Polar Geometry

  1. What are the rectangular coordinates \((x,y)\) of the point \((r,\theta) = \left(3,\frac{\pi}{6}\right)\) expressed in polar coordinates?
  2. What are a pair of polar coordinates \((r,\theta)\) of the point \((x,y) = \left(-3,7\right)\) expressed in rectangular coordinates?
  3. For each of the following functions \(f,\) manually sketch the graph \(r = f(\theta)\) for \(\theta\) ranging from \(0\) to \(2\pi\) in polar coordinates. Imagine what the graph would look like if \(\theta\) were extended beyond \(2\pi\) to \(\infty.\) Imagine what the graph would look like if \(\theta\) were extended back past \(0\) negative values. Eventually use technology to plot these graphs to compare with your sketch.
    \[ f(t) = 3 \]
    \[ f(t) = t \]
    \[ f(t) = t+3 \]
    \[ f(t) = t-3 \]
    \[ f(t) = t^3 \]
    \[ f(t) = \cos(t) \]
    \[ f(t) = \cos(2t) \]
    \[ f(t) = 2\cos(t) \]
    \[ f(t) = 2\big(1+\cos(t)\big) \]
    \[ f(t) = \ln(t) \]
    \[ f(t) = 4+2\sec(t) \\ {\color{Dimgrey}\footnotesize\text{conchoid}} \]
    \[ f(t) = \sin(t)\tan(t) \\ {\color{Dimgrey}\footnotesize\text{cissoid of Diocles}} \]
    \[ f(t) = 1+2\sin\left(t/2\right) \\ {\color{Dimgrey}\footnotesize\text{nephroid of Freeth}} \]
    \[ f(t) = \sqrt{1 - (4/5)\sin^2(t)} \\ {\color{Dimgrey}\footnotesize\text{hippopede}} \]
  4. Notice that the polar curve \(r = \cos(2\theta)\) has four “petals.” What’s a formula for a polar curve that has five petals?
  5. Consider the straight line \(y = 7\) in rectangular coordinates. How do you express this line as the graph of a function \(r = f(\theta)\) in polar coordinates?