### Practice Test

Assume $\alpha $ is opposite side $a,\beta $ is opposite side $b,$ and $\gamma $ is opposite side $c.$ Solve the triangle, if possible, and round each answer to the nearest tenth, given $\beta =\mathrm{68\xb0},b=21,c=16.$

A pilot flies in a straight path for 2 hours. He then makes a course correction, heading 15° to the right of his original course, and flies 1 hour in the new direction. If he maintains a constant speed of 575 miles per hour, how far is he from his starting position?

Convert $\left(2,2\right)$ to polar coordinates, and then plot the point.

Convert the polar equation to a Cartesian equation: ${x}^{2}+{y}^{2}=5\mathrm{y.}$

Test the equation for symmetry: $r=-4\mathrm{sin}(2\theta ).$

Graph $r=3-5\text{sin}\phantom{\rule{0.3em}{0ex}}\theta .$

Write the complex number in polar form: $4+i\text{.}$

Convert the complex number from polar to rectangular form: $z=5\text{cis}\left(\frac{2\pi}{3}\right).$

Given ${z}_{1}=8\mathrm{cis}\left(\mathrm{36\xb0}\right)$ and ${z}_{2}=2\mathrm{cis}\left(\mathrm{15\xb0}\right),$ evaluate each expression.

${z}_{1}{z}_{2}$

${\left({z}_{2}\right)}^{3}$

Plot the complex number $\mathrm{-5}-i$ in the complex plane.

Eliminate the parameter $t$ to rewrite the following parametric equations as a Cartesian equation: $\{\begin{array}{l}x(t)=t+1\hfill \\ y(t)=2{t}^{2}\hfill \end{array}.$

Parameterize (write a parametric equation for) the following Cartesian equation by using $x\left(t\right)=a\mathrm{cos}\phantom{\rule{0.3em}{0ex}}t$ and $y(t)=b\mathrm{sin}\phantom{\rule{0.3em}{0ex}}t:$ $\frac{{x}^{2}}{36}+\frac{{y}^{2}}{100}=1.$

Graph the set of parametric equations and find the Cartesian equation: $\{\begin{array}{l}x(t)=-2\mathrm{sin}\phantom{\rule{0.3em}{0ex}}t\hfill \\ y(t)=5\mathrm{cos}\phantom{\rule{0.3em}{0ex}}t\hfill \end{array}.$

A ball is launched with an initial velocity of 95 feet per second at an angle of 52° to the horizontal. The ball is released at a height of 3.5 feet above the ground.

- Find the parametric equations to model the path of the ball.
- Where is the ball after 2 seconds?
- How long is the ball in the air?

For the following exercises, use the vectors ** u** =

**− 3**

*i***and**

*j***= 2**

*v***+ 3**

*i***.**

*j*Calculate** $u\cdot v.$ **

Given vector** $v$ **has an initial point ${P}_{1}=\left(2,2\right)$ and terminal point ${P}_{2}=\left(-1,0\right),$ write the vector** $v$ **in terms of** $i$ **and** $j.$ **On the graph, draw** $v,$ **and** $-v.$ **