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Section 4.3 Polar Coordinates

Use polar coordinates to solve the following problems,

  1. Express the polar equation \(r=\cos 2\theta\) in rectangular coordinates.

  2. Sketch polar graphs of:

    1. \(r=1+\sin \theta\text{.}\)

    2. \(r=\cos 3\theta\text{.}\)

  3. For the each of the following circles find a polar equation, i.e. an equation in \(r\) and \(\theta\text{:}\)

    1. \(\displaystyle x^2+y^2=4\)

    2. \(\displaystyle (x-1)^2+y^2=1\)

    3. \(\displaystyle x^2+(y-0.5)^2=0.25\)

  4. Find the maximum height above the \(x\)-axis of the cardioid \(r=1+\cos \theta\text{.}\)

  5. Sketch the graph of the curve whose equation in polar coordinates is \(r=1-2\cos\theta\text{,}\) \(0\leq \theta \lt 2\pi\text{.}\)

  6. Sketch the graph of the curve whose equation in polar coordinates is \(r=3\cos 3\theta\text{.}\)

  7. Sketch the curve whose polar equation is \(r=-1+\cos \theta\text{,}\) indicating any symmetries. Mark on your sketch the polar coordinates of all points where the curve intersects the polar axis.

  8. Sketch a polar coordinate plot of:

    1. \(\displaystyle \ds r=\frac{1}{2}+\sin \theta\)

    2. \(\displaystyle r=2\cos 3\theta\)

    3. \(\displaystyle r^2=-4\sin 2\theta\)

    4. \(\displaystyle r=2\sin \theta\)

    5. \(\displaystyle r=2\cos \theta\)

    6. \(\displaystyle r=4+7\cos \theta\)

  9. Consider the curve given by the polar equation \(r=1-\cos \theta\text{,}\) for \(0\leq \theta \lt 2\pi\text{.}\)

    1. Given a point \(P\) on this curve with polar coordinates \((r,\theta)\text{,}\) represent its Cartesian coordinates \((x,y)\) in terms of \(\theta\text{.}\)

    2. Find the slope of the tangent line to the curve where \(\ds \theta = \frac{\pi }{2}\text{.}\)

    3. Find the points on this curve where the tangent line is horizontal or vertical.

  10. Consider the curve given by the polar equation \(r=\cos (2\theta)\text{,}\) for \(0\leq \theta \lt 2\pi\text{.}\)

    1. Find \(\ds \frac{dy}{dx}\) in terms of \(\theta\text{.}\)

    2. Find the Cartesian coordinates for the point on the curve corresponding to \(\ds \theta = \frac{\pi }{8}\text{.}\)

    3. Find the tangent line to the curve at the point corresponding to \(\ds \theta = \frac{\pi }{8}\text{.}\)

    4. Sketch this curve for \(\displaystyle 0\leq \theta \leq \frac{\pi}{4}\) and label the point from part (b) on your curve.

  11. Consider the curve given by the polar equation \(r=4\cos (3\theta)\text{,}\) for \(0\leq \theta \lt 2\pi\text{.}\)

    1. Find the Cartesian coordinates for the point on the curve corresponding to \(\ds \theta = \frac{\pi }{3}\text{.}\)

    2. One of graphs in Figure 4.3.1 is the graph of \(r=4\cos(3\theta)\text{.}\) Indicate which one by circling it.

    3. Find the slope of the tangent line to the curve where \(\ds \theta = \frac{\pi }{3}\text{.}\)

  12. Consider the curve given by the polar equation \(r=4\sin (3\theta)\text{,}\) for \(0\leq \theta \lt 2\pi\text{.}\)

    1. Find the Cartesian coordinates for the point on the curve corresponding to \(\ds \theta = \frac{\pi }{6}\text{.}\)

    2. One of graphs in Figure 4.3.1 is the graph of \(r=4\sin(3\theta)\text{.}\) Indicate which one by circling it.

    3. Find the slope of the tangent line to the curve where \(\ds \theta = \frac{\pi }{3}\text{.}\)

    Figure 4.3.1. Graph of \(r=4\cos (3\theta)\text{?}\) Graph of \(r=4\sin (3\theta)\text{?}\)

  13. Consider the curve given by the polar equation \(r=1+3\cos(2\theta)\text{,}\) for \(0\leq \theta \lt 2\pi\text{.}\)

    1. Find the Cartesian coordinates for the point on the curve corresponding to \(\ds \theta = \frac{\pi }{6}\text{.}\)

    2. One of graphs in Figure 4.3.2 is the graph of \(r=1+3\cos(2\theta)\text{.}\) Indicate which one by putting a checkmark in the box below the graph you chose.

    3. Find the slope of the tangent line to the curve where \(\ds \theta = \frac{\pi }{6}\text{.}\)

    Figure 4.3.2. Graph of \(r=1+3\cos (2\theta)\text{?}\)

  14. Consider the curve given by the polar equation \(r=1-2\sin \theta\text{,}\) for \(0\leq \theta \lt 2\pi\text{.}\)

    1. Find the Cartesian coordinates for the point on the curve corresponding to \(\ds \theta = \frac{3\pi }{2}\text{.}\)

    2. The curve intersects the \(x\)-axis at two points other than the pole. Find polar coordinates for these other points.

    3. On Figure 4.3.3 identify the graphs that correspond to the following two polar curves.

      \begin{equation*} \begin{array}{cc} \fbox { } \ r=1-2\sin \theta \amp \fbox { } \ r=1+2\sin \theta \end{array} \end{equation*}
      Figure 4.3.3. Which Graph Is Extra?

