# 10.8 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 $\text{\hspace{0.17em}}z=5\mathrm{cis}\left(\frac{3\pi }{4}\right)$

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

sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
cosA\1+sinA=secA-tanA
Wrong question
why two x + seven is equal to nineteen.
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
simplify each radical by removing as many factors as possible (a) √75
how is infinity bidder from undefined?
what is the value of x in 4x-2+3
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb