Question 1

The initial point for the vector is the origin, and $\theta$ denotes the angle (measured counterclockwise) from the $X$ -axis to the vector. The magnitude of $\mathrm { V }$ is $30$ cm/sec, and $\theta = 120 ^ { \circ }$ Compute the horizontal and vertical components of the given vector. (Round your answers to two decimal places.)
A) $V _ { x } = 25.98$ cm/sec $V _ { y } \approx - 15$ cm/sec
B) $V _ { x } = 15$ cm/sec $V _ { y } \approx 25.98$ cm/sec
C) $V _ { z } = - 15$ cm/sec $V _ { y } \approx 25.98$ cm/sec
D) $V _ { x } = 21.21$ cm/sec $V _ { y } \approx 21.21$ cm/sec
E) $V _ { x } \approx - 25.98 \mathrm {~cm} / \mathrm { sec } V _ { y } \approx 15 \mathrm {~cm} / \mathrm { sec }$

Accepted Answer

C

Question 2

On a sheet of paper, graph the parametric equation after eliminating the parameter $t$ $( 0 \leq t \leq 2 \pi$ ). Specify the approximate direction on the curve corresponding to increasing values of $t$ . $x = 6 \sin 2 t , y = 5 \cos 2 t$
A) counterclockwise
B) clockwise

Accepted Answer

To eliminate the parameter $t$, we use the trigonometric identities $\sin^2 \theta + \cos^2 \theta = 1$. Dividing $x$ by 6 and $y$ by 5, we get $\sin 2t = \frac{x}{6}$ and $\cos 2t = \frac{y}{5}$. Squaring both and adding them gives $\frac{x^2}{36} + \frac{y^2}{25} = \sin^2 2t + \cos^2 2t = 1$, which is the equation of an ellipse. For increasing values of $t$, the direction of motion on the ellipse is determined by the signs of $\frac{dx}{dt}$ and $\frac{dy}{dt}$. Since $\frac{dx}{dt} = 12\cos 2t$ and $\frac{dy}{dt} = -10\sin 2t$, when $t$ increases from 0, the initial motion is to the left (since $\cos(0) = 1$ makes $\frac{dx}{dt}$ positive and $\sin(0) = 0$ makes $\frac{dy}{dt}$ zero), indicating a clockwise direction.

Question 3

Convert the given rectangular coordinates to polar coordinates. Express the answer in such a way that r is nonnegative and $0 \leq \theta < 2 \pi$ . $( - 5 , - 5 )$
A) $\left( 5 , - \frac { 5 \pi } { 4 } \right)$
B) $\left( 5 \sqrt { 2 } , \frac { 5 \pi } { 4 } \right)$
C) $\left( 5 , \frac { \pi } { 4 } \right)$
D) $\left( 5 \sqrt { 2 } , \frac { \pi } { 4 } \right)$
E) $\left( - 5 \sqrt { 2 } , \frac { 5 \pi } { 4 } \right)$

Accepted Answer

B

Question 4

Convert the given rectangular coordinates to polar coordinates. Express the answer in such a way that r is nonnegative and $0 \leq \theta < 2 \pi$ . $( 7,0 )$
A) $( 0,7 )$
B) $( 7,0 )$
C) $\left( - 7 , \frac { 3 \pi } { 2 } \right)$
D) $( - 7,0 )$
E) $\left( 7 , \frac { 3 \pi } { 2 } \right)$

Accepted Answer

To convert rectangular coordinates to polar coordinates, we use the formulas
\begin{align*}
x &= r \cos 	heta \
y &= r \sin 	heta
\end{align*}
Since the given point is $(7,0)$, we have $x=7$ and $y=0$. This tells us that $r \cos 	heta = 7$ and $r \sin 	heta = 0$. Since $\sin 	heta$ is zero when $	heta$ is an even multiple of $\pi$, we have two possible values for $	heta$: $	heta = 0$ or $	heta = \pi$ (however, we are asked to express $	heta$ in the interval $0 \leq 	heta < 2\pi$, so we choose $	heta = 0$). Solving for $r$, we get $r = \frac{x}{\cos 	heta} = \frac{7}{\cos 0} = 7$. Therefore, the polar coordinates are $(r, 	heta) = (7,0)$.

