9. Polar Equations

Plot the point on the polar coordinate system.

$(5,210°)$

126

Plot the point on the polar coordinate system.

$(-3,-90°)$

223

Plot the point on the polar coordinate system.

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

168

Plot the point on the polar coordinate system.

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

179

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) $r≥0,-2π≤θ≤0$ ,

138

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

125

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

113

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°)

275

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)

186

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)

187

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°)

204

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

290

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)

261

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°)

252

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

231

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°)

233

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

232

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)

231

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

197

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)

183

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

231

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)

204

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

193

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)

217

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

196

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°)

183

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

184

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

202

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

187

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

199

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

240

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

310

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

227

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

378

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

225

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

277

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

276

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

201

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

238

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)

206

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

210

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

185

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

281

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

269

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

217

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

202

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

272

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

249

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

279

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

232

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

294

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

238

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

198

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

347

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

287

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

193

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)

209

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

229

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

269

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

224

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

176