Question 1

Graph the complex number.
-$1 - \mathrm { i }$

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 2

Use the figure to find the specified vector.
-Find -a.

Accepted Answer

A)  $\langle 8 , - 6 \rangle$ 
B)  $\langle 6 , - 8 \rangle$ 
C)  $\langle - 6 , - 8 \rangle$ 
D)  $\langle - 8,6 \rangle$

Question 3

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal
places.
-$\left( 15 \operatorname { cis } 18.5 ^ { \circ } \right) \left( 5 \operatorname { cis } 7.5 ^ { \circ } \right)$

Accepted Answer

A)  $3.8128 + 3.2346 \mathrm { i }$ 
B)  $57.1919 + 67.4096 \mathrm { i }$ 
C)  $9.7038 + 11.4384 \mathrm { i }$ 
D)  $67.4096 + 32.8778 \mathrm { i }$

Question 4

Sketch the vectors u and w with angle θ between them and sketch the resultant.
-$$| \mathbf { u } | = 10 , | \mathbf { w } | = 4 , \theta = 40 ^ { \circ }$$

Accepted Answer

A) 
 
B)

Question 5

Find the polar coordinates of the point(s) of intersection of the given curves for 0 ≤ θ < 2π.
-$$r = 10 + 3 \sin \theta , r = 10 + 3 \cos \theta$$

Accepted Answer

A)  $\left( 10 + \frac { 3 \sqrt { 2 } } { 2 } , \frac { 3 \pi } { 4 } \right) , \left( 10 - \frac { 3 \sqrt { 2 } } { 2 } , \frac { 7 \pi } { 4 } \right)$ 
B)  $\left( 10 - \frac { 3 \sqrt { 2 } } { 2 } , \frac { \pi } { 4 } \right) , \left( 10 + \frac { 3 \sqrt { 2 } } { 2 } , \frac { 7 \pi } { 4 } \right)$ 
C)  $\left( 10 + \frac { 3 \sqrt { 2 } } { 2 } , \frac { \pi } { 4 } \right) , \left( 10 - \frac { 3 \sqrt { 2 } } { 2 } , \frac { 5 \pi } { 4 } \right)$ 
D)  $\left\{ 10 - \frac { 3 \sqrt { 2 } } { 2 } , \frac { \pi } { 4 } , \left( 10 + \frac { 3 \sqrt { 2 } } { 2 } , \frac { 5 \pi } { 4 } \right) \right.$

Question 6

The rectangular coordinates of a point are given. Express the point in polar coordinates with r ≥ 0 and 0° ≤ θ < 360°.
-$( 0 , - 2 )$

Accepted Answer

A)  $\left( 2,270 ^ { \circ } \right)$ 
B)  $\left( 2,0 ^ { \circ } \right)$ 
C)  $\left( 2,180 ^ { \circ } \right)$ 
D)  $\left( 2,90 ^ { \circ } \right)$

Question 7

Write the complex number in rectangular form.
-$\operatorname { cis } 210$

Accepted Answer

A)  $- \frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i }$ 
B)  $\frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i }$ 
C)  $- \frac { 1 } { 2 } + \frac { - 1 } { 2 } \mathrm { i }$ 
D)  $- \frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \mathrm { i }$

Question 8

Graph the polar equation for θ in [0°, 360°).
-$$r = 4 \cos 6 \theta$$

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 9

Solve the problem.
-Suppose you would like to cross a 202-foot wide river in a boat. Assume that the boat can travel $39 \mathrm { mph }$ relative to the water and that the current is flowing west at the rate of $6 \mathrm { mph }$. What bearing should be chosen so that the boat will land at a point exactly across from its starting point? Give your answer to the nearest tenth of a degree.

Accepted Answer

A)  $81.2 ^ { \circ }$ 
B)  $5.9 ^ { \circ }$ 
C)  $7.1 ^ { \circ }$ 
D)  $8.8 ^ { \circ }$

Question 10

Find the area of triangle ABC with the given parts. Round to the nearest tenth when necessary.
-$$\begin{array} { l } 
\mathrm { b } = 15.5 \mathrm { ft } \
\mathrm { A } = 36 ^ { \circ } 10 ^ { \prime } \
\mathrm { C } = 95 ^ { \circ } 50 ^ { \prime }
\end{array}$$

Accepted Answer

A)  $189.8 \mathrm { ft } ^ { 2 }$ 
B)  $194.8 \mathrm { ft } ^ { 2 }$ 
C)  $94.9 \mathrm { ft } ^ { 2 }$ 
D)  $89.9 \mathrm { ft } ^ { 2 }$

Question 11

Find sum of the pair of complex numbers.
-$5 - 5 \mathrm { i } , 10 - 11 \mathrm { i }$

Accepted Answer

A)  $16 - 15 \mathrm { i }$ 
B)  $15 - 6 i$ 
C)  $15 - 16 i$ 
D)  $15 + 6 \mathrm { i }$

Question 12

Solve the problem.
-One rope pulls a barge directly east with a force of 68 newtons, and another rope pulls the barge directly north with a force of 62 newtons. Find the magnitude of the resultant force acting on the barge.

