Question 1

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

Accepted Answer

A)   
B)   
C)   
D)   
E)     A

Question 2

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 $\frac { 1 } { | 3 b - 4 d | } ( 3 b - 4 c )$

Accepted Answer

A)    ⟨−5673673,20673673⟩\left\langle - \frac { 5 \sqrt { 673 } } { 673 } , \frac { 20 \sqrt { 673 } } { 673 } \right\rangle⟨−6735673c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​,67320673c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​⟩ 
B)    ⟨20673673,5673673⟩\left\langle \frac { 20 \sqrt { 673 } } { 673 } , \frac { 5 \sqrt { 673 } } { 673 } \right\rangle⟨67320673c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​,6735673c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​⟩ 
C)    ⟨−20673673,−5673673⟩\left\langle - \frac { 20 \sqrt { 673 } } { 673 } , - \frac { 5 \sqrt { 673 } } { 673 } \right\rangle⟨−67320673c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​,−6735673c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​⟩ 
D)    ⟨−5673673,−20673673⟩\left\langle - \frac { 5 \sqrt { 673 } } { 673 } , - \frac { 20 \sqrt { 673 } } { 673 } \right\rangle⟨−6735673c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​,−67320673c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​⟩ 
E)    ⟨5673673,20673673⟩\left\langle \frac { 5 \sqrt { 673 } } { 673 } , \frac { 20 \sqrt { 673 } } { 673 } \right\rangle⟨6735673c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​,67320673c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​⟩ A

Question 3

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

Accepted Answer

A)    (0,7) ( 0,7 ) (0,7)  
B)    (7,0) ( 7,0 ) (7,0)  
C)    (−7,3π2) \left( - 7 , \frac { 3 \pi } { 2 } \right) (−7,23π​)  
D)    (−7,0) ( - 7,0 ) (−7,0)  
E)    (7,3π2) \left( 7 , \frac { 3 \pi } { 2 } \right) (7,23π​)  B

Question 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.

Accepted Answer

A)   54,900 ft
B)   2,050 ft
C)   20,050 ft
D)   80,430 ft
E)   4,200 ft

Question 5

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

Accepted Answer

A)    y2+x2(y2−9) =0y ^ { 2 } + x ^ { 2 } \left( y ^ { 2 } - 9 \right)  = 0y2+x2(y2−9) =0 
B)    y4+x2(y2−9) =0y ^ { 4 } + x ^ { 2 } \left( y ^ { 2 } - 9 \right)  = 0y4+x2(y2−9) =0 
C)    y4+x2(y2−1) −9=0y ^ { 4 } + x ^ { 2 } \left( y ^ { 2 } - 1 \right)  - 9 = 0y4+x2(y2−1) −9=0 
D)    x2+y2(x2−3) =0x ^ { 2 } + y ^ { 2 } \left( x ^ { 2 } - 3 \right)  = 0x2+y2(x2−3) =0 
E)    x4+y2(x2−9) =0x ^ { 4 } + y ^ { 2 } \left( x ^ { 2 } - 9 \right)  = 0x4+y2(x2−9) =0

Question 6

Use the equation to determine polar coordinates of the point B. $r = - \frac { 3 } { 2 } - \frac { 3 } { 2 } \sin \theta$

Accepted Answer

A)    (1,5π4) \left( 1 , \frac { 5 \pi } { 4 } \right) (1,45π​)  
B)    (−1.5,5π4) \left( - 1.5 , \frac { 5 \pi } { 4 } \right) (−1.5,45π​)  
C)    (0,5π4) \left( 0 , \frac { 5 \pi } { 4 } \right) (0,45π​)  
D)    (−0.44,5π4) \left( - 0.44 , \frac { 5 \pi } { 4 } \right) (−0.44,45π​)  
E)    (2.56,5π4) \left( 2.56 , \frac { 5 \pi } { 4 } \right) (2.56,45π​)

Question 7

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.)

Accepted Answer

A)    Vx=25.98V _ { x } = 25.98Vx​=25.98  cm/sec  Vy≈−15V _ { y } \approx - 15Vy​≈−15  cm/sec
B)    Vx=15V _ { x } = 15Vx​=15  cm/sec  Vy≈25.98V _ { y } \approx 25.98Vy​≈25.98  cm/sec
C)    Vz=−15V _ { z } = - 15Vz​=−15  cm/sec  Vy≈25.98V _ { y } \approx 25.98Vy​≈25.98  cm/sec
D)    Vx=21.21V _ { x } = 21.21Vx​=21.21  cm/sec  Vy≈21.21V _ { y } \approx 21.21Vy​≈21.21  cm/sec
E)    Vx≈−25.98 cm/secVy≈15 cm/secV _ { x } \approx - 25.98 \mathrm {~cm} / \mathrm { sec } V _ { y } \approx 15 \mathrm {~cm} / \mathrm { sec }Vx​≈−25.98 cm/secVy​≈15 cm/sec

Question 8

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

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 9

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.

Accepted Answer

A)   Distance from lighthouse  AAA  : 0.95 miles, Distance from lighthouse  BBB  : 1.28 miles
B)   Distance from lighthouse  AAA  : 1.35 miles, Distance from lighthouse  BBB  : 0.97 miles
C)   Distance from lighthouse  AAA  : 0.92 miles, Distance from lighthouse  BBB  : 0.76 miles
D)   Distance from lighthouse  AAA  : 1.43 miles, Distance from lighthouse  BBB  : 0.84 miles
E)   Distance from lighthouse  AAA  : 0.97miles, Distance from lighthouse  BBB  : 1.19 miles

Question 10

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

Accepted Answer

A)    (5,−5π4) \left( 5 , - \frac { 5 \pi } { 4 } \right) (5,−45π​)  
B)    (52,5π4) \left( 5 \sqrt { 2 } , \frac { 5 \pi } { 4 } \right) (52c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​,45π​)  
C)    (5,π4) \left( 5 , \frac { \pi } { 4 } \right) (5,4π​)  
D)    (52,π4) \left( 5 \sqrt { 2 } , \frac { \pi } { 4 } \right) (52c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​,4π​)  
E)    (−52,5π4) \left( - 5 \sqrt { 2 } , \frac { 5 \pi } { 4 } \right) (−52c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​,45π​)

Question 11

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

Accepted Answer

A)    12(cos⁡2π3+isin⁡2π3) \frac { 1 } { 2 } \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) 21​(cos32π​+isin32π​)  
B)    12(sin⁡2π3+icos⁡2π3) \frac { 1 } { 2 } \left( \sin \frac { 2 \pi } { 3 } + i \cos \frac { 2 \pi } { 3 } \right) 21​(sin32π​+icos32π​)  
C)    (cos⁡2π3−isin⁡2π3) \left( \cos \frac { 2 \pi } { 3 } - i \sin \frac { 2 \pi } { 3 } \right) (cos32π​−isin32π​)  
D)    (sin⁡2π3−icos⁡2π3) \left( \sin \frac { 2 \pi } { 3 } - i \cos \frac { 2 \pi } { 3 } \right) (sin32π​−icos32π​)  
E)    (cos⁡2π3+isin⁡2π3) \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) (cos32π​+isin32π​)

Question 12

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

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 13

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.)

Accepted Answer

A)    PQ=221P Q = 221PQ=221  ft.
B)    PQ=354P Q = 354PQ=354  ft.
C)    PQ=246P Q = 246PQ=246  ft.
D)    PQ=524P Q = 524PQ=524  ft.
E)    PQ=694P Q = 694PQ=694  ft.

Question 14

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 $c + d$ .

Accepted Answer

A)    ⟨11,−4⟩\langle 11 , - 4 \rangle⟨11,−4⟩ 
B)    ⟨0,7⟩\langle 0,7 \rangle⟨0,7⟩ 
C)    ⟨1,−5⟩\langle 1 , - 5 \rangle⟨1,−5⟩ 
D)    ⟨7,−7⟩\langle 7 , - 7 \rangle⟨7,−7⟩ 
E)    ⟨−4,4⟩\langle - 4,4 \rangle⟨−4,4⟩

Question 15

Refer to the figure. If $\angle B = 45 ^ { \circ }$ and $A C = 40 \mathrm {~cm}$ , find $A B$ .

