Polar coordinates

Polar coordinates, conversions, polar equations and polar curves. 4 problems with step-by-step solutions.

Solve each problem in your notebook first, then open the solution.

Problem 1 Polar coordinates

a) Plot the points \(A\left(2, \frac{\pi}{4}\right)\), \(B(3, 240^\circ)\), \(C(4, -120^\circ)\), \(D(1, 450^\circ)\).

b) Give three other polar coordinates for \(P(4, 150^\circ)\) with \(-360^\circ \le \theta \le 360^\circ\).

c) Find the symmetric point of \(P\left(3, \frac{\pi}{6}\right)\) with respect to the polar axis and to the pole.

Show solution

a) \(D(1, 450^\circ) = D(1, 90^\circ)\); \(C(4, -120^\circ) = C(4, 240^\circ)\), the same direction as \(B\) but further out.

b) \((4, -210^\circ)\), \((-4, 330^\circ)\), \((-4, -30^\circ)\).

c) Polar axis: \(\left(3, -\frac{\pi}{6}\right)\). Pole: \(\left(-3, \frac{\pi}{6}\right)\), i.e. \(\left(3, \frac{7\pi}{6}\right)\).

Problem 2 Rectangular and polar

a) Convert to polar coordinates: \((3, 3\sqrt3)\), \((-2, -2)\), \((0, -5)\), \((-\sqrt3, 1)\).

b) Convert to rectangular coordinates: \((6, \pi)\), \(\left(2, \frac{\pi}{3}\right)\), \((10, -45^\circ)\), \(\left(-4, \frac{5\pi}{6}\right)\).

Show solution

\(r = \sqrt{x^2 + y^2}\), \(\tan\theta = \frac{y}{x}\) (choose the right quadrant); \(x = r\cos\theta\), \(y = r\sin\theta\).

a) \(\left(6, \frac{\pi}{3}\right)\); \(\left(2\sqrt2, \frac{5\pi}{4}\right)\); \(\left(5, \frac{3\pi}{2}\right)\); \(\left(2, \frac{5\pi}{6}\right)\).

b) \((-6, 0)\); \((1, \sqrt3)\); \((5\sqrt2, -5\sqrt2)\); \((2\sqrt3, -2)\).

Problem 3 Polar equations

a) Write in rectangular form: \(r = 4\), \(r = 6\cos\theta\), \(\theta = \frac{\pi}{3}\), \(r = \dfrac{2}{\cos\theta - \sin\theta}\).

b) Write in polar form: \(x = -2\), \(y^2 = 4x\), \(x^2 + y^2 - 6y = 0\), \(2x + y = 5\).

Show solution

a) \(x^2 + y^2 = 16\); \(r^2 = 6r\cos\theta\): \(x^2 + y^2 = 6x\); \(y = \sqrt3 x\) (for \(x \ge 0\), or the whole line if \(r\) can be negative); \(r\cos\theta - r\sin\theta = 2\): \(x - y = 2\).

b) \(r\cos\theta = -2\); \(r^2\sin^2\theta = 4r\cos\theta\), i.e. \(r = \frac{4\cos\theta}{\sin^2\theta}\); \(r = 6\sin\theta\); \(r(2\cos\theta + \sin\theta) = 5\).

Problem 4 Polar curves

Sketch: a) \(r = 2 + 2\sin\theta\) b) \(r = 3 + \cos\theta\) c) \(r^2 = 4\cos 2\theta\).

Show solution

a) A cardioid: \(r = 4\) at \(\theta = \frac{\pi}{2}\), \(r = 2\) at \(0\) and \(\pi\), \(r = 0\) at \(\frac{3\pi}{2}\) (the cusp at the pole).

b) A limaçon without a loop: \(r\) changes between 2 (\(\theta = \pi\)) and 4 (\(\theta = 0\)).

c) A lemniscate: defined where \(\cos 2\theta \ge 0\), i.e. \(-\frac{\pi}{4} \le \theta \le \frac{\pi}{4}\) and \(\frac{3\pi}{4} \le \theta \le \frac{5\pi}{4}\); the largest \(r = 2\) at \(\theta = 0, \pi\). It looks like the figure eight lying on its side.