# 8.8 Vectors  (Page 10/22)

 Page 10 / 22

Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it 30° from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.

5.1583 pounds, 75.8° from the horizontal

## Non-right Triangles: Law of Sines

For the following exercises, assume $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}a,\beta \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\gamma \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$\beta =50°,a=105,b=45$

Not possible

$\alpha =43.1°,a=184.2,b=242.8$

Solve the triangle.

$C=120°,a=23.1,c=34.1$

Find the area of the triangle.

A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 2.1 km apart, to be 25° and 49°, as shown in [link] . Find the distance of the plane from point $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and the elevation of the plane.

distance of the plane from point $\text{\hspace{0.17em}}A:\text{\hspace{0.17em}}$ 2.2 km, elevation of the plane: 1.6 km

## Non-right Triangles: Law of Cosines

Solve the triangle, rounding to the nearest tenth, assuming $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}a,\beta \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\gamma \text{\hspace{0.17em}}$ s opposite side

Solve the triangle in [link] , rounding to the nearest tenth.

$B=71.0°,C=55.0°,a=12.8$

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

To find the distance between two cities, a satellite calculates the distances and angle shown in [link] (not to scale). Find the distance between the cities. Round answers to the nearest tenth.

40.6 km

## Polar Coordinates

Plot the point with polar coordinates $\text{\hspace{0.17em}}\left(3,\frac{\pi }{6}\right).$

Plot the point with polar coordinates $\text{\hspace{0.17em}}\left(5,-\frac{2\pi }{3}\right)$

Convert $\text{\hspace{0.17em}}\left(6,-\frac{3\pi }{4}\right)\text{\hspace{0.17em}}$ to rectangular coordinates.

Convert $\text{\hspace{0.17em}}\left(-2,\frac{3\pi }{2}\right)\text{\hspace{0.17em}}$ to rectangular coordinates.

$\text{\hspace{0.17em}}\left(0,2\right)\text{\hspace{0.17em}}$

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

Convert $\left(-9,-4\right)$ to polar coordinates.

$\left(9.8489,203.96°\right)$

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

$x=-2$

${x}^{2}+{y}^{2}=64$

$r=8$

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

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

$r=7\text{cos}\text{\hspace{0.17em}}\theta$

${x}^{2}+{y}^{2}=7x$

$r=\frac{-2}{4\mathrm{cos}\text{\hspace{0.17em}}\theta +\mathrm{sin}\text{\hspace{0.17em}}\theta }$

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

$\theta =\frac{3\pi }{4}$

$y=-x$

$r=5\mathrm{sec}\text{\hspace{0.17em}}\theta$

## Polar Coordinates: Graphs

For the following exercises, test each equation for symmetry.

$r=4+4\mathrm{sin}\text{\hspace{0.17em}}\theta$

symmetric with respect to the line $\theta =\frac{\pi }{2}$

$r=7$

Sketch a graph of the polar equation $\text{\hspace{0.17em}}r=1-5\mathrm{sin}\text{\hspace{0.17em}}\theta .\text{\hspace{0.17em}}$ Label the axis intercepts.

Sketch a graph of the polar equation $\text{\hspace{0.17em}}r=5\mathrm{sin}\left(7\theta \right).$

Sketch a graph of the polar equation $\text{\hspace{0.17em}}r=3-3\mathrm{cos}\text{\hspace{0.17em}}\theta$

## Polar Form of Complex Numbers

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

$-2+6i$

$4-\text{​}3i$

5

Write the complex number in polar form.

$5+9i$

$\frac{1}{2}-\frac{\sqrt{3}}{2}\text{​}i$

$\mathrm{cis}\left(-\frac{\pi }{3}\right)$

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

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

$z=3\mathrm{cis}\left(40°\right)$

$2.3+1.9i$

For the following exercises, find the product $\text{\hspace{0.17em}}{z}_{1}{z}_{2}\text{\hspace{0.17em}}$ in polar form.

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

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

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

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

$60\mathrm{cis}\left(\frac{\pi }{2}\right)$

For the following exercises, find the quotient $\text{\hspace{0.17em}}\frac{{z}_{1}}{{z}_{2}}\text{\hspace{0.17em}}$ in polar form.

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

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

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

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

$3\mathrm{cis}\left(\frac{4\pi }{3}\right)$

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

Find $\text{\hspace{0.17em}}{z}^{4}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=2\mathrm{cis}\left(70°\right)$

Find $\text{\hspace{0.17em}}{z}^{2}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=5\mathrm{cis}\left(\frac{3\pi }{4}\right)$

$25\mathrm{cis}\left(\frac{3\pi }{2}\right)$

#### Questions & Answers

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich