91Ó°ÊÓ

In Exercises 1-12, write each expression as a complex number in standard form. If an expression simplifies to either a real number or a pure imaginary number, leave in that form. $$ \sqrt{-16} $$

In Exercises 1-20, graph the curve defined by the following sets of parametric equations. Be sure to indicate the direction of movement along the curve. $$ x=\tan t, y=1, t \text { in }-\frac{\pi}{4}, \frac{\pi}{4} $$

In Exercises 13-28, express each complex number in polar form. $$ 3+0 i $$

In Exercises 21-40, convert each point given in polar coordinates to exact rectangular coordinates. $$ \left(2, \frac{3 \pi}{4}\right) $$

In Exercises 21-40, convert each point given in polar coordinates to exact rectangular coordinates. $$ \left(0, \frac{11 \pi}{6}\right) $$

In Exercises 13-40, perform the indicated operation, simplify, and express in standard form. $$ (-3-2 i)(7-4 i) $$

In Exercises 29-44, use a calculator to express each complex number in polar form. Express Exercises 29-36 in degrees and Exercises 37-44 in radians. $$ -3+4 i $$

For Exercises 37-46, recall that the flight of a projectile can be modeled with the parametric equations $$ x=\left(v_{0} \cos \theta\right) t \quad y=-16 t^{2}+\left(v_{0} \sin \theta\right) t+h $$ where \(t\) is in seconds, \(v_{0}\) is the initial velocity in feet per second, \(\theta\) is the initial angle with the horizontal, and \(h\) is the initial height above ground, where \(x\) and \(y\) are in feet. Flight of a Projectile. A projectile is launched from the ground at a speed of 400 feet per second at an angle of \(45^{\circ}\) with the horizontal. How far does the projectile travel (what is the horizontal distance), and what is its maximum altitude? (Note the symmetry of the projectile path.)

In Exercises 21-40, convert each point given in polar coordinates to exact rectangular coordinates. $$ \left(5,270^{\circ}\right) $$

For Exercises \(41-48\), for each complex number \(z\), write the complex conjugate \(\bar{z}\), and find \(z \bar{z}\). $$ z=2-3 i $$