Accepted Answer

A)    808080 
B)    8033\frac { 80 \sqrt { 3 } } { 3 }3803c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
C)    40240 \sqrt { 2 }402c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​ 
D)    404040 
E)    8032\frac { 80 \sqrt { 3 } } { 2 }2803c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​

Question 16

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

Accepted Answer

A)    6(cos⁡π+isin⁡π) 6 ( \cos \pi + i \sin \pi ) 6(cosπ+isinπ)  
B)    6(sin⁡0) 6 ( \sin 0 ) 6(sin0)  
C)    6(sin⁡π+icos⁡π) 6 ( \sin \pi + i \cos \pi ) 6(sinπ+icosπ)  
D)    6(cos⁡0+isin⁡0) 6 ( \cos 0 + i \sin 0 ) 6(cos0+isin0)  
E)    6(sin⁡0+icos⁡0) 6 ( \sin 0 + i \cos 0 ) 6(sin0+icos0)

Question 17

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

Accepted Answer

A)    cos⁡A=0.689\cos A = 0.689cosA=0.689  ,  cos⁡B=0.749\cos B = 0.749cosB=0.749  ,  cos⁡C=0.082\cos C = 0.082cosC=0.082 
B)    cos⁡A=0.859,cos⁡B=0.601,cos⁡C=0.993\cos A = 0.859 , \cos B = 0.601 , \cos C = 0.993cosA=0.859,cosB=0.601,cosC=0.993 
C)    cos⁡A=0.929\cos A = 0.929cosA=0.929  ,  cos⁡B=0.779\cos B = 0.779cosB=0.779  ,  cos⁡C=0.082\cos C = 0.082cosC=0.082 
D)    cos⁡A=0\cos A = 0cosA=0  ,  cos⁡B=0.385\cos B = 0.385cosB=0.385  ,  cos⁡C=0.923\cos C = 0.923cosC=0.923 
E)    cos⁡A=0.005\cos A = 0.005cosA=0.005  ,  cos⁡B=0.539\cos B = 0.539cosB=0.539  ,  cos⁡C=0.854\cos C = 0.854cosC=0.854

Question 18

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

Accepted Answer

A)    3310−310i\frac { 3 \sqrt { 3 } } { 10 } - \frac { 3 } { 10 } i1033c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​−103​i 
B)    310−3310i\frac { 3 } { 10 } - \frac { 3 \sqrt { 3 } } { 10 } i103​−1033c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​i 
C)    310+3310i\frac { 3 } { 10 } + \frac { 3 \sqrt { 3 } } { 10 } i103​+1033c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​i 
D)    −310−3310i- \frac { 3 } { 10 } - \frac { 3 \sqrt { 3 } } { 10 } i−103​−1033c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​i 
E)    3310+310i\frac { 3 \sqrt { 3 } } { 10 } + \frac { 3 } { 10 } i1033c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​+103​i

Question 19

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 $t$ . $x = 6 \sin 2 t , y = 5 \cos 2 t$

Accepted Answer

A)   counterclockwise
B)   clockwise

Question 20

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

Accepted Answer

A)   
B)   
C)   
D)   
E)

Exam 10: Additional Topics in Trigonometry

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

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

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.

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

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

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

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

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 $c + d$ .

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

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

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

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

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 $t$ . $x = 6 \sin 2 t , y = 5 \cos 2 t$

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

Exam 10: Additional Topics in Trigonometry

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

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)

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.

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

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

Determine the graph of the equation. r=2+3sin⁡θr = 2 + 3 \sin \thetar=2+3sinθ

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 from rectangular to trigonometric form. (Choose an argument θ\thetaθ such that 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π .) −12+123i- \frac { 1 } { 2 } + \frac { 1 } { 2 } \sqrt { 3 } i−21​+21​3​i

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

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 .

Convert from rectangular to trigonometric form. (Choose an argument θ\thetaθ such that 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π .) 6

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

Simplify. 3(cos⁡π3+isin⁡π3) 5(cos⁡π6+isin⁡π6) \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) }5(cos6π​+isin6π​) 3(cos3π​+isin3π​) ​

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

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

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

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

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.

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

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

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

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

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

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

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

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

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 $t$ . $x = 6 \sin 2 t , y = 5 \cos 2 t$

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