9. Polar Equations

Plot the point on the polar coordinate system.

$(5,210°)$

123

Plot the point on the polar coordinate system.

$(-3,-90°)$

214

Plot the point on the polar coordinate system.

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

158

Plot the point on the polar coordinate system.

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

167

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) $r≥0,-2π≤θ≤0$ ,

134

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

117

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

108

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°)

268

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)

183

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)

184

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°)

199

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

279

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)

255

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°)

248

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

226

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°)

231

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

226

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

192

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)

225

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

187

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)

182

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

224

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)

200

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)

210

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

191

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

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

177

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

181

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

198

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

183

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

192

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

234

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

290

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

220

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

372

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

216

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

268

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

272

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

196

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

231

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)

203

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

196

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

180

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

274

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

264

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

212

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

197

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

262

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

190

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

239

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

272

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

229

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

290

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

231

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

191

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

339

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

281

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)

204

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

224

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

262

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

220

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

173