Question 5

Two points P and Q are on opposite sides of a river (see the sketch). From P to another point R on the same side is 340 ft. Angles $P R Q$ and $R P Q$ are found to be $23 ^ { \circ }$ and $120 ^ { \circ }$ , respectively. Compute the distance from P to Q, across the river. (Round your answer to the nearest foot.) 
A) $P Q = 221$ ft.
B) $P Q = 354$ ft.
C) $P Q = 246$ ft.
D) $P Q = 524$ ft.
E) $P Q = 694$ ft.

Accepted Answer

The answer of Two points P and Q are on...

Question 6

Determine the graph that reflects the polar equation. $r = - 5 - 5 \cos \theta$
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Determine the graph that reflects the polar...

Question 7

An airplane crashes in a lake and is spotted by observers at lighthouses A and B along the coast. Lighthouse B is 1.10 miles due east of lighthouse A. The bearing of the airplane from lighthouse A is $\mathrm { S } 20 ^ { \circ } \mathrm { E }$ ; the bearing of the plane from lighthouse B is $\mathrm { S } 40 ^ { \circ } \mathrm { W }$ . Find the distance from each lighthouse to the crash site.
A) Distance from lighthouse $A$ : 0.95 miles, Distance from lighthouse $B$ : 1.28 miles
B) Distance from lighthouse $A$ : 1.35 miles, Distance from lighthouse $B$ : 0.97 miles
C) Distance from lighthouse $A$ : 0.92 miles, Distance from lighthouse $B$ : 0.76 miles
D) Distance from lighthouse $A$ : 1.43 miles, Distance from lighthouse $B$ : 0.84 miles
E) Distance from lighthouse $A$ : 0.97miles, Distance from lighthouse $B$ : 1.19 miles

Accepted Answer

The answer of An airplane crashes in a lake and...

Question 8

Assume that the vectors $\mathbf { a }$ , $\mathbf { b }$ , $\text { C }$ and $\text { d }$ are defined as follows: $\mathbf { a } = \langle 1,1 \rangle$ $\mathbf { b } = \langle 5,4 \rangle$ $c = \langle 6 , - 8 \rangle$ $\mathbf { d } = \langle - 5,3 \rangle$ Compute $c + d$ .
A) $\langle 11 , - 4 \rangle$
B) $\langle 0,7 \rangle$
C) $\langle 1 , - 5 \rangle$
D) $\langle 7 , - 7 \rangle$
E) $\langle - 4,4 \rangle$

Accepted Answer

To compute $c + d$, add the corresponding components of vectors $c = \langle 6 , - 8 \rangle$ and $d = \langle - 5,3 \rangle$. This results in $\langle 6 + (-5), -8 + 3 \rangle = \langle 1, -5 \rangle$.

Question 9

Convert to rectangular form. $r = 3 \tan \theta$
A) $y ^ { 2 } + x ^ { 2 } \left( y ^ { 2 } - 9 \right) = 0$
B) $y ^ { 4 } + x ^ { 2 } \left( y ^ { 2 } - 9 \right) = 0$
C) $y ^ { 4 } + x ^ { 2 } \left( y ^ { 2 } - 1 \right) - 9 = 0$
D) $x ^ { 2 } + y ^ { 2 } \left( x ^ { 2 } - 3 \right) = 0$
E) $x ^ { 4 } + y ^ { 2 } \left( x ^ { 2 } - 9 \right) = 0$

Accepted Answer

The answer of Convert to rectangular form. \(r = 3...

Question 10

Determine the graph that reflects the polar equation. $r = - 2 + 4 \sin \theta$
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Determine the graph that reflects the polar...

Question 11

Assume that the vectors $\mathbf { a }$ , $\mathbf { b }$ , $\text { C }$ and $\text { d }$ are defined as follows: $\mathbf { a } = \langle 1,2 \rangle$ $$\mathbf { b } = \langle 5,4 \rangle$$ $\mathbf { c } = \langle 5 , - 2 \rangle$ 
$\mathbf { d } = \langle - 2,0 \rangle$ Compute $\frac { 1 } { | 3 b - 4 d | } ( 3 b - 4 c )$
A) $\left\langle - \frac { 5 \sqrt { 673 } } { 673 } , \frac { 20 \sqrt { 673 } } { 673 } \right\rangle$
B) $\left\langle \frac { 20 \sqrt { 673 } } { 673 } , \frac { 5 \sqrt { 673 } } { 673 } \right\rangle$
C) $\left\langle - \frac { 20 \sqrt { 673 } } { 673 } , - \frac { 5 \sqrt { 673 } } { 673 } \right\rangle$
D) $\left\langle - \frac { 5 \sqrt { 673 } } { 673 } , - \frac { 20 \sqrt { 673 } } { 673 } \right\rangle$
E) $\left\langle \frac { 5 \sqrt { 673 } } { 673 } , \frac { 20 \sqrt { 673 } } { 673 } \right\rangle$

Accepted Answer

First, calculate $3\mathbf{b} - 4\mathbf{d}$ and $3\mathbf{b} - 4\mathbf{c}$. For $3\mathbf{b} - 4\mathbf{d}$, we have $3\langle 5,4 \rangle - 4\langle -2,0 \rangle = \langle 15,12 \rangle + \langle 8,0 \rangle = \langle 23,12 \rangle$. The magnitude $|\langle 23,12 \rangle|$ is $\sqrt{23^2 + 12^2} = \sqrt{529 + 144} = \sqrt{673}$. For $3\mathbf{b} - 4\mathbf{c}$, we have $3\langle 5,4 \rangle - 4\langle 5,-2 \rangle = \langle 15,12 \rangle - \langle 20,-8 \rangle = \langle -5,20 \rangle$. Thus, $\frac{1}{|\langle 23,12 \rangle|}\langle -5,20 \rangle = \frac{1}{\sqrt{673}}\langle -5,20 \rangle = \left\langle -\frac{5\sqrt{673}}{673}, \frac{20\sqrt{673}}{673} \right\rangle$.

Question 12

Refer to the figure. If $\angle B = 45 ^ { \circ }$ and $A C = 40 \mathrm {~cm}$ , find $A B$ . 
A) $80$
B) $\frac { 80 \sqrt { 3 } } { 3 }$
C) $40 \sqrt { 2 }$
D) $40$
E) $\frac { 80 \sqrt { 3 } } { 2 }$

Accepted Answer

The answer of Refer to the figure. If \(\angle B...

Question 13

Graph the parametric equations using the given range for the parameter $t$ . Begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by $x = \cos t$ and $y = \sin t$ it would be natural to choose a viewing rectangle extending from -1 to 1 in both the $x$ - and $y$ -directions. Graph the parametric equations on a graphing utility. Sketch the result. $x = 5 \cos t$ and $y = 2 \sin t$ , $0 \leq t \leq \frac { \pi } { 2 }$ (one-quarter of an ellipse)
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Graph the parametric equations using the given...

Question 14

Assume that the coordinates of the points $P$ and $e$ are as follows: $P ( - 2,4 ) \quad Q ( 3,6 )$ Draw the vector $\overrightarrow { Q P }$ (using graph paper) and compute its magnitude.
A) $| \overrightarrow { Q P } | = \sqrt { 43 }$
B) $| \overrightarrow { Q P } | = \sqrt { 37 }$
C) $| \overrightarrow { Q P } | = \sqrt { 57 }$
D) $| \overrightarrow { Q P } | = \sqrt { 29 }$
E) $| \overrightarrow { Q P } | = \sqrt { 59 }$

Accepted Answer

The answer of Assume that the coordinates of the points...

Question 15

Use the given information to find the cosines of angles in $A B C$ . $a = 13$ cm, $b = 12$ cm, $c = 5$ cm
A) $\cos A = 0.689$ , $\cos B = 0.749$ , $\cos C = 0.082$
B) $\cos A = 0.859 , \cos B = 0.601 , \cos C = 0.993$
C) $\cos A = 0.929$ , $\cos B = 0.779$ , $\cos C = 0.082$
D) $\cos A = 0$ , $\cos B = 0.385$ , $\cos C = 0.923$
E) $\cos A = 0.005$ , $\cos B = 0.539$ , $\cos C = 0.854$

Accepted Answer

The answer of Use the given information to find the...

Question 16

Use the equation to determine polar coordinates of the point B. $r = - \frac { 3 } { 2 } - \frac { 3 } { 2 } \sin \theta$ 
A) $\left( 1 , \frac { 5 \pi } { 4 } \right)$
B) $\left( - 1.5 , \frac { 5 \pi } { 4 } \right)$
C) $\left( 0 , \frac { 5 \pi } { 4 } \right)$
D) $\left( - 0.44 , \frac { 5 \pi } { 4 } \right)$
E) $\left( 2.56 , \frac { 5 \pi } { 4 } \right)$

Accepted Answer

The answer of Use the equation to determine polar coordinates...

Question 17

On a sheet of paper, graph the parametric equation after eliminating the parameter $t$ ( $0 \leq t \leq 2 \pi$ ). Specify the approximate direction on the curve corresponding to increasing values of $t$ . $x = \cos t , y = 2 \sin t$
A) clockwise
B) counterclockwise

Accepted Answer

To eliminate the parameter $t$, we can square both equations and get $x^2+\dfrac{y^2}{4}=1$, which is the equation of an ellipse centered at the origin with semi-axes $a=1$ and $b=2$.

Since the equation is an ellipse, we know that as $t$ increases, the point moves counterclockwise around the origin, tracing out the ellipse in that direction. Therefore, the approximate direction on the curve corresponding to increasing values of $t$ is counterclockwise, so the answer is B.

Question 18

Determine the graph that reflects the polar equation. $r = 2 \cos 5 \theta$ (five-leafed rose)
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Determine the graph that reflects the polar...

Question 19

Convert to rectangular form. $$r \sin \left( \theta + \frac { \pi } { 4 } \right) = 4$$
A) $y = - x + 8 \sqrt { 2 }$
B) $y = - x + 4 \sqrt { 2 }$
C) $y = - x \sqrt { 2 } + 4 \sqrt { 2 }$
D) $y = x + 4 \sqrt { 2 }$
E) $y = 4 - \frac { \sqrt { 2 } } { 2 }$

Accepted Answer

To convert the given polar equation $r \sin \left( \theta + \frac { \pi } { 4 } \right) = 4$ to rectangular form, we use the identities $r^2 = x^2 + y^2$, $r\cos(\theta) = x$, and $r\sin(\theta) = y$. The given equation can be rewritten using the sine addition formula: $\sin(\theta + \frac{\pi}{4}) = \sin(\theta)\cos(\frac{\pi}{4}) + \cos(\theta)\sin(\frac{\pi}{4})$. Since $\sin(\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, the equation becomes $r(\frac{\sqrt{2}}{2}y + \frac{\sqrt{2}}{2}x) = 4$, or $r\sqrt{2}(y + x) = 8$. Dividing both sides by $\sqrt{2}$ and rearranging, we get $y = -x + 4\sqrt{2}$, which matches choice B.

Question 20

The accompanying figure shows two ships at points $P$ and $Q$ , which are in the same vertical plane as an airplane at point $R$ . When the height of the airplane is 4,000 ft, the angle of depression to $P$ is 38° and that to $e$ is 15°. Find the distance between the two ships. 
A) 54,900 ft
B) 2,050 ft
C) 20,050 ft
D) 80,430 ft
E) 4,200 ft

Accepted Answer

Let's call the distance between the airplane and ship P "x" and the distance between the airplane and ship Q "y". We can create two equations based on the angles of depression:
$$\tan(38^\circ)=\frac{4000}{x}$$
$$\tan(15^\circ)=\frac{4000}{y}$$
We can solve for x and y in terms of each other:
$$x=\frac{4000}{\tan(38^\circ)}\approx 5066.83\text{ ft}$$
$$y=\frac{4000}{\tan(15^\circ)}\approx 13763.75\text{ ft}$$
Finally, we can use the Pythagorean Theorem to find the distance between the two ships:
$$\sqrt{(y-x)^2+4000^2}\approx \boxed{\textbf{(C) }20,050\text{ ft}}.$$

