9. Polar Equations

Plot the point on the polar coordinate system.

$(5,210°)$

125

Plot the point on the polar coordinate system.

$(-3,-90°)$

218

Plot the point on the polar coordinate system.

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

166

Plot the point on the polar coordinate system.

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

176

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) $r≥0,-2π≤θ≤0$ ,

136

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

123

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

111

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°)

273

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)

185

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)

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, −135°)

203

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

289

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)

259

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.

230

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

231

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

193

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)

229

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

182

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

229

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)

202

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

191

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)

215

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

195

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°)

181

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

183

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

200

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

185

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

197

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

238

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

304

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

226

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

376

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

192

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

222

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

272

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

275

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

199

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

232

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)

205

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

209

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

183

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

279

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

268

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

215

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

201

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

268

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

193

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

246

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

277

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

231

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

293

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

236

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

196

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

345

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

284

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

191

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)

208

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

228

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

268

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

223

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)

190

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

175