9-3 Polar and Rectangular Forms of Equations
Identify the graph of each rectangular equation. Then write the equation in polar form. Support your
answer by graphing the polar form of the equation.
33. y =
x
SOLUTION: The graph of y =
Then simplify.
x is a line. To find the polar form of this equation, replace y with r sin
Evaluate the function for several
polar equation is a line.
and x with r cos
.
-values in its domain and use these points to graph the function. The graph of this
35. x2 + (y − 8)2 = 64
SOLUTION: 2
2
The graph of x + (y − 8) = 64 is a circle with radius 8 centered at (0, 8). To find the polar form of this equation,
replace x with r cos and y with r sin . Then simplify.
Evaluate the function for several
polar equation is a circle.
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-values in its domain and use these points to graph the function. The graph of this
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9-3 Polar and Rectangular Forms of Equations
35. x2 + (y − 8)2 = 64
SOLUTION: 2
2
The graph of x + (y − 8) = 64 is a circle with radius 8 centered at (0, 8). To find the polar form of this equation,
replace x with r cos and y with r sin . Then simplify.
Evaluate the function for several
polar equation is a circle.
-values in its domain and use these points to graph the function. The graph of this
Write each equation in rectangular form, and then identify its graph. Support your answer by graphing the
polar form of the equation.
37. SOLUTION: The graph of this equation is a line through the origin with slope −
its domain and use these points to graph the function.
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. Evaluate the function for several
-values in
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9-3 Polar and Rectangular Forms of Equations
Write each equation in rectangular form, and then identify its graph. Support your answer by graphing the
polar form of the equation.
37. SOLUTION: The graph of this equation is a line through the origin with slope −
its domain and use these points to graph the function.
. Evaluate the function for several
-values in
39. r = 4 cos
SOLUTION: The graph of this equation as a circle centered at (2, 0) with radius 2. Evaluate the function for several
its domain and use these points to graph the function.
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-values in
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9-3 Polar and Rectangular Forms of Equations
39. r = 4 cos
SOLUTION: The graph of this equation as a circle centered at (2, 0) with radius 2. Evaluate the function for several
its domain and use these points to graph the function.
-values in
41. r = 8 csc
SOLUTION: The graph of this equation is a horizontal line through the y-intercept 8 with slope 0. Evaluate the function for several
-values in its domain and use these points to graph the function.
43. cot
= −7
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SOLUTION: Page 4
9-3 Polar and Rectangular Forms of Equations
43. cot
= −7
SOLUTION: The graph of this equation is a line through the origin with slope
its domain and use these points to graph the function.
. Evaluate the function for several
-values in
45. r = sec
SOLUTION: The graph of this equation is a vertical line through the x-intercept 1 with an undefined slope. Evaluate the function
for several -values in its domain and use these points to graph the function.
47. MICROPHONE The polar pattern for a directional microphone at a football game is given by r = 2 + 2 cos θ.
a. Graph the polar pattern.
b. Will the microphone detect a sound that originates from the point with rectangular coordinates (−2, 0)? Explain.
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SOLUTION: Page 5
a. This graph is symmetric with respect to the polar axis, so you can find points on the interval [0, π] and then use
polar axis symmetry to complete the graph.
9-3 Polar and Rectangular Forms of Equations
47. MICROPHONE The polar pattern for a directional microphone at a football game is given by r = 2 + 2 cos θ.
a. Graph the polar pattern.
b. Will the microphone detect a sound that originates from the point with rectangular coordinates (−2, 0)? Explain.
SOLUTION: a. This graph is symmetric with respect to the polar axis, so you can find points on the interval [0, π] and then use
polar axis symmetry to complete the graph.
b. Convert the rectangular coordinates (−2, 0) to polar coordinates.
The sound originates from the point with polar coordinates (2, π). This point does not lie within the polar region that
is graphed. Thus, the microphone will not detect the sound.
Write each equation in rectangular form, and then identify its graph. Support your answer by graphing the
polar form of the equation.
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sound
originates
from the point
with polar
coordinates (2, π). This point does not lie within the polar region that
9-3 The
Polar
and
Rectangular
Forms
of Equations
is graphed. Thus, the microphone will not detect the sound.
Write each equation in rectangular form, and then identify its graph. Support your answer by graphing the
polar form of the equation.
49. SOLUTION: The graph of this equation is a line through the point
with slope 1. Evaluate the function for several -values in its domain and use these points to graph the function.
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9-3 Polar and Rectangular Forms of Equations
51. SOLUTION: The graph of this equation is a line through the point (0, 4) with slope
values in its domain and use these points to graph the function.
. Evaluate the function for several
-
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9-3 Polar and Rectangular Forms of Equations
53. SOLUTION: The graph of this equation is a line through the point (0, −5) with slope 1. Evaluate the function for several
in its domain and use these points to graph the function.
-values
55. SOLUTION: The graph of this equation is a circle with a center at (0, −2) and radius 2. Evaluate the function for several
values in its domain and use these points to graph the function.
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9-3 Polar and Rectangular Forms of Equations
55. SOLUTION: The graph of this equation is a circle with a center at (0, −2) and radius 2. Evaluate the function for several
values in its domain and use these points to graph the function.
-
Identify the graph of each rectangular equation. Then write the equation in polar form. Support your
answer by graphing the polar form of the equation.
57. 6x − 3y = 4
SOLUTION: Rewrite 6x − 3y = 4 in slope intercept form.
The graph of
with r cos
is a line with point and y with r sin
and slope 2. To find the polar form of the equation, replace x
in the original equation. Then simplify.
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9-3 Polar and Rectangular Forms of Equations
Identify the graph of each rectangular equation. Then write the equation in polar form. Support your
answer by graphing the polar form of the equation.
57. 6x − 3y = 4
SOLUTION: Rewrite 6x − 3y = 4 in slope intercept form.
is a line with point The graph of
with r cos
and y with r sin
and slope 2. To find the polar form of the equation, replace x
in the original equation. Then simplify.
Evaluate the function for several
-values in its domain and use these points to graph the function. Note that this graph will be similar to
.
59. (x − 6)2 + (y − 8)2 = 100
SOLUTION: 2
2
The graph of (x − 6) + (y − 8) = 100 is a circle with radius 10 centered at (6, 8). To find the polar form of this
equation, replace x with r cos and y with r sin . Then simplify.
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9-3 Polar and Rectangular Forms of Equations
59. (x − 6)2 + (y − 8)2 = 100
SOLUTION: 2
2
The graph of (x − 6) + (y − 8) = 100 is a circle with radius 10 centered at (6, 8). To find the polar form of this
equation, replace x with r cos and y with r sin . Then simplify.
Evaluate the function for several
polar equation is a circle.
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-values in its domain and use these points to graph the function. The graph of this
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