9. Polar Equations

Plot the point on the polar coordinate system.

$(5,210°)$

124

Plot the point on the polar coordinate system.

$(-3,-90°)$

216

Plot the point on the polar coordinate system.

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

161

Plot the point on the polar coordinate system.

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

169

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.

121

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

110

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°)

271

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)

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, 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°)

201

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

284

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)

256

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°)

250

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

227

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

229

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)

226

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)

182

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

227

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)

211

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

194

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°)

180

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

182

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)

184

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

193

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

237

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

292

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

222

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

374

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

220

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

269

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

198

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)

204

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

200

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

181

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

277

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

267

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

214

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

199

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

265

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

192

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

241

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

276

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

233

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

194

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

340

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

283

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)

206

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

226

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

267

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

222

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)

189

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

174