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%  Polar Coordinates Review                 %
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\problemgrouptitle{Polar Coordinates Review}

\item Find Cartesian coordinates of the point $\displaystyle(r,\theta)=\left(7,\frac{5\pi}{3}\right)$\,.
\answer{
$(x,y)=\left(\frac{7}{2},-\frac{7\sqrt{3}}{2}\right)$
}

\item Find polar coordinates of the point $\displaystyle(x,y)=\left(0,\pi\right)$\,.
\answer{
$(r,\theta)=\left(\pi,\frac{\pi}{2}\right)$
}

\begin{problemblock}[2]{Find a Cartesian equation for the given curve and identify it.}


\item $r=\cos\theta-6\sin\theta$

\answer{
circle: $\displaystyle\left(x-\frac{1}{2}\right)^2+\left(y-3\right)^2=\frac{37}{4}$
}

\item $r^2 \left( \cos^2\!\theta-3\sin^2\!\theta \right)=16$
\answer{
hyperbola: $x^2-3y^2=16$
}

\end{problemblock}

\begin{problemblock}[2]{Find a polar equation that has the same graph as the given Cartesian equation.}

\item $xy+y^2=8$
\answer{
$\displaystyle r^2\left[\sin(2\theta)+2\sin^2\!\theta \right]=16$
}

\item $(x-1)^2+y^2=4$
\answer{
$r^2-2r\cos\theta=3$
}

\end{problemblock}

\begin{problemblock}[2]{Find the slope of the tangent line to the given
curve at the indicated value of $\theta$.}

\item $r=3\cos^2\!\theta;\,\,\,\, \theta=\frac{\pi}{4}$
\answer{
$\displaystyle  -\frac{1}{3}$
}

\item $r=3\cos\theta+\sin\theta-2;\,\,\,\, \theta=\frac{\pi}{2}$
\answer{
$-3$
}

\end{problemblock}

\begin{problemblock}[2]{Find the points on the curve at which the tangent line is horizontal or vertical.}

\item $\displaystyle r=e^\theta$

\answer{
Horizontal if $\displaystyle \theta=\frac{3\pi}{4}+2n\pi,\,\,\, \theta=\frac{7\pi}{4}+2n\pi$ \\ Vertical if $\displaystyle \theta=\frac{\pi}{4}+2n\pi,\,\,\,\, \theta=\frac{5\pi}{4}+2n\pi$
}

\item $r=1+\sin\theta$
\answer{
Horizontal if $\displaystyle \theta=\frac{\pi}{2}+2n\pi, \,\,\, \theta=\frac{7\pi}{6}+2n\pi,\,\,\, \theta=\frac{11\pi}{6}+2n\pi$ \\  Vertical if $\displaystyle \theta=\frac {3\pi}{2}+2n\pi,\,\,\, \theta=\frac{\pi}{6}+2n\pi,\,\,\, \theta=\frac{5\pi}{6}+2n\pi$
}

\end{problemblock}

\begin{problemblock}[2]{Sketch the graph of the equation and find the area of the region bounded by the graph.}

\item $r=\sin\theta-2 \cos \theta$  
\answer{
$\displaystyle A=\frac {5\pi}{4}$
}

\item $r=3 \cos \theta-4$
\answer{
$\displaystyle A=\frac{41\pi}{2}$
}

\end{problemblock}

\begin{problemblock}[2]{Find the area of the region that lies inside both curves.}

\item $r=4\sin\theta,\,\,\, r=4\cos \theta$
\answer{
$A=2\pi-4$
}

\item $r=3\sin\theta+4,\,\,\, r=3\cos\theta+4$
\answer{
$\displaystyle A=\frac{41\pi}{2}-24\sqrt{2}$
}

\end{problemblock}

\begin{problemblock}[2]{Find the length of the given polar curve.}

\item $\displaystyle r=\sin^2\!\left(\frac{\theta}{2}\right)$
\answer{
$L=4$
}

\item $r= \theta^4;\,\,\,0\le\theta\le2\pi$
\answer{
$\displaystyle L=\frac{32}{5}\left(\pi^2+4\right)^{5/2}-\frac{128}{3}\left(\pi^2+4\right)^{3/2}+\frac{2048}{15}$
}

\end{problemblock}

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