Problem 86
True or False? Justify your answer with a proof or a counterexample. The rectangular coordinates of the point \(\left(4, \frac{5 \pi}{6}\right)\) are \((2 \sqrt{3},-2)\).
Problem 87
True or False? Justify your answer with a proof or a counterexample. The equations \(x=\cosh (3 t), y=2 \sinh (3 t)\) represent a hyperbola.
Problem 88
True or False? Justify your answer with a proof or a counterexample. The arc length of the spiral given by \(r=\frac{\theta}{2}\) for \(0 \leq \theta \leq 3 \pi\) is \(\frac{9}{4} \pi^{3}\).
Problem 90
For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. $$ x=1+t, y=t^{2}-1,-1 \leq t \leq 1 $$
Problem 91
For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. $$ x=e^{t}, y=1-e^{3 t}, 0 \leq t \leq 1 $$
Problem 92
For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. $$ x=\sin \theta, y=1-\csc \theta, 0 \leq \theta \leq 2 \pi $$
Problem 93
For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. $$ x=4 \cos \phi, y=1-\sin \phi, 0 \leq \phi \leq 2 \pi $$
Problem 94
For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any. $$ r=4 \sin \left(\frac{\theta}{3}\right) $$
Problem 95
For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any. $$ r=5 \cos (5 \theta) $$
Problem 96
For the following exercises, find the polar equation for the curve given as a Cartesian equation. $$ x+y=5 $$
