91Ó°ÊÓ

To plot points in polar coordinates, we use a grid consisting of _______ centered at the pole and _________ emanating from the pole.

Let \(P\) be a point in the plane. (a) If \(P\) has polar coordinates \((r, \theta)\) then it has rectangular coordinates \((x, y)\) where \(x=\) _________ and \(y=\) _________. (b) If \(P\) has rectangular coordinates \((x, y)\) then it has polar coordinates \((r, \theta)\) where \(r^{2}=\) _________ and \(\tan \theta=\) _________.

Sketching a Curve by Eliminating the Parameter \(A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. Use arrows to indicate the direction of the curve as \(t\) increases. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=4 t^{2}, \quad y=8 t^{3}$$

In this section we stated that parametric equations contain more information than just the shape of a curve. Write a short paragraph explaining this statement. Use the following example and your answers to parts (a) and (b) below in your explanation. The position of a particle is given by the parametric equations $$x=\sin t \quad y=\cos t$$ where \(t\) represents time. We know that the shape of the path of the particle is a circle. (a) How long does it take the particle to go once around the circle? Find parametric equations if the particle moves twice as fast around the circle. (b) Does the particle travel clockwise or counterclockwise around the circle? Find parametric equations if the particle moves in the opposite direction around the circle.

The quadratic formula works whether the coefficients of the equation are real or complex. Solve the following equations using the quadratic formula and, if necessary, De Moivre's Theorem. $$z^{2}-2 i z-2=0$$