Accepted Answer

A)  92 newtons 
B)  6 newtons 
C)  130 newtons 
D)  4200 newtons

Question 13

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

Accepted Answer

A)  $\langle 8,2 \rangle$ 
B)  $\langle 1,7 \rangle$ 
C)  $\langle 9,1 \rangle$ 
D)  $\langle 7 , - 1 \rangle$

Question 14

Assume a triangle ABC has standard labeling. Determine whether SAA, ASA, SSA, SAS, or SSS is given. Then
whether the law of sines or the law of cosines should be used to begin solving the triangle.
-$b , c$, and $A$

Accepted Answer

A)  SAS; law of sines 
B)  SSS; law of sines 
C)  SSS; law of cosines 
D)  SAS; law of cosines

Question 15

Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as
-$$\begin{array} { l } 
a = 28 \mathrm { ft } \
\mathrm { b } = 38 \mathrm { ft } \
\mathrm { c } = 42 \mathrm { ft }
\end{array}$$

Accepted Answer

A)  No triangle satisfies the given conditions. 
B)  $\mathrm { A } = 40.6 ^ { \circ } , \mathrm { B } = 31 ^ { \circ } , \mathrm { C } = 108.4 ^ { \circ }$ 
C)  $\mathrm { A } = 62 ^ { \circ } , \mathrm { B } = 40.6 ^ { \circ } , \mathrm { C } = 77.4 ^ { \circ }$ 
D)  $\mathrm { A } = 40.6 ^ { \circ } , \mathrm { B } = 62 ^ { \circ } , \mathrm { C } = 77.4 ^ { \circ }$

Question 16

Determine whether the pair of vectors is orthogonal.
-$$\langle 8 , - 1 \rangle , \langle 4,32 \rangle$$

Accepted Answer

A)  Yes 
B)  $\mathrm { No }$

Question 17

Find an equivalent equation in rectangular coordinates.
-$$r ( 1 - 2 \cos \theta ) = 1$$

Accepted Answer

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

Question 18

Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
-A trigonometry student makes the statement, ʺIf we know any two sides and one angle of
a triangle, then the triangle is uniquely determined.ʺ Is this a valid statement? Explain,
referring to the congruence axioms.

Accepted Answer

The answer of Write the word or phrase that best...

Question 19

Solve the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate.
-$$\begin{array} { l } 
A = 48 ^ { \circ } 10 ^ { \prime } \
B = 36 ^ { \circ } 10 ^ { \prime } \
a = 48.1
\end{array}$$

Accepted Answer

A)  $\mathrm { C } = 95 ^ { \circ } 40 ^ { \prime } , \mathrm { b } = 38.1 , \mathrm { c } = 64.2$ 
B)  $\mathrm { C } = 95 ^ { \circ } 40 ^ { \prime } , \mathrm { b } = 64.2 , \mathrm { c } = 38.1$ 
C)  $\mathrm { C } = 96 ^ { \circ } 40 ^ { \prime } , \mathrm { b } = 64.2 , \mathrm { c } = 38.1$ 
D)  $\mathrm { C } = 96 ^ { \circ } 40 ^ { \prime } , \mathrm { b } = 38.1 , \mathrm { c } = 64.2$

Question 20

Find an equivalent equation in rectangular coordinates.
-$\mathrm { r } \sin \theta = 10$

Accepted Answer

A)  $y = 10 x$ 
B)  $x = 10 y$ 
C)  $x = 10$ 
D)  $\mathrm { y } = 10$

Exam 8: Applications of Trigonometry

Graph the complex number. - $1 - \mathrm { i }$ $Graph the complex number. - 1 - \mathrm { i } A) B) C) D)$

Use the figure to find the specified vector. -Find -a. $Use the figure to find the specified vector. -Find -a. A) \langle 8 , - 6 \rangle B) \langle 6 , - 8 \rangle C) \langle - 6 , - 8 \rangle D) \langle - 8,6 \rangle$

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal places. - $\left( 15 \operatorname { cis } 18.5 ^ { \circ } \right) \left( 5 \operatorname { cis } 7.5 ^ { \circ } \right)$

Sketch the vectors u and w with angle θ between them and sketch the resultant. - $| \mathbf { u } | = 10 , | \mathbf { w } | = 4 , \theta = 40 ^ { \circ }$

Find the polar coordinates of the point(s) of intersection of the given curves for 0 ≤ θ < 2π. - $r = 10 + 3 \sin \theta , r = 10 + 3 \cos \theta$

The rectangular coordinates of a point are given. Express the point in polar coordinates with r ≥ 0 and 0° ≤ θ < 360°. - $( 0 , - 2 )$

