Deck 10: Additional Topics in Trigonometry

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) $V _ { x } \approx - 25.98 \mathrm {~cm} / \mathrm { sec } V _ { y } \approx 15 \mathrm {~cm} / \mathrm { sec }$

Answer 1

$V _ { z } = - 15$ cm/sec $V _ { y } \approx 25.98$ cm/sec

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

Answer 2

clockwise

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) $\left( - 5 \sqrt { 2 } , \frac { 5 \pi } { 4 } \right)$

Answer 3

$\left( 5 \sqrt { 2 } , \frac { 5 \pi } { 4 } \right)$

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) $\left( 7 , \frac { 3 \pi } { 2 } \right)$

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.

Question 6

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

A)
B)
C)
D)
E)

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

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) $\langle - 4,4 \rangle$

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) $x ^ { 4 } + y ^ { 2 } \left( x ^ { 2 } - 9 \right) = 0$

Question 10

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

A)
B)
C)
D)
E)

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) $\left\langle \frac { 5 \sqrt { 673 } } { 673 } , \frac { 20 \sqrt { 673 } } { 673 } \right\rangle$

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) $\frac { 80 \sqrt { 3 } } { 2 }$

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)

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) $| \overrightarrow { Q P } | = \sqrt { 59 }$

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) $\cos A = 0.005$ , $\cos B = 0.539$ , $\cos C = 0.854$

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) $\left( 2.56 , \frac { 5 \pi } { 4 } \right)$

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

Question 18

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

A)
B)
C)
D)
E)

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) $y = 4 - \frac { \sqrt { 2 } } { 2 }$

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

Question 21

Convert from rectangular to trigonometric form. (Choose an argument $\theta$ such that $0 \leq \theta < 2 \pi$ .) $- \frac { 1 } { 2 } + \frac { 1 } { 2 } \sqrt { 3 } i$

A) $\frac { 1 } { 2 } \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$
B) $\frac { 1 } { 2 } \left( \sin \frac { 2 \pi } { 3 } + i \cos \frac { 2 \pi } { 3 } \right)$
C) $\left( \cos \frac { 2 \pi } { 3 } - i \sin \frac { 2 \pi } { 3 } \right)$
D) $\left( \sin \frac { 2 \pi } { 3 } - i \cos \frac { 2 \pi } { 3 } \right)$
E) $\left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$

Question 22

Determine the graph of the equation. $r = 2 + 3 \sin \theta$

A)
B)
C)
D)
E)

Question 23

Convert from rectangular to trigonometric form. (Choose an argument $\theta$ such that $0 \leq \theta < 2 \pi$ .) 6

A) $6 ( \cos \pi + i \sin \pi )$
B) $6 ( \sin 0 )$
C) $6 ( \sin \pi + i \cos \pi )$
D) $6 ( \cos 0 + i \sin 0 )$
E) $6 ( \sin 0 + i \cos 0 )$

Question 24

Carry out the indicated operations. $\left[ \frac { \cos \frac { \pi } { 11 } + i \sin \frac { \pi } { 11 } } { \cos \left( - \frac { \pi } { 11 } \right) + i \sin \left( - \frac { \pi } { 11 } \right) } \right] ^ { 4 }$

A) $\left( \cos \frac { - 4 \pi } { 11 } + i \sin \frac { - 4 \pi } { 11 } \right)$
B) $\left( \cos \frac { 8 \pi } { 11 } + i \sin \frac { 8 \pi } { 11 } \right)$
C) $( \cos 0 + i \sin 0 )$
D) $\left( \cos \frac { 16 \pi } { 11 } + i \sin \frac { 16 \pi } { 11 } \right)$
E) $\left( \cos \frac { 4 \pi } { 11 } + i \sin \frac { 4 \pi } { 11 } \right)$

Question 25

Simplify. $$\frac { 3 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right) } { 5 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right) }$$

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