  15. Consider the curve \(C\) given by the polar equation \(r=1+2\cos \theta\text{,}\) for \(0\leq \theta \lt 2\pi\text{.}\)

    1. Find the Cartesian coordinates for the point on the curve corresponding to \(\ds \theta = \frac{\pi }{3}\text{.}\)

    2. Find the slope of the tangent line where \(\ds \theta = \frac{\pi }{3}\text{.}\)

    3. On Figure 4.3.4 identify the graph of \(C\text{.}\)

    Figure 4.3.4. Graph of \(r=1+2\cos \theta\text{?}\)

    1. Sketch a polar coordinate plot of

      \begin{equation*} r=1+2\sin 3\theta, \ 0\leq \theta \leq 2\pi\text{.} \end{equation*}

    2. How many points lie in the intersection of the two polar graphs

      \begin{equation*} r=1+2\sin 3\theta, \ 0\leq \theta \leq 2\pi \end{equation*}

      and

      \begin{equation*} r=1? \end{equation*}

    3. Algebraically find all values of \(\theta\) that

      \begin{equation*} 1=1+2\sin 3\theta, \ 0\leq \theta \leq 2\pi\text{.} \end{equation*}

    4. Explain in a sentence or two why the answer to part (b) differs from (or is the same as) the number of solutions you found in part (c).

  17. Consider the following curve \(C\) given in polar coordinates as

    \begin{equation*} r(\theta )=1+\sin \theta +e^{\sin \theta }, \ 0\leq \theta \leq 2\pi\text{.} \end{equation*}

    1. Calculate the value of \(r(\theta )\) for \(\ds \theta =0, \frac{\pi }{2}, \frac{3\pi }{2}\text{.}\)

    2. Sketch a graph of \(C\text{.}\)

    3. What is the minimum distance from a point on the curve \(C\) to the origin? (i.e. determine the minimum of \(|r(\theta )|=r(\theta )=1+\sin \theta +e^{\sin \theta }\) for \(\theta \in [0,2\pi ]\))

    1. Give polar coordinates for each of the points \(A\text{,}\) \(B\text{,}\) \(C\) and \(D\) on Figure 4.3.5.

      Figure 4.3.5. Polar coordinates of \(A, B, C\) and \(D\)

    2. On Figure 4.3.6 identify the graphs that correspond to the following three polar curves.

      \begin{equation*} \fbox { } \ r=1-2\sin \theta \ \ \fbox { } \ r^2\theta =1 \ \ \fbox { } \ r=\frac{1}{1-2\sin \theta} \end{equation*}
      Figure 4.3.6. Match Polar Expressions And Graphs

    1. Sketch the curve defined by \(r=1+2\sin \theta\text{.}\)

    2. For what values of \(\theta\text{,}\) \(\theta \in [-\pi ,\pi )\text{,}\) is the radius \(r\) positive?

    3. For what values of \(\theta\text{,}\) \(\theta \in [-\pi ,\pi )\text{,}\) is the radius \(r\) maximum and for what values is it minimum?

    1. Sketch the graph described in polar coordinates by the equation \(r=\theta\) where \(-\pi \leq \theta \leq 3\pi\text{.}\)

    2. Find the slope of this curve when \(\ds \theta =\frac{5\pi }{2}\text{.}\) Simplify your answer for full credit.

    3. Express the polar equation \(r=\theta\) in cartesian coordinates, as an equation in \(x\) and \(y\text{.}\)

    1. Let \(C\) denote the graph of the polar equation \(r=5\sin \theta\text{.}\) Find the rectangular coordinates of the point on \(C\) corresponding to \(\ds \theta =\frac{3\pi }{2}\text{.}\)

    2. Write a rectangular equation (i.e. using the variables \(x\) and \(y\)) for \(C\text{.}\) (in other words, convert the equation for \(C\) into rectangular coordinates.)

    3. Rewrite the equation of \(C\) in parametric form, i.e. express both \(x\) and \(y\) as functions of \(\theta\text{.}\)

    4. Find an expression for \(\ds \frac{dy}{dx}\) in terms of \(\theta\text{.}\)

    5. Find the equation of the tangent line to \(C\) at the point corresponding to \(\ds \theta =\frac{\pi }{6}\text{.}\)

  22. Find the slope of the tangent line to the polar curve \(r=2\) at the points where it intersects the polar curve \(r=4\cos \theta\text{.}\) (Hint: After you find the intersection points, convert one of the curves to a pair of parametric equations with \(\theta\) as the perimeter.

  23. A bee goes out from its hive in a spiral path given in polar coordinates by \(r=be^{kt}\) and \(\theta =ct\text{,}\) where \(b\text{,}\) \(k\text{,}\) and \(c\) are positive constants. Show that the angle between the bee's velocity and acceleration remains constant as the bee moves outward.