Question 1

Determine whether the pair of vectors is orthogonal.
-$$\langle 16 , - 1 \rangle , \langle 4,32 \rangle$$

Accepted Answer

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

Question 2

Answer the question.
-What complex number is the common cube root of $- \mathrm { i }$ and fourth root of 1 ?

Accepted Answer

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

Question 3

Write the vector in the form <a, b>. If necessary, round values to the nearest hundredth.
-forces of 89 and 63 newtons, forming an angle of $90 ^ { \circ }$ (round to the nearest newton)

Accepted Answer

A)  152 newtons 
B)  26 newtons 
C)  5607 newtons 
D)  109 newtons 
A)  152 newtons 
B)  26 newtons 
C)  5607 newtons 
D)  109 newtons

Question 4

Find all cube roots of the complex number. Leave answers in trigonometric form.
-64

Accepted Answer

A)  $4 \operatorname { cis } 0 ^ { \circ } , 4 \operatorname { cis } 120 ^ { \circ } , 4 \operatorname { cis } 240 ^ { \circ }$ 
B)  $4 \operatorname { cis } 0 ^ { \circ } , 4 \operatorname { cis } 60 ^ { \circ } , 4 \operatorname { cis } 120 ^ { \circ }$ 
C)  $4 \operatorname { cis } 45 ^ { \circ } , 4 \operatorname { cis } 165 ^ { \circ } , 4 \operatorname { cis } 325 ^ { \circ }$ 
D)  $4 \operatorname { cis } 90 ^ { \circ } , 4 \operatorname { cis } 210 ^ { \circ } , 4 \operatorname { cis } 330 ^ { \circ }$ 
A)  $4 \operatorname { cis } 0 ^ { \circ } , 4 \operatorname { cis } 120 ^ { \circ } , 4 \operatorname { cis } 240 ^ { \circ }$ 
B)  $4 \operatorname { cis } 0 ^ { \circ } , 4 \operatorname { cis } 60 ^ { \circ } , 4 \operatorname { cis } 120 ^ { \circ }$ 
C)  $4 \operatorname { cis } 45 ^ { \circ } , 4 \operatorname { cis } 165 ^ { \circ } , 4 \operatorname { cis } 325 ^ { \circ }$ 
D)  $4 \operatorname { cis } 90 ^ { \circ } , 4 \operatorname { cis } 210 ^ { \circ } , 4 \operatorname { cis } 330 ^ { \circ }$

Question 5

Find the product. Write the product in rectangular form, using exact values.
-[4 cis $300 ^ { \circ }$ ] [5 $\left. \operatorname { cis } 120 ^ { \circ } \right]$

Accepted Answer

A)  $- 10 - 10 \sqrt { 3 } \mathrm { i }$ 
B)  $10 + 10 \sqrt { 3 } i$ 
C)  $10 \sqrt { 3 } - 10 \mathrm { i }$ 
D)  $- 10 \sqrt { 3 } + 10 \mathrm { i }$ 
A)  $- 10 - 10 \sqrt { 3 } \mathrm { i }$ 
B)  $10 + 10 \sqrt { 3 } i$ 
C)  $10 \sqrt { 3 } - 10 \mathrm { i }$ 
D)  $- 10 \sqrt { 3 } + 10 \mathrm { i }$

Question 6

Find all specified roots.
-Sixth roots of i.

Accepted Answer

A)  $\operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 } , \operatorname { cis } \frac { 11 \pi } { 4 } , \mathrm { i }$ 
B)  $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 }$ 
C)  $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 } , \operatorname { cis } \frac { 11 \pi } { 4 } , 1$ 
D)  $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 5 \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 7 \pi } { 4 } , \operatorname { cis } \frac { 13 \pi } { 12 } , \frac { 17 \pi } { 12 }$ 
A)  $\operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 } , \operatorname { cis } \frac { 11 \pi } { 4 } , \mathrm { i }$ 
B)  $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 }$ 
C)  $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 17 \pi } { 12 } , \operatorname { cis } \frac { 25 \pi } { 12 } , \operatorname { cis } \frac { 11 \pi } { 4 } , 1$ 
D)  $\operatorname { cis } \frac { \pi } { 12 } , \operatorname { cis } \frac { 5 \pi } { 12 } , \operatorname { cis } \frac { 3 \pi } { 4 } , \operatorname { cis } \frac { 7 \pi } { 4 } , \operatorname { cis } \frac { 13 \pi } { 12 } , \frac { 17 \pi } { 12 }$

Question 7

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°].
-$\langle 9 \sqrt { 3 } , 9 \rangle$

Accepted Answer

A)  $36 ; 30 ^ { \circ }$ 
B)  $18 \sqrt { 2 } ; 60 ^ { \circ }$ 
C)  $18 ; 150 ^ { \circ }$ 
D)  $18 ; 30 ^ { \circ }$ 
A)  $36 ; 30 ^ { \circ }$ 
B)  $18 \sqrt { 2 } ; 60 ^ { \circ }$ 
C)  $18 ; 150 ^ { \circ }$ 
D)  $18 ; 30 ^ { \circ }$

Question 8

Find the dot product for the pair of vectors.
-$- 5 \mathbf { j } , 7 \mathbf { i }$

Accepted Answer

A)  2 
B)  12 
C)  0 
D)  $- 12$ 
A)  2 
B)  12 
C)  0 
D)  $- 12$

Question 9

Answer the question.
-If $\mathrm { n }$ is an even number, then which of the following statements is true?

Accepted Answer

A)  The equation $\mathrm { x } ^ { \mathrm { n } } + 1 = 0$ has exactly one real root. 
B)  The equation $\mathrm { x } ^ { \mathrm { n } } + 1 = 0$ has $\mathrm { n } + 1$ roots. 
C)  The equation $\mathrm { x } ^ { \mathrm { n } } + 1 = 0$ has $\mathrm { n }$ nonreal roots. 
D)  The equation $x _ { n } + 1 = 0$ has at least one real root. 
A)  The equation $\mathrm { x } ^ { \mathrm { n } } + 1 = 0$ has exactly one real root. 
B)  The equation $\mathrm { x } ^ { \mathrm { n } } + 1 = 0$ has $\mathrm { n } + 1$ roots. 
C)  The equation $\mathrm { x } ^ { \mathrm { n } } + 1 = 0$ has $\mathrm { n }$ nonreal roots. 
D)  The equation $x _ { n } + 1 = 0$ has at least one real root.

Question 10

Solve the problem.
-A room in the shape of a triangle has sides of length 18.1 yards, 14.8 yards, and 15.8 yards. If carpeting costs $18.88 per square yard, padding costs $2.00 per square yard, and there is no charge
For installation, how much, to the nearest dollar, will it cost to carpet the room?

Accepted Answer

A)  $472 
B)  $2327 
C)  $4233 
D)  $111 
A)  $472 
B)  $2327 
C)  $4233 
D)  $111

Question 11

Give the rectangular coordinates for the point.
-$\left( - 4,45 ^ { \circ } \right)$

Accepted Answer

A)  $( 2 \sqrt { 2 } , 2 \sqrt { 2 } )$ 
B)  $( - 2 \pi , - 2 \pi )$ 
C)  $( - 2 \sqrt { 2 } , - 2 \sqrt { 2 } )$ 
D)  $( - 2 \sqrt { 3 } , - 2 \sqrt { 3 } )$ 
A)  $( 2 \sqrt { 2 } , 2 \sqrt { 2 } )$ 
B)  $( - 2 \pi , - 2 \pi )$ 
C)  $( - 2 \sqrt { 2 } , - 2 \sqrt { 2 } )$ 
D)  $( - 2 \sqrt { 3 } , - 2 \sqrt { 3 } )$

Question 12

Find the missing parts of the triangle.
-$$\begin{array} { l } 
\mathrm { C } = 110 ^ { \circ } \
\mathrm { c } = 43 \mathrm {~km} \
\mathrm { a } = 53 \mathrm {~km}
\end{array}$$
If necessary, round angles to the nearest degree and side lengths to the nearest $\mathrm { km }$.

Accepted Answer

A)  $\mathrm { A } = 43 ^ { \circ } , \mathrm { B } = 84 ^ { \circ } , \mathrm { b } = 59 \mathrm {~km}$ 
B)  $\mathrm { A } = 44 ^ { \circ } , \mathrm { B } = 83 ^ { \circ } , \mathrm { b } = 65 \mathrm {~km}$ 
C)  no such triangle 
D)  $A = 43 ^ { \circ } , B = 84 ^ { \circ } , b = 62 \mathrm {~km}$ 
A)  $\mathrm { A } = 43 ^ { \circ } , \mathrm { B } = 84 ^ { \circ } , \mathrm { b } = 59 \mathrm {~km}$ 
B)  $\mathrm { A } = 44 ^ { \circ } , \mathrm { B } = 83 ^ { \circ } , \mathrm { b } = 65 \mathrm {~km}$ 
C)  no such triangle 
D)  $A = 43 ^ { \circ } , B = 84 ^ { \circ } , b = 62 \mathrm {~km}$

Question 13

Write the complex number in rectangular form.
-$3 \operatorname { cis } 240 ^ { \circ }$

Accepted Answer

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

Question 14

Solve the problem.
-A stained glass window is composed of 20 triangular sections, each with sides 6, 8, and 7 in. Find the total area of the window (to the nearest square inch).

Accepted Answer

A)  $20 \mathrm { in } ^ { 2 }$ 
B)  407 in. 2 
C)  $125 \mathrm { in } .2$ 
D)  4737 in. 2 
A)  $20 \mathrm { in } ^ { 2 }$ 
B)  407 in. 2 
C)  $125 \mathrm { in } .2$ 
D)  4737 in. 2

Question 15

Find the dot product for the pair of vectors.
-$\mathbf { i } + 4 \mathbf { j } , 6 \mathbf { i } - \mathbf { j }$

Accepted Answer

A)  $- 2$ 
B)  10 
C)  0 
D)  2 
A)  $- 2$ 
B)  10 
C)  0 
D)  2

Question 16

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal
places.
-$\left[ 25.4 \operatorname { cis } \frac { 5 \pi } { 12 } \right] ^ { 2 }$

Accepted Answer

A)  $- 558.7249 + 322.5800 \mathrm { i }$ 
B)  $- 21.9970 + 12.7000 \mathrm { i }$ 
C)  $322.5800 - 558.7249 \mathrm { i }$ 
D)  $166.9797 + 623.1767 \mathrm { i }$ 
A)  $- 558.7249 + 322.5800 \mathrm { i }$ 
B)  $- 21.9970 + 12.7000 \mathrm { i }$ 
C)  $322.5800 - 558.7249 \mathrm { i }$ 
D)  $166.9797 + 623.1767 \mathrm { i }$

Question 17

Solve the problem.
-Given that the polar equation $\mathrm { r } = \frac { \mathrm { a } \left( 1 - \mathrm { e } ^ { 2 } \right) } { 1 + \mathrm { e } \cos \theta }$ models the orbits of the planets, compare Earth's orbit relative to the sun with that of Pluto and describe the difference. For Earth, $\mathrm { a } = 1.00$ and $\mathrm { e } = 0.017$. For Pluto, $\mathrm { a } = 39.4$ and $\mathrm { e } = 0.249$.

Accepted Answer

A)  Earth's orbit closely approximates a circle with the sun positioned at its center. Pluto's orbit is more elliptical with the sun positioned on its long axis about 10 astronomical units away from the center. 
B)  Pluto's orbit closely approximates a circle with the sun positioned at its center. Earth's orbit is more elliptical with the sun positioned on its long axis about 10 astronomical units away from the center. 
C)  Earth circles the sun, but Pluto does not. 
D)  The orbits of Earth and Pluto are both circular, with the sun positioned at their centers, but the diameter of Pluto's orbit is about 40 times that of Earth. 
A)  Earth's orbit closely approximates a circle with the sun positioned at its center. Pluto's orbit is more elliptical with the sun positioned on its long axis about 10 astronomical units away from the center. 
B)  Pluto's orbit closely approximates a circle with the sun positioned at its center. Earth's orbit is more elliptical with the sun positioned on its long axis about 10 astronomical units away from the center. 
C)  Earth circles the sun, but Pluto does not. 
D)  The orbits of Earth and Pluto are both circular, with the sun positioned at their centers, but the diameter of Pluto's orbit is about 40 times that of Earth.

Question 18

Write the complex number in trigonometric form $$r ( \cos \theta + i \sin \theta )$$ with θ in the interval [0°, 360°).
-$4 i$

Accepted Answer

A)  $4 \left( \cos 270 ^ { \circ } + i \sin 270 ^ { \circ } \right)$ 
B)  $4 \left( \cos 0 ^ { \circ } + i \sin 0 ^ { \circ } \right)$ 
C)  $4 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right)$ 
D)  $4 \left( \cos 90 ^ { \circ } + \mathrm { i } \sin 90 ^ { \circ } \right)$ 
A)  $4 \left( \cos 270 ^ { \circ } + i \sin 270 ^ { \circ } \right)$ 
B)  $4 \left( \cos 0 ^ { \circ } + i \sin 0 ^ { \circ } \right)$ 
C)  $4 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right)$ 
D)  $4 \left( \cos 90 ^ { \circ } + \mathrm { i } \sin 90 ^ { \circ } \right)$

Question 19

Find the component form of the indicated vector.
-Let $\mathbf { u } = \langle - 5,9 \rangle$. Find $- 7 \mathbf { u }$.

Accepted Answer

A)  $\langle 35 , - 63 \rangle$ 
B)  $\langle - 35,63 \rangle$ 
C)  $\langle - 35 , - 63 \rangle$ 
D)  $\langle 35,63 \rangle$ 
A)  $\langle 35 , - 63 \rangle$ 
B)  $\langle - 35,63 \rangle$ 
C)  $\langle - 35 , - 63 \rangle$ 
D)  $\langle 35,63 \rangle$

Question 20

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

Accepted Answer

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

Determine whether the pair of vectors is orthogonal. - $\langle 16 , - 1 \rangle , \langle 4,32 \rangle$

Answer the question. -What complex number is the common cube root of $- \mathrm { i }$ and fourth root of 1 ?

Write the vector in the form <a, b>. If necessary, round values to the nearest hundredth. -forces of 89 and 63 newtons, forming an angle of $90 ^ { \circ }$ (round to the nearest newton)

Find all cube roots of the complex number. Leave answers in trigonometric form. -64

Find the product. Write the product in rectangular form, using exact values. -[4 cis $300 ^ { \circ }$ ] [5 $\left. \operatorname { cis } 120 ^ { \circ } \right]$

Find all specified roots. -Sixth roots of 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°]. - $\langle 9 \sqrt { 3 } , 9 \rangle$

Find the dot product for the pair of vectors. - $- 5 \mathbf { j } , 7 \mathbf { i }$

Answer the question. -If $\mathrm { n }$ is an even number, then which of the following statements is true?

Solve the problem. -A room in the shape of a triangle has sides of length 18.1 yards, 14.8 yards, and 15.8 yards. If carpeting costs $18.88 per square yard, padding costs $2.00 per square yard, and there is no charge For installation, how much, to the nearest dollar, will it cost to carpet the room?

Give the rectangular coordinates for the point. - $\left( - 4,45 ^ { \circ } \right)$

Find the missing parts of the triangle. - =11 =43 =53 If necessary, round angles to the nearest degree and side lengths to the nearest $\mathrm { km }$ .

Write the complex number in rectangular form. - $3 \operatorname { cis } 240 ^ { \circ }$

Solve the problem. -A stained glass window is composed of 20 triangular sections, each with sides 6, 8, and 7 in. Find the total area of the window (to the nearest square inch).

Find the dot product for the pair of vectors. - $\mathbf { i } + 4 \mathbf { j } , 6 \mathbf { i } - \mathbf { j }$

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal places. - $\left[ 25.4 \operatorname { cis } \frac { 5 \pi } { 12 } \right] ^ { 2 }$

Write the complex number in trigonometric form $r ( \cos \theta + i \sin \theta )$ with θ in the interval [0°, 360°). - $4 i$

Find the component form of the indicated vector. -Let $\mathbf { u } = \langle - 5,9 \rangle$ . Find $- 7 \mathbf { u }$ .

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

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

Determine whether the pair of vectors is orthogonal. - ⟨16,−1⟩,⟨4,32⟩\langle 16 , - 1 \rangle , \langle 4,32 \rangle⟨16,−1⟩,⟨4,32⟩

Answer the question. -What complex number is the common cube root of −i- \mathrm { i }−i and fourth root of 1 ?

Write the vector in the form <a, b>. If necessary, round values to the nearest hundredth. -forces of 89 and 63 newtons, forming an angle of 90∘90 ^ { \circ }90∘ (round to the nearest newton)