Polar Coordinates

(requires JavaScript)

1. Sketch the region in the plane consisting of points whose polar coordinates satisfy $-1\le r\le 1$ and ${\displaystyle \frac{\pi }{4}\le \theta \le \frac{3\pi }{4}}$.

2. Find a polar equation for the curve represented by the Cartesian equation

   $x^2+y^2=2cx$

   $r=2c\cos <!--FUNCTION\; APPLICATION-->\left(\theta \right)$
3. Sketch the curve given by $r=\sin <!--FUNCTION\; APPLICATION-->\left(\theta \right)$.
   A circle through the origin and ${\displaystyle \left(1,\frac{\pi }{2}\right)}$.
4. Sketch the curve given by $r=2\cos <!--FUNCTION\; APPLICATION-->\left(4\theta \right)$.

5. Match the polar equations with the graphs without using a computer. Explain.

   1. ${\displaystyle r=\sin <!--FUNCTION\; APPLICATION-->\left(\frac{\theta }{2}\right)}$
   2. ${\displaystyle r=\sin <!--FUNCTION\; APPLICATION-->\left(\frac{\theta }{4}\right)}$
   3. ${\displaystyle r=\frac{1}{\cos <!--FUNCTION\; APPLICATION-->\left(3\theta \right)}}$
   4. $r=\theta \sin <!--FUNCTION\; APPLICATION-->\left(\theta \right)$
   5. $r=1+4\cos <!--FUNCTION\; APPLICATION-->\left(5\theta \right)$
   6. ${\displaystyle r=\frac{1}{\sqrt{\theta }}}$

   (a) (b)

   (c) (d)

   (e) (f)