### Review Exercises

##### Non-right Triangles: Law of Sines

For the following exercises, assume $\alpha $ is opposite side $a,\beta $ is opposite side $b,$ and $\gamma $ is opposite side $c.$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$\alpha =\mathrm{43.1\xb0},a=184.2,b=242.8$

Find the area of the triangle.

##### Non-right Triangles: Law of Cosines

Solve the triangle, rounding to the nearest tenth, assuming $\alpha $ is opposite side $a,\beta $ is opposite side $b,$ and $\gamma $ s opposite side $c:\phantom{\rule{0.3em}{0ex}}a=4,\phantom{\rule{0.5em}{0ex}}b=6,c=8.$

Find the area of a triangle with sides of length 8.3, 6.6, and 9.1.

##### Polar Coordinates

Plot the point with polar coordinates $\left(3,\frac{\pi}{6}\right).$

Convert $\left(6,-\frac{3\pi}{4}\right)$ to rectangular coordinates.

Convert $\left(7,-2\right)$ to polar coordinates.

For the following exercises, convert the given Cartesian equation to a polar equation.

$x=-2$

${x}^{2}+{y}^{2}=-2y$

For the following exercises, convert the given polar equation to a Cartesian equation.

$r=\frac{-2}{4\mathrm{cos}\phantom{\rule{0.3em}{0ex}}\theta +\mathrm{sin}\phantom{\rule{0.3em}{0ex}}\theta}$

For the following exercises, convert to rectangular form and graph.

$r=5\mathrm{sec}\phantom{\rule{0.3em}{0ex}}\theta $

##### Polar Coordinates: Graphs

For the following exercises, test each equation for symmetry.

$r=7$

Sketch a graph of the polar equation $r=1-5\mathrm{sin}\phantom{\rule{0.3em}{0ex}}\theta .$ Label the axis intercepts.

Sketch a graph of the polar equation $r=5\mathrm{sin}\left(7\theta \right).$

##### Polar Form of Complex Numbers

For the following exercises, find the absolute value of each complex number.

$-2+6i$

Write the complex number in polar form.

$5+9i$

For the following exercises, convert the complex number from polar to rectangular form.

$z=5\mathrm{cis}\left(\frac{5\pi}{6}\right)$

For the following exercises, find the product ${z}_{1}{z}_{2}$ in polar form.

${z}_{1}=2\mathrm{cis}\left(\mathrm{89\xb0}\right)$

${z}_{2}=5\mathrm{cis}\left(\mathrm{23\xb0}\right)$

${z}_{1}=10\mathrm{cis}\left(\frac{\pi}{6}\right)$

${z}_{2}=6\mathrm{cis}\left(\frac{\pi}{3}\right)$

For the following exercises, find the quotient $\frac{{z}_{1}}{{z}_{2}}$ in polar form.

${z}_{1}=12\mathrm{cis}\left(\mathrm{55\xb0}\right)$

${z}_{2}=3\mathrm{cis}\left(\mathrm{18\xb0}\right)$

${z}_{1}=27\mathrm{cis}\left(\frac{5\pi}{3}\right)$

${z}_{2}=9\mathrm{cis}\left(\frac{\pi}{3}\right)$

For the following exercises, find the powers of each complex number in polar form.

Find ${z}^{4}$ when $z=2\mathrm{cis}\left(\mathrm{70\xb0}\right)$

For the following exercises, evaluate each root.

Evaluate the cube root of $z$ when $z=64\mathrm{cis}\left(\mathrm{210\xb0}\right).$

For the following exercises, plot the complex number in the complex plane.

$6-2i$

##### Parametric Equations

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

$\{\begin{array}{l}x\left(t\right)=3t-1\hfill \\ y\left(t\right)=\sqrt{t}\hfill \end{array}$

$\{\begin{array}{l}x(t)=-\mathrm{cos}\phantom{\rule{0.3em}{0ex}}t\hfill \\ y(t)=2{\mathrm{sin}}^{2}t\hfill \end{array}$

Parameterize (write a parametric equation for) each Cartesian equation by using $x\left(t\right)=a\mathrm{cos}\phantom{\rule{0.3em}{0ex}}t$ and $y(t)=b\mathrm{sin}\phantom{\rule{0.3em}{0ex}}t$ for $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{16}=1.$

Parameterize the line from $(-2,3)$ to $(4,7)$ so that the line is at $(-2,3)$ at $t=0$ and $(4,7)$ at $t=1.$

##### Parametric Equations: Graphs

For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.

$\{\begin{array}{l}x\left(t\right)=3{t}^{2}\hfill \\ y\left(t\right)=2t-1\hfill \end{array}$

$\{\begin{array}{l}x(t)={e}^{t}\hfill \\ y(t)=-2{e}^{5\phantom{\rule{0.3em}{0ex}}t}\hfill \end{array}$

$\{\begin{array}{l}x(t)=3\mathrm{cos}\phantom{\rule{0.3em}{0ex}}t\hfill \\ y(t)=2\mathrm{sin}\phantom{\rule{0.3em}{0ex}}t\hfill \end{array}$

A ball is launched with an initial velocity of 80 feet per second at an angle of 40° to the horizontal. The ball is released at a height of 4 feet above the ground.

- ⓐ Find the parametric equations to model the path of the ball.
- ⓑ Where is the ball after 3 seconds?
- ⓒ How long is the ball in the air?

##### Vectors

For the following exercises, determine whether the two vectors,** $u$ **and** $v,$ **are equal, where** $u$ **has an initial point ${P}_{1}$ and a terminal point ${P}_{2},$ and** $v$ **has an initial point ${P}_{3}$ and a terminal point ${P}_{4}.$

${P}_{1}=\left(-1,4\right),{P}_{2}=\left(3,1\right),{P}_{3}=\left(5,5\right)$ and ${P}_{4}=\left(9,2\right)$

${P}_{1}=\left(6,11\right),{P}_{2}=\left(-2,8\right),{P}_{3}=\left(0,-1\right)$ and ${P}_{4}=\left(-8,2\right)$

For the following exercises, use the vectors** $u=2i-j\text{,}v=4i-3j\text{,}$ **and** $w=-2i+5j$ **to evaluate the expression.

** u** −

*v*For the following exercises, find a unit vector in the same direction as the given vector.

** a** = 8

**− 6**

*i*

*j*For the following exercises, find the magnitude and direction of the vector.

$\langle 6,\mathrm{-2}\rangle $

For the following exercises, calculate** $u\cdot v\text{.}$ **

** u** = −2

**+**

*i***and**

*j***= 3**

*v***+ 7**

*i*

*j*Given ** v** $=\u3008\mathrm{-3},4\u3009$ draw

**, 2**

*v***, and $\frac{1}{2}$**

*v***.**

*v*Given initial point ${P}_{1}=\left(3,2\right)$ and terminal point ${P}_{2}=\left(-5,-1\right),$ write the vector** $v$ **in terms of** $i$ **and** $j.$ **Draw the points and the vector on the graph.