91Ó°ÊÓ

Sketch the curve with the polar equation. \(r=e^{\theta}, \quad \theta \geq 0 \quad\) (logarithmic spiral)

Sketch the curve with the polar equation. \(r=2 \sin 5 \theta \quad\) (five-leaved rose)

Use the parametric representation of a circle in Exercise 73 to show that the circumference of a circle of radius \(a\) is \(2 \pi a .\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The curve with parametric equations \(x=f(t)\) and \(y=g(t)\) is a line if and only if \(f\) and \(g\) are both linear functions of \(t\).

a. Plot the graphs of the cardioids \(r=a(1+\cos \theta)\) and \(r=a(1-\cos \theta)\). b. Show that the cardioids intersect at right angles except at the pole.

Let \(\psi\) be the angle between the radial line \(O P\) and the tangent line to the curve with polar equation \(r=f(\theta)\) at \(P\) (see the figure). Show that $$ \tan \psi=r \frac{d \theta}{d r} $$ Hint: Observe that \(\psi=\phi-\theta\). Then use the trigonometric identity $$ \tan (a-b)=\frac{\tan a-\tan b}{1+\tan a \tan b} $$

Use a graphing utility to plot the curve with the polar equation. \(r=\frac{\sin \theta}{\theta}, \quad-6 \pi \leq \theta<6 \pi \quad\) (cochleoid)

A transmitter \(B\) is located 200 miles due east of a transmit- ter \(A\) on a straight coastline. The two transmitters send out signals simultaneously to a ship that is located at \(P\). Suppose that the ship receives the signal from \(B, 800\) microseconds \((\mu\) sec \()\) before it receives the signal from \(A .\) a. Assuming that radio waves travel at a speed of \(980 \mathrm{ft} / \mu\) sec, find an equation of the hyperbola on which the ship lies (see page 838 ). Hint: \(d(P, A)-d(P, B)=2 a\). b. If the ship is sailing in a direction parallel to and \(20 \mathrm{mi}\) north of the coastline, locate the position of the ship at that instant of time.