Question 1

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
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)  $\left\langle - \frac { 5 \sqrt { 673 } } { 673 } , \frac { 20 \sqrt { 673 } } { 673 } \right\rangle$ 
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)  $\left\langle \frac { 5 \sqrt { 673 } } { 673 } , \frac { 20 \sqrt { 673 } } { 673 } \right\rangle$ 
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)  $\left\langle \frac { 5 \sqrt { 673 } } { 673 } , \frac { 20 \sqrt { 673 } } { 673 } \right\rangle$ 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)  $( 0,7 )$ 
B)  $( 7,0 )$ 
C)  $\left( - 7 , \frac { 3 \pi } { 2 } \right)$ 
D)  $( - 7,0 )$ 
E)  $\left( 7 , \frac { 3 \pi } { 2 } \right)$ 
A)  $( 0,7 )$ 
B)  $( 7,0 )$ 
C)  $\left( - 7 , \frac { 3 \pi } { 2 } \right)$ 
D)  $( - 7,0 )$ 
E)  $\left( 7 , \frac { 3 \pi } { 2 } \right)$ 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 
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)  $y ^ { 2 } + x ^ { 2 } \left( y ^ { 2 } - 9 \right) = 0$ 
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)  $x ^ { 4 } + y ^ { 2 } \left( x ^ { 2 } - 9 \right) = 0$ 
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 6

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

Accepted Answer

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)  $\left( 2.56 , \frac { 5 \pi } { 4 } \right)$ 
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 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)  $V _ { x } = 25.98$ 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)  $V _ { x } \approx - 25.98 \mathrm {~cm} / \mathrm { sec } V _ { y } \approx 15 \mathrm {~cm} / \mathrm { sec }$ 
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)  $V _ { x } \approx - 25.98 \mathrm {~cm} / \mathrm { sec } V _ { y } \approx 15 \mathrm {~cm} / \mathrm { sec }$

Question 8

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
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 $A$ : 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 
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 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)  $\left( 5 , - \frac { 5 \pi } { 4 } \right)$ 
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)  $\left( - 5 \sqrt { 2 } , \frac { 5 \pi } { 4 } \right)$ 
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)  $\left( - 5 \sqrt { 2 } , \frac { 5 \pi } { 4 } \right)$

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)  $\frac { 1 } { 2 } \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$ 
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)  $\left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$ 
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 12

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
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)  $P Q = 221$ ft. 
B)  $P Q = 354$ ft. 
C)  $P Q = 246$ ft. 
D)  $P Q = 524$ ft. 
E)  $P Q = 694$ ft. 
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 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)  $\langle 11 , - 4 \rangle$ 
B)  $\langle 0,7 \rangle$ 
C)  $\langle 1 , - 5 \rangle$ 
D)  $\langle 7 , - 7 \rangle$ 
E)  $\langle - 4,4 \rangle$ 
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 15

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

Accepted Answer

A)  $80$ 
B)  $\frac { 80 \sqrt { 3 } } { 3 }$ 
C)  $40 \sqrt { 2 }$ 
D)  $40$ 
E)  $\frac { 80 \sqrt { 3 } } { 2 }$ 
A)  $80$ 
B)  $\frac { 80 \sqrt { 3 } } { 3 }$ 
C)  $40 \sqrt { 2 }$ 
D)  $40$ 
E)  $\frac { 80 \sqrt { 3 } } { 2 }$

Question 16

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

Accepted Answer

A)  $6 ( \cos \pi + i \sin \pi )$ 
B)  $6 ( \sin 0 )$ 
C)  $6 ( \sin \pi + i \cos \pi )$ 
D)  $6 ( \cos 0 + i \sin 0 )$ 
E)  $6 ( \sin 0 + i \cos 0 )$ 
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 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)  $\cos A = 0.689$ , $\cos B = 0.749$ , $\cos C = 0.082$ 
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)  $\cos A = 0.005$ , $\cos B = 0.539$ , $\cos C = 0.854$ 
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 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)  $\frac { 3 \sqrt { 3 } } { 10 } - \frac { 3 } { 10 } i$ 
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)  $\frac { 3 \sqrt { 3 } } { 10 } + \frac { 3 } { 10 } i$ 
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)  $\frac { 3 \sqrt { 3 } } { 10 } + \frac { 3 } { 10 } i$

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 
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)    
A)    
B)    
C)    
D)    
E)

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. $r = 3 \tan \theta$

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

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

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. $r = - 5 - 5 \cos \theta$

Fundamentals

Equations and Inequalities

Functions

Polynomial and Rational Functions Applications to Optimization

Exponential and Logarithmic Functions

An Introduction to Trigonometry Via Right Triangles

The Trigonometric Functions

Graphs of the Trigonometric Functions

Analytical Trigonometry

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The Conic Sections

Roots of Polynomial Equations

Additional Topics in Algebra

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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

Fundamentals

Equations and Inequalities

Functions

Polynomial and Rational Functions Applications to Optimization

Exponential and Logarithmic Functions

An Introduction to Trigonometry Via Right Triangles

The Trigonometric Functions

Graphs of the Trigonometric Functions

Analytical Trigonometry

Systems of Equations

The Conic Sections

Roots of Polynomial Equations

Additional Topics in Algebra

Filters

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. $r = 3 \tan \theta$

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

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

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. $r = - 5 - 5 \cos \theta$