 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)$