Question 1

A force of 50 pounds is exerted along a rope attached to a crate at an angle of 30° above the horizontal. The crate is moved 43 feet. How much work has been Accomplished? Round answer to one decimal place.
A) 2,150.0 foot-pounds
B) 2, 482.6 foot-pounds
C) 2,006.0 foot-pounds
D) 2, 244.3 foot-pounds
E) 1,862.0 foot-pounds

Accepted Answer

The answer of A force of 50 pounds is exerted...

Question 2

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.
$$\mathbf { u } \cdot \mathbf { w } + \mathbf { v } \cdot \mathbf { w }$$
A) vector
B) scalar

Accepted Answer

The expression involves dot products ($\mathbf { u } \cdot \mathbf { w }$ and $\mathbf { v } \cdot \mathbf { w }$), each of which results in a scalar. The sum of two scalars is also a scalar.

Question 3

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. $\cos \left( \frac { \pi } { 4 } \right) + i \sin \left( \frac { \pi } { 4 } \right)$
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.

Accepted Answer

I is a fourth root of 1 because it represents $e^{i\frac{3\pi}{2}}$, which raised to the fourth power equals 1. II is not a root because it's a fraction and not in the form of $e^{ix}$. III is not a fourth root of 1 because $e^{i\frac{\pi}{4}}$ raised to the fourth power equals $i$, not 1.

Question 4

Given $A = 54 ^ { \circ } , B = 69 ^ { \circ }$, and $a = 7.1$, use the Law of Sines to solve the triangle for the value of $b$. Round answer to two decimal places.

A) b = 6.38
B) b = 7.36
C) b = 6.85
D) b = 8.19
E) b = 6.15

Accepted Answer

The answer of Given \(A = 54 ^ { \circ...

Question 5

In the figure below, $a = 8 , b = 12$, and $c = 13$. Use this information to solve the parallelogram for $\theta$. The diagonals of the parallelogram are represented by $c$ and $d$. Round answer to two decimal places.

A) 101.72°
B) 95.86
C) 90.00°
D) 91.46
E) 78.28°

Accepted Answer

We know that the diagonals of a parallelogram bisect each other. Therefore, we can find half the diagonal by dividing c by 2: $\frac{c}{2} = \frac{13}{2} = 6.5$. Using this value as the opposite side and 8 as the adjacent side, we can find the value of $\theta$ using the inverse tangent function: $\theta = \tan^{-1}\left(\frac{6.5}{8}\right) \approx \boxed{101.72^\circ}$ (rounded to two decimal places).

Question 6

Given $a = 12 , b = 5$, and $c = 15$, use the Law of Cosines to solve the triangle for the value of $C$. Round answer to two decimal places.

A) $60.34 ^ { \circ }$
B) $117.82 ^ { \circ }$
C) $80.45 ^ { \circ }$
D) $17.15 ^ { \circ }$
E) $45.04 ^ { \circ }$

Accepted Answer

The answer of Given \(a = 12 , b =...

Question 7

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.
$( w \cdot w ) w$
A) vector
B) scalar

Accepted Answer

The expression $( w \cdot w ) w$ involves the dot product of $\mathbf{w}$ with itself, which yields a scalar, and then multiplies this scalar by the vector $\mathbf{w}$. Multiplying a scalar by a vector results in a vector, hence the expression is a vector.

Question 8

Find the component form of $\mathbf { v }$ if $\| \mathbf { v } \| = 4$ and the angle it makes with the $x$-axis is $120 ^ { \circ }$.
A) $\langle - 2 \sqrt { 3 } , 2 \rangle$
B) $\langle - 4,4 \sqrt { 3 } \rangle$
C) $\langle - 4 \sqrt { 3 } , - 4 \rangle$
D) $\langle - 2,2 \sqrt { 3 } \rangle$
E) $\langle - 2 \sqrt { 2 } , 2 \sqrt { 2 } \rangle$

Accepted Answer

We can start by sketching the vector in the coordinate plane. Since it makes an angle of $120^\circ$ with the $x$-axis, it will lie in the second quadrant, and will form a $30^\circ$ angle with the $y$-axis.

[asy]
size(6cm);
import olympiad;
import markers;

draw((-1,0)--(2.8,0),Arrow);
draw((0,-1)--(0,2.8),Arrow);

draw((0,0)--dir(120),red,Arrow);

draw(rightanglemark((0,0),dir(120),(0,1)));
draw((0,0.2)--(0.2,0.2)--(0.2,0));

label("$	heta$", (0.3,0.3), NE, red);
label("$x$", (2.8,0), E);
label("$y$", (0,2.8), N);

label("$\mathbf{v}$", dir(120), W, red);
[/asy]

We can see that the $y$-component of $\mathbf{v}$ is $\|\mathbf{v}\|\sin 30^\circ=2$, and that the $x$-component of $\mathbf{v}$ is $\|\mathbf{v}\|\cos 30^\circ=2\sqrt{3}$. Therefore, the component form of $\mathbf{v}$ is $\boxed{\langle -2,2\sqrt{3}angle}$.

Question 9

Find the trigonometric form of the complex number shown below. $- 3 i$
A) $3 ( \cos 0 + i \sin 0 )$
B) $3 ( \cos \pi + i \sin \pi )$
C) $3 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
D) $3 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$
E) $3 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$

Accepted Answer

The complex number $-3i$ has a magnitude of 3 and an angle of $-\frac{\pi}{2}$ (measured counterclockwise from the positive real axis). Using the definitions of cosine and sine in the unit circle, we have $\cos \left(-\frac{\pi}{2}\right) = 0$ and $\sin \left(-\frac{\pi}{2}\right) = -1$. Thus, the complex number can be written in trigonometric form as $3\left(\cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2}\right)$. Option C matches this form.

Question 10

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

A)
$$w _ { 1 } = \cos \left( 20 ^ { \circ } \right) + i \sin \left( 20 ^ { \circ } \right)$$
$$\begin{array} { l } 
w _ { 2 } = \cos \left( 110 ^ { \circ } \right) + i \sin \left( 110 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 200 ^ { \circ } \right) + i \sin \left( 200 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 290 ^ { \circ } \right) + i \sin \left( 290 ^ { \circ } \right)
\end{array}$$

B)
$w _ { 1 } = \cos \left( 30 ^ { \circ } \right) + i \sin \left( 30 ^ { \circ } \right)$
$$\begin{array} { l } 
w _ { 2 } = \cos \left( 120 ^ { \circ } \right) + i \sin \left( 120 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 210 ^ { \circ } \right) + i \sin \left( 210 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 300 ^ { \circ } \right) + i \sin \left( 300 ^ { \circ } \right)
\end{array}$$

C)
$$w _ { 1 } = \cos \left( 35 ^ { \circ } \right) + i \sin \left( 35 ^ { \circ } \right)$$
$$\begin{array} { l } 
w _ { 2 } = \cos \left( 125 ^ { \circ } \right) + i \sin \left( 125 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 215 ^ { \circ } \right) + i \sin \left( 215 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 305 ^ { \circ } \right) + i \sin \left( 305 ^ { \circ } \right)
\end{array}$$

D)
$w _ { 1 } = \cos \left( 25 ^ { \circ } \right) + i \sin \left( 25 ^ { \circ } \right)$
$$\begin{array} { l } 
w _ { 2 } = \cos \left( 115 ^ { \circ } \right) + i \sin \left( 115 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 205 ^ { \circ } \right) + i \sin \left( 205 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 295 ^ { \circ } \right) + i \sin \left( 295 ^ { \circ } \right)
\end{array}$$ E)
$$\begin{array} { l } 
w _ { 1 } = \cos \left( 40 ^ { \circ } \right) + i \sin \left( 40 ^ { \circ } \right) \
w _ { 2 } = \cos \left( 130 ^ { \circ } \right) + i \sin \left( 130 ^ { \circ } \right) \
w _ { 3 } = \cos \left( 220 ^ { \circ } \right) + i \sin \left( 220 ^ { \circ } \right) \
w _ { 4 } = \cos \left( 310 ^ { \circ } \right) + i \sin \left( 310 ^ { \circ } \right)
\end{array}$$

Accepted Answer

The complex number $- \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } i$ is in the form of $\cos(120^\circ) + i\sin(120^\circ)$. The fourth roots are found by dividing the angle by 4 and adding $90^\circ$ increments: $30^\circ, 120^\circ, 210^\circ, 300^\circ$.

Question 11

Given $\mathbf { u } = - 3 \mathbf { i } - 2 \mathbf { j }$ and $\mathbf { v } = - \mathbf { i } - 5 \mathbf { j }$, determine $- 9 \mathbf { u } + 6 \mathbf { v }$.
A) $- 9 \mathbf { u } + 6 \mathbf { v } = 21 \mathbf { i } - 12 \mathbf { j }$
B) $- 9 \mathbf { u } + 6 \mathbf { v } = 27 \mathbf { i } - 21 \mathbf { j }$
C) $- 9 \mathbf { u } + 6 \mathbf { v } = - 9 \mathbf { i } - 3 \mathbf { j }$
D) $- 9 \mathbf { u } + 6 \mathbf { v } = 12 \mathbf { i } - 3 \mathbf { j }$
E) $- 9 \mathbf { u } + 6 \mathbf { v } = - 3 \mathbf { i } + 3 \mathbf { j }$

Accepted Answer

The answer of Given \(\mathbf { u } = -...

Question 12

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form.
$( 3 + 3 i ) ( 5 + 5 i )$
A) $- 15 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$
B) $- 30 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
C) $15 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
D) $30 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
E) $15 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$

Accepted Answer

To multiply complex numbers in trigonometric form, you multiply their magnitudes and add their angles. The magnitude of $3 + 3i$ is $\sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}$, and its angle is $\frac{\pi}{4}$ because it forms a 45-degree angle in the first quadrant. The magnitude of $5 + 5i$ is $\sqrt{5^2 + 5^2} = \sqrt{50} = 5\sqrt{2}$, and its angle is also $\frac{\pi}{4}$. Multiplying the magnitudes gives $3\sqrt{2} \times 5\sqrt{2} = 30$, and adding the angles gives $\frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2}$. Therefore, the product in trigonometric form is $30 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right)$.

Question 13

A straight road makes an angle, $A$, of $16 ^ { \circ }$ with the horizontal. When the angle of elevation, $B$, of the sun is $58 ^ { \circ }$, a vertical pole beside the road casts a shadow 6 feet long parallel to the road. Approximate the length of the pole. Round answer to two decimal places.

A) 3.12 feet
B) 4.75 feet
C) 7.58 feet
D) 9.60 feet
E) 4.73 feet

Accepted Answer

The answer of A straight road makes an angle, $A$,...

Question 14

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

Accepted Answer

1. Draw a diagram:

[asy]
pair A, B, C;

A = (0,0);
B = (9,0);
C = (4.66,4.19);

draw(A--C--B);
draw(rightanglemark(A,C,B,15));

label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
[/asy]

2. Calculate the angles:

$$\begin{aligned} \angle ACB &= 180^\circ - \angle BCA - \angle CAB \ &= 180^\circ - (54^\circ 28' - 27^\circ 27') - 27^\circ 27' \ &= 72^\circ 55' \end{aligned}$$

$$\angle ABC = 180^\circ - 90^\circ - \angle ACB = 17^\circ 05'$$

3. Use the Law of Sines to find the length of BC:

$$\frac{BC}{\sin \angle ACB} = \frac{AB}{\sin \angle ABC}$$

$$BC = \frac{AB \sin \angle ACB}{\sin \angle ABC} = \frac{9\sin 72^\circ 55'}{\sin 17^\circ 05'} \approx 8.07$$

Therefore, the answer is (C) 8.07 miles.

Question 15

In the figure below, $a = 8 , b = 12$, and $d = 13$. Use this information to solve the parallelogram for $\beta$. The diagonals of the parallelogram are represented by $c$ and $d$. Round answer to two decimal places.

A) 91.50°
B) 78.28°
C) 96.61°
D) 101.72°
E) 92.03°

Accepted Answer

The answer of In the figure below, \(a = 8...

Question 16

Represent the complex number below graphically.
$\sqrt { 3 } - i$

A)

B)

C)

D)

E)

Accepted Answer

The answer of Represent the complex number below graphically.
\(\sqrt {...

Question 17

Perform the operation below and leave the result in trigonometric form.
$$\frac { 17 \left( \cos 59 ^ { \circ } + i \sin 59 ^ { \circ } \right) } { 21 \left( \cos 12 ^ { \circ } + i \sin 12 ^ { \circ } \right) }$$
A) $\frac { 17 } { 21 } \left( \cos 47 ^ { \circ } + i \sin 47 ^ { \circ } \right)$
B) $- 4 \left( \cos \frac { 59 ^ { \circ } } { 12 } + i \sin \frac { 59 ^ { \circ } } { 12 } \right)$
C) $38 \left( \cos 71 ^ { \circ } + i \sin 71 ^ { \circ } \right)$
D) $357 \left( \cos \frac { 59 ^ { \circ } } { 12 } + i \sin \frac { 59 ^ { \circ } } { 12 } \right)$
E) $\frac { 17 } { 21 } \left( \cos \frac { 59 ^ { \circ } } { 12 } + i \sin \frac { 59 ^ { \circ } } { 12 } \right)$

Accepted Answer

The division of complex numbers in trigonometric form is performed by dividing their magnitudes and subtracting their angles. Thus, the magnitude is $17/21$ and the angle is $59^\circ - 12^\circ = 47^\circ$, resulting in $\frac{17}{21}(\cos 47^\circ + i\sin 47^\circ)$.

Question 18

Find the trigonometric form of the complex number shown below.
$8 i$
A) $8 ( \cos \pi + i \sin \pi )$
B) $8 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
C) $8 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
D) $8 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$
E) $8 ( \cos 0 + i \sin 0 )$

Accepted Answer

The trigonometric form of a complex number is $r(\cos \theta + i\sin \theta)$, where $r$ is the magnitude of the complex number and $\theta$ is the argument. For $8i$, $r = 8$ and $\theta = \frac{\pi}{2}$, because the complex number lies on the positive imaginary axis.

Question 19

Use Heron's area formula to find the area of the triangle pictured below, if $a = 10$ inches, $\mathrm { b } = 12$ inches, and $\mathrm { c } = 4$ inches.

A) $3 \sqrt { 3 }$
B) 20
C) 60
D) $8 \sqrt { 77 }$
E) $3 \sqrt { 39 }$

Accepted Answer

The answer of Use Heron's area formula to find the...

Question 20

After a severe storm, three sisters, April, May, and June, stood on their front porch and noticed that the tree in their front yard was leaning 6° from vertical toward the house. From the porch, which is 110 feet away from the base of the tree, they noticed that the Angle of elevation to the top of the tree was 27°. Approximate the height of the tree. Round answer to two decimal places.
A) 57.97 feet
B) 59.55 feet
C) 69.55 feet
D) 53.49 feet
E) 60.86 feet

Accepted Answer

The answer of After a severe storm, three sisters, April,...

Quiz 6: Additional Topics in Trigonometry

A force of 50 pounds is exerted along a rope attached to a crate at an angle of 30° above the horizontal. The crate is moved 43 feet. How much work has been Accomplished? Round answer to one decimal place.

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. $\cos \left( \frac { \pi } { 4 } \right) + i \sin \left( \frac { \pi } { 4 } \right)$

Find the component form of $\mathbf { v }$ if $\| \mathbf { v } \| = 4$ and the angle it makes with the $x$ -axis is $120 ^ { \circ }$ .

Find the trigonometric form of the complex number shown below. $- 3 i$

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

Given $\mathbf { u } = - 3 \mathbf { i } - 2 \mathbf { j }$ and $\mathbf { v } = - \mathbf { i } - 5 \mathbf { j }$ , determine $- 9 \mathbf { u } + 6 \mathbf { v }$ .

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. $( 3 + 3 i ) ( 5 + 5 i )$

Represent the complex number below graphically. $\sqrt { 3 } - i$ $Represent the complex number below graphically. \sqrt { 3 } - i A) B) C) D) E)$

Perform the operation below and leave the result in trigonometric form. $\frac { 17 \left( \cos 59 ^ { \circ } + i \sin 59 ^ { \circ } \right) } { 21 \left( \cos 12 ^ { \circ } + i \sin 12 ^ { \circ } \right) }$

Find the trigonometric form of the complex number shown below. $8 i$

Quiz 6: Additional Topics in Trigonometry

A force of 50 pounds is exerted along a rope attached to a crate at an angle of 30° above the horizontal. The crate is moved 43 feet. How much work has been Accomplished? Round answer to one decimal place.

Given A=54∘,B=69∘A = 54 ^ { \circ } , B = 69 ^ { \circ }A=54∘,B=69∘ , and a=7.1a = 7.1a=7.1 , use the Law of Sines to solve the triangle for the value of bbb . Round answer to two decimal places.

In the figure below, a=8,b=12a = 8 , b = 12a=8,b=12 , and c=13c = 13c=13 . Use this information to solve the parallelogram for θ\thetaθ . The diagonals of the parallelogram are represented by ccc and ddd . Round answer to two decimal places.

Given a=12,b=5a = 12 , b = 5a=12,b=5 , and c=15c = 15c=15 , use the Law of Cosines to solve the triangle for the value of CCC . Round answer to two decimal places.

Find the component form of v\mathbf { v }v if ∥v∥=4\| \mathbf { v } \| = 4∥v∥=4 and the angle it makes with the xxx -axis is 120∘120 ^ { \circ }120∘ .

Find the trigonometric form of the complex number shown below. −3i- 3 i−3i

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

Given u=−3i−2j\mathbf { u } = - 3 \mathbf { i } - 2 \mathbf { j }u=−3i−2j and v=−i−5j\mathbf { v } = - \mathbf { i } - 5 \mathbf { j }v=−i−5j , determine −9u+6v- 9 \mathbf { u } + 6 \mathbf { v }−9u+6v .

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. (3+3i) (5+5i) ( 3 + 3 i ) ( 5 + 5 i ) (3+3i) (5+5i)

In the figure below, a=8,b=12a = 8 , b = 12a=8,b=12 , and d=13d = 13d=13 . Use this information to solve the parallelogram for β\betaβ . The diagonals of the parallelogram are represented by ccc and ddd . Round answer to two decimal places.

Represent the complex number below graphically. 3−i\sqrt { 3 } - i3​−i

Find the trigonometric form of the complex number shown below. 8i8 i8i

Use Heron's area formula to find the area of the triangle pictured below, if a=10a = 10a=10 inches, b=12\mathrm { b } = 12b=12 inches, and c=4\mathrm { c } = 4c=4 inches.