Quiz 10: Additional Topics in Trigonometry

On a sheet of paper, graph the parametric equation after eliminating the parameter $t$ $( 0 \leq t \leq 2 \pi$ ) . Specify the approximate direction on the curve corresponding to increasing values of $t$ . $x = 6 \sin 2 t , y = 5 \cos 2 t$

Convert the given rectangular coordinates to polar coordinates. Express the answer in such a way that r is nonnegative and $0 \leq \theta < 2 \pi$ . $( - 5 , - 5 )$

Convert the given rectangular coordinates to polar coordinates. Express the answer in such a way that r is nonnegative and $0 \leq \theta < 2 \pi$ . $( 7,0 )$

Determine the graph that reflects the polar equation. $r = - 5 - 5 \cos \theta$

Assume that the vectors $\mathbf { a }$ , $\mathbf { b }$ , $\text { C }$ and $\text { d }$ are defined as follows: $\mathbf { a } = \langle 1,1 \rangle$ $\mathbf { b } = \langle 5,4 \rangle$ $c = \langle 6 , - 8 \rangle$ $\mathbf { d } = \langle - 5,3 \rangle$ Compute $c + d$ .

Convert to rectangular form. $r = 3 \tan \theta$

Determine the graph that reflects the polar equation. $r = - 2 + 4 \sin \theta$

Refer to the figure. If $\angle B = 45 ^ { \circ }$ and $A C = 40 \mathrm {~cm}$ , find $A B$ . $Refer to the figure. If \angle B = 45 ^ { \circ } and A C = 40 \mathrm {~cm} , find A B . A) 80 B) \frac { 80 \sqrt { 3 } } { 3 } C) 40 \sqrt { 2 } D) 40 E) \frac { 80 \sqrt { 3 } } { 2 }$

Assume that the coordinates of the points $P$ and $e$ are as follows: $P ( - 2,4 ) \quad Q ( 3,6 )$ Draw the vector $\overrightarrow { Q P }$ (using graph paper) and compute its magnitude.

Use the given information to find the cosines of angles in $A B C$ . $a = 13$ cm, $b = 12$ cm, $c = 5$ cm

On a sheet of paper, graph the parametric equation after eliminating the parameter $t$ ( $0 \leq t \leq 2 \pi$ ) . Specify the approximate direction on the curve corresponding to increasing values of $t$ . $x = \cos t , y = 2 \sin t$

Determine the graph that reflects the polar equation. $r = 2 \cos 5 \theta$ (five-leafed rose)

Convert to rectangular form. $r \sin \left( \theta + \frac { \pi } { 4 } \right) = 4$

The accompanying figure shows two ships at points $P$ and $Q$ , which are in the same vertical plane as an airplane at point $R$ . When the height of the airplane is 4,000 ft, the angle of depression to $P$ is 38° and that to $e$ is 15°. Find the distance between the two ships.

Quiz 10: Additional Topics in Trigonometry

On a sheet of paper, graph the parametric equation after eliminating the parameter ttt (0≤t≤2π( 0 \leq t \leq 2 \pi(0≤t≤2π ) . Specify the approximate direction on the curve corresponding to increasing values of ttt . x=6sin⁡2t,y=5cos⁡2tx = 6 \sin 2 t , y = 5 \cos 2 tx=6sin2t,y=5cos2t

Convert the given rectangular coordinates to polar coordinates. Express the answer in such a way that r is nonnegative and 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π . (−5,−5) ( - 5 , - 5 ) (−5,−5)

Convert the given rectangular coordinates to polar coordinates. Express the answer in such a way that r is nonnegative and 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π . (7,0) ( 7,0 ) (7,0)

Determine the graph that reflects the polar equation. r=−5−5cos⁡θr = - 5 - 5 \cos \thetar=−5−5cosθ

Convert to rectangular form. r=3tan⁡θr = 3 \tan \thetar=3tanθ

Determine the graph that reflects the polar equation. r=−2+4sin⁡θr = - 2 + 4 \sin \thetar=−2+4sinθ

Refer to the figure. If ∠B=45∘\angle B = 45 ^ { \circ }∠B=45∘ and AC=40 cmA C = 40 \mathrm {~cm}AC=40 cm , find ABA BAB .