Write the complex number in rectangular form. - $\operatorname { cis } 210$

Graph the polar equation for θ in [0°, 360°) . - $r = 4 \cos 6 \theta$ $Graph the polar equation for θ in [0°, 360°) . - r = 4 \cos 6 \theta A) B) C) D)$

Find the area of triangle ABC with the given parts. Round to the nearest tenth when necessary. - $\begin{array} { l } \mathrm { b } = 15.5 \mathrm { ft } \\\mathrm { A } = 36 ^ { \circ } 10 ^ { \prime } \\\mathrm { C } = 95 ^ { \circ } 50 ^ { \prime }\end{array}$

Find sum of the pair of complex numbers. - $5 - 5 \mathrm { i } , 10 - 11 \mathrm { i }$

Solve the problem. -One rope pulls a barge directly east with a force of 68 newtons, and another rope pulls the barge directly north with a force of 62 newtons. Find the magnitude of the resultant force acting on the barge.

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

Assume a triangle ABC has standard labeling. Determine whether SAA, ASA, SSA, SAS, or SSS is given. Then whether the law of sines or the law of cosines should be used to begin solving the triangle. - $b , c$ , and $A$

Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as - $\begin{array} { l } a = 28 \mathrm { ft } \\\mathrm { b } = 38 \mathrm { ft } \\\mathrm { c } = 42 \mathrm { ft }\end{array}$

Determine whether the pair of vectors is orthogonal. - $\langle 8 , - 1 \rangle , \langle 4,32 \rangle$

Find an equivalent equation in rectangular coordinates. - $r ( 1 - 2 \cos \theta ) = 1$

Solve the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate. - $\begin{array} { l } A = 48 ^ { \circ } 10 ^ { \prime } \\B = 36 ^ { \circ } 10 ^ { \prime } \\a = 48.1\end{array}$

Find an equivalent equation in rectangular coordinates. - $\mathrm { r } \sin \theta = 10$

Exam 8: Applications of Trigonometry

Graph the complex number. - 1−i1 - \mathrm { i }1−i

Use the figure to find the specified vector. -Find -a.

Sketch the vectors u and w with angle θ between them and sketch the resultant. - ∣u∣=10,∣w∣=4,θ=40∘| \mathbf { u } | = 10 , | \mathbf { w } | = 4 , \theta = 40 ^ { \circ }∣u∣=10,∣w∣=4,θ=40∘

Find the polar coordinates of the point(s) of intersection of the given curves for 0 ≤ θ < 2π. - r=10+3sin⁡θ,r=10+3cos⁡θr = 10 + 3 \sin \theta , r = 10 + 3 \cos \thetar=10+3sinθ,r=10+3cosθ

The rectangular coordinates of a point are given. Express the point in polar coordinates with r ≥ 0 and 0° ≤ θ < 360°. - (0,−2) ( 0 , - 2 ) (0,−2)

Write the complex number in rectangular form. - cis⁡210\operatorname { cis } 210cis210

Graph the polar equation for θ in [0°, 360°) . - r=4cos⁡6θr = 4 \cos 6 \thetar=4cos6θ

Find sum of the pair of complex numbers. - 5−5i,10−11i5 - 5 \mathrm { i } , 10 - 11 \mathrm { i }5−5i,10−11i

Solve the problem. -One rope pulls a barge directly east with a force of 68 newtons, and another rope pulls the barge directly north with a force of 62 newtons. Find the magnitude of the resultant force acting on the barge.

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

Assume a triangle ABC has standard labeling. Determine whether SAA, ASA, SSA, SAS, or SSS is given. Then whether the law of sines or the law of cosines should be used to begin solving the triangle. - b,cb , cb,c , and AAA

Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as - a=28ftb=38ftc=42ft\begin{array} { l } a = 28 \mathrm { ft } \\\mathrm { b } = 38 \mathrm { ft } \\\mathrm { c } = 42 \mathrm { ft }\end{array}a=28ftb=38ftc=42ft​

Determine whether the pair of vectors is orthogonal. - ⟨8,−1⟩,⟨4,32⟩\langle 8 , - 1 \rangle , \langle 4,32 \rangle⟨8,−1⟩,⟨4,32⟩

Find an equivalent equation in rectangular coordinates. - r(1−2cos⁡θ) =1r ( 1 - 2 \cos \theta ) = 1r(1−2cosθ) =1

Solve the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate. - A=48∘10′B=36∘10′a=48.1\begin{array} { l } A = 48 ^ { \circ } 10 ^ { \prime } \\B = 36 ^ { \circ } 10 ^ { \prime } \\a = 48.1\end{array}A=48∘10′B=36∘10′a=48.1​

Find an equivalent equation in rectangular coordinates. - rsin⁡θ=10\mathrm { r } \sin \theta = 10rsinθ=10