9. Polar Equations

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Plot the point on the polar coordinate system.

$(5,210°)$

Plot the point on the polar coordinate system.

$(-3,-90°)$

Plot the point on the polar coordinate system.

$(6,-\frac{11\pi}{6})$

Plot the point on the polar coordinate system.

$(-2,\frac{2\pi}{3})$

Plot the point $(3,\frac{\pi}{2})$ & find another set of coordinates, $(r,θ)$ , for this point, where:

(A) $r≥0,2π≤θ≤4π$ ,

(B) $r≥0,-2π≤θ≤0$ ,

Plot the point $(5,-\frac{\pi}{3})$ , then identify which of the following sets of coordinates is the same point.

Plot the point $(-3,-\frac{\pi}{6})$ , then identify which of the following sets of coordinates is the same point.

In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (3, 225°)

In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (−3, 5π/4)

In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (3, π)

In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (3, −135°)

In Exercises 7–12, test for symmetry with respect to a. the polar axis. b. the line θ=π2. c. the pole. r = 4 + 3 cos θ

In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (−3, −3π/4)

In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (2, 45°)

Convert x² + (y + 8)² = 64 to a polar equation that expresses r in terms of θ.

In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (3, 90°)

In Exercises 13–14, graph each polar equation. r = 1 + sin θ

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 2 cos θ

In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (3, 4π/3)

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − sin θ

In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (−1, π)

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 2 + 2 cos θ

In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (−2, − π/2)

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 2 + cos θ

In Exercises 21–26, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. Then find another representation of this point in which a. r>0, 2π < θ < 4π. b. r<0, 0. < θ < 2π. c. r>0, −2π. < θ < 0. (5, π/6)

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 + 2 cos θ

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 2 − 3 sin θ

123

In Exercises 27–32, select the representations that do not change the location of the given point. (7, 140°) (−7, 320°)

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 4 sin 3θ

In Exercises 27–32, select the representations that do not change the location of the given point. (4, 120°) (−4, 300°)

In Exercises 27–32, select the representations that do not change the location of the given point. (2, − 3π/4) (2, − 7π/4)

In Exercises 27–32, select the representations that do not change the location of the given point. (−2, 7π/6) (−2, −5π/6)

In Exercises 27–32, select the representations that do not change the location of the given point. (−5, − π/4) (−5, 7π/4)

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − 3 sin θ

In Exercises 27–32, select the representations that do not change the location of the given point. (−6, 3π) (6, −π)

In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (4, 90°)

147

In Exercises 13–34, test for symmetry and then graph each polar equation. r cos θ = −3

In Exercises 35–44, test for symmetry and then graph each polar equation. r = cos θ/2

In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (−4, π/2)

In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (7.4, 2.5)

In Exercises 35–44, test for symmetry and then graph each polar equation. r = 1 / 1−cos θ

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (−2, 2)

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. _ (2,−2√3)

In Exercises 35–44, test for symmetry and then graph each polar equation. r = 2 + 3 sin 2θ

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. _ (−√3,−1)

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (5, 0)

108

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. 3x + y = 7

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. x = 7

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. x² + y² = 9

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. (x − 2)² + y² = 4

In Exercises 54–60, convert each polar equation to a rectangular equation. Then use your knowledge of the rectangular equation to graph the polar equation in a polar coordinate system. θ = 3π/4

In Exercises 54–60, convert each polar equation to a rectangular equation. Then use your knowledge of the rectangular equation to graph the polar equation in a polar coordinate system. r = 5 csc θ

In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 8

In Exercises 61–63, test for symmetry with respect to a. the polar axis. b. the line θ = π/2. c. the pole. r = 5 + 3 cos θ

In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 4 csc θ

101

In Exercises 64–70, graph each polar equation. Be sure to test for symmetry. r = 2 + 2 sin θ

In Exercises 64–70, graph each polar equation. Be sure to test for symmetry. r = 2 + cos θ

In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 12 cos θ

130

In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 6 cos θ + 4 sin θ

In Exercises 79–80, convert each polar equation to a rectangular equation. Then determine the graph's slope and y-intercept. r sin (θ − π/4) = 2

In Exercises 81–82, find the rectangular coordinates of each pair of points. Then find the distance, in simplified radical form, between the points. (2, 2π/3) and (4, π/6)

In Exercises 81–86, solve each equation in the complex number system. Express solutions in polar and rectangular form. x⁶ − 1 = 0

In Exercises 81–86, solve each equation in the complex number system. Express solutions in polar and rectangular form. x⁴ + 16i = 0

In Exercises 81–86, solve each equation in the complex number system. Express solutions in polar and rectangular form. _ x³ − (1 + i√3 = 0

In calculus, it can be shown that e^(iθ) = cos θ + i sin θ. In Exercises 87–90, use this result to plot each complex number. e^(πi/4)

In calculus, it can be shown that e^(iθ) = cos θ + i sin θ. In Exercises 87–90, use this result to plot each complex number. -e^-πi