91Ó°ÊÓ

Plot the point having the given set of polar coordinates; then give two other sets of polar coordinates of the same point, one with the same value of \(r\) and one with an \(r\) having opposite sign.\((-3,-\pi)\)

Find the area of the intersection of the regions enclosed by the graphs of the two given equations.\(\left\\{\begin{array}{l}r=4 \sin \theta \\ r=4 \cos \theta\end{array}\right.\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}r=4(1+\sin \theta) \\ r(1-\sin \theta)=3\end{array}\right.\)

Draw a sketch of the graph of the given equation.\(r=e^{\theta}\) (logarithmic spiral)

Find a measurement of the angle between the tangent lines of the given pair of curves at all points of intersection.\(\left\\{\begin{array}{l}r=1-\sin \theta \\\ r=1+\sin \theta\end{array}\right.\)

Find the area of the intersection of the regions enclosed by the graphs of the two given equations.\(\left\\{\begin{array}{l}r=3 \sin 2 \theta \\ r=3 \cos 2 \theta\end{array}\right.\)

Find the rectangular cartesian coordinates of each of the following points whose polar coordinates are given: (a) \((3, \pi)\); (b) \(\left(\sqrt{2},-\frac{3}{4} \pi\right) ;\) (c) \(\left(-4, \frac{2}{3} \pi\right) ;\) (d) \(\left(-2,-\frac{1}{2} \pi\right) ;\) (e) \(\left(-2, \frac{7}{4} \pi\right) ;\) (f) \(\left(-1,-\frac{\tau}{6} \pi\right)\).

Find the area of the intersection of the regions enclosed by the graphs of the two given equations.\(\left\\{\begin{array}{c}r^{2}=2 \cos 2 \theta \\\ r=1\end{array}\right.\)

Find a set of polar coordinates for each of the following points whose rectangular cartesian coordinates are given. Take \(r>0\) and \(0 \leq \theta<2 \pi\). (a) \((1,-1) ;\) (b) \((-\sqrt{3}, 1) ;(\mathrm{c})(2,2) ;\) (d) \((-5,0) ;\) (e) \((0,-2) ;\) (f) \((-2,-2 \sqrt{3})\).

Find a measurement of the angle between the tangent lines of the given pair of curves at all points of intersection.\(\left\\{\begin{array}{l}r=3 \cos \theta \\\ r=1+\cos \theta\end{array}\right.\)