Assume that the coordinates of the points PPP and eee are as follows: P(−2,4) Q(3,6) P ( - 2,4 ) \quad Q ( 3,6 ) P(−2,4) Q(3,6) Draw the vector QP→\overrightarrow { Q P }QP​ (using graph paper) and compute its magnitude.

Use the given information to find the cosines of angles in ABCA B CABC . a=13a = 13a=13 cm, b=12b = 12b=12 cm, c=5c = 5c=5 cm

Use the equation to determine polar coordinates of the point B. r=−32−32sin⁡θr = - \frac { 3 } { 2 } - \frac { 3 } { 2 } \sin \thetar=−23​−23​sinθ

On a sheet of paper, graph the parametric equation after eliminating the parameter ttt ( 0≤t≤2π0 \leq t \leq 2 \pi0≤t≤2π ) . Specify the approximate direction on the curve corresponding to increasing values of ttt . x=cos⁡t,y=2sin⁡tx = \cos t , y = 2 \sin tx=cost,y=2sint

Determine the graph that reflects the polar equation. r=2cos⁡5θr = 2 \cos 5 \thetar=2cos5θ (five-leafed rose)

Convert to rectangular form. rsin⁡(θ+π4) =4r \sin \left( \theta + \frac { \pi } { 4 } \right) = 4rsin(θ+4π​) =4

The accompanying figure shows two ships at points PPP and QQQ , which are in the same vertical plane as an airplane at point RRR . When the height of the airplane is 4,000 ft, the angle of depression to PPP is 38° and that to eee is 15°. Find the distance between the two ships.

On a sheet of paper, graph the parametric equation after eliminating the parameter $t$ $( 0 \leq t \leq 2 \pi$ ) . Specify the approximate direction on the curve corresponding to increasing values of $t$ . $x = 6 \sin 2 t , y = 5 \cos 2 t$

Convert the given rectangular coordinates to polar coordinates. Express the answer in such a way that r is nonnegative and $0 \leq \theta < 2 \pi$ . $( - 5 , - 5 )$

Convert the given rectangular coordinates to polar coordinates. Express the answer in such a way that r is nonnegative and $0 \leq \theta < 2 \pi$ . $( 7,0 )$

Determine the graph that reflects the polar equation. $r = - 5 - 5 \cos \theta$

Convert to rectangular form. $r = 3 \tan \theta$

Determine the graph that reflects the polar equation. $r = - 2 + 4 \sin \theta$

Refer to the figure. If $\angle B = 45 ^ { \circ }$ and $A C = 40 \mathrm {~cm}$ , find $A B$ . $Refer to the figure. If \angle B = 45 ^ { \circ } and A C = 40 \mathrm {~cm} , find A B . A) 80 B) \frac { 80 \sqrt { 3 } } { 3 } C) 40 \sqrt { 2 } D) 40 E) \frac { 80 \sqrt { 3 } } { 2 }$

Assume that the coordinates of the points $P$ and $e$ are as follows: $P ( - 2,4 ) \quad Q ( 3,6 )$ Draw the vector $\overrightarrow { Q P }$ (using graph paper) and compute its magnitude.

Use the given information to find the cosines of angles in $A B C$ . $a = 13$ cm, $b = 12$ cm, $c = 5$ cm

On a sheet of paper, graph the parametric equation after eliminating the parameter $t$ ( $0 \leq t \leq 2 \pi$ ) . Specify the approximate direction on the curve corresponding to increasing values of $t$ . $x = \cos t , y = 2 \sin t$

Determine the graph that reflects the polar equation. $r = 2 \cos 5 \theta$ (five-leafed rose)

Convert to rectangular form. $r \sin \left( \theta + \frac { \pi } { 4 } \right) = 4$

The accompanying figure shows two ships at points $P$ and $Q$ , which are in the same vertical plane as an airplane at point $R$ . When the height of the airplane is 4,000 ft, the angle of depression to $P$ is 38° and that to $e$ is 15°. Find the distance between the two ships.