Reflection #24

alright for this week we learned all about chapter 10 and learned how to use formulas to help us condense, expand, and solve problems with trig functions.

formulas:Cos(α +/- β)=cos α cos β -/+ sin α sin βsin(α +/- β)=sin α cos β -/+ cos α sin βsin x + sin y= 2 sin x + y/2 cos x-y/2sin x - sin y= 2 cos x + y/2 sin x-y/2cos x + cos y= 2 cos x + y/2 cos x-y/2cos x - cos y= 2 sin x + y/2 sin x-y/2

tan (α + β)=tan α + tan β/1-tan α tan βtan (α - β)=tan α - tan β/1+tan α tan β

sin2α=2sin α cos αcos 2α=cos^2 α –sin^2 α = 1-2 sin^2 α= 2 cos^2 α -1tan 2α = 2tan α /1-tan^2 αsin α/2= +/- √1-cos α/2cos α/2= +/- √1+ cos α/2tan α/2= +/- √1-cos α or 1 + cos α=sin α/1+cos α=1-cos α/sin α

something that i understood the most was section 2 here's some examples:
tan α = 2 and tan β=1
find tan (α - β)
= tan α + tan β/1-tan α tan β
=2+1/1-(2)(6)
=3/-1
=-3

Find the exact value of: tan 15+tan 30/1-tan 15 tan 30
tan α = 2 and tan β=1
find tan (α - β)
= tan (15 + 30)
=tan (45)
=1

for something that i didn't understand from this chapter was for example was in section 4 when you use the formulas as identities. i don't understand how you use the unit circle or the four quadrants to determine the degrees like in the homework on page 389 numbers 12 & 13.

relfection

welll since i barely attended school this week and no one wants to tutor me on the weekends i dont really know much except for whats obvious in the notes..

limacon...r=a+b sin(theta) OR r=a+b cos(theta)
rose...r=a sin(n theta) OR r=a cos (n theta)
**n=how many petals
Archimedes spiral...r=a theta+b
logarithmic spiral...r=a b^theta
circle with its center at the pole...r=c
circle that intersects with the pole...r=a sin(theta) OR r=a cos(theta)

examples:
1) r=theta+2
2) r=2+3cos(theta)
3) r=5
4) r=3sin(4 theta)
5) r=1/2(3^theta)
6) r=2sin(theta)

1) archimedes spiral
2) limacon
3) circle with its center at the pole
4) rose with 4 petals
5) logarithmic spiral
6) circle that intersects with the pole

reflection 23

so, this week was deff not my favorite week in class, this stuff is really confusing
but i do understand it somewhat

ok, i get the stuff from the beginning of the week,
you know, stuff like:

give the polar coordinates for (3,4)

r=±√(3^2+4^2)=±5
Ø=tan^(-1)(4/3)=53.130°

l
l √ 53.130°
...............l................
l
√ l
233.130° l

so the answer's gonna be (5,53.130°) (-5,133.130°)

u kno, the simple stuff

and then the other way around:

give the rectangular coordinates for (3,30°)

x=rcosØ y=rsinØ
x=3cos30° y=3sin30°
x=3(√(3)/2) y=3(1/2)

so the answer’s gonna be (((3√(3))/2)),3/2)

and i also get the conversions and stuff, like:

x^2+y^2=1 convert to polar

(rcosØ)^2+(rsinØ)^2=1
r^2cos^2Ø+r^2sin^2Ø=1
r^2(cos^2Ø+sin^2Ø)=1
r^2=1
r=±1
r=1 or r=-1

i understand all of that, even though it took me a while to get it right, lol

but the new stuff is really confusing,
like the stuff we had to do for homework this weekend, i had a lot of trouble with it, and i highly doubt that i did it right :(

and one more thing

WHO DAT!

REFLECTION

Rectangular and Polar.

To convert from rectangular to polar, Plug x=rcos(theta) and y=rsin(theta) into the equations, and solve for r.

To convert from polar to rectangular, Use identites to get rid of the number in front of theta. Plug in y/r for sin(theta) and x/r for cos(theta). Plug in square root of x^2 + y^2 for r. Solve for y if possible.

and here are the formulas for the different shapes on the polar graph.

Limacon-r=a+b sin/cos(theta)
Rose-r=a sin/cos(N theta)
Archimedes spiral-r=a theta+b
Logarithmic spiral-r=a b^theta
Circle with its center at the pole-r=c
Circle that intersects with the pole-r=a sin/cos(theta)
Cardioid- same as limacon except a=b

and remember that if b>a in a limacon it can be minus instead of plus.
Can someone please verify that for me because im not sure if thats correct?

Reflection #23

okay, this week was really easy for me! I love chapter 11 because i actually understand what is going on. All of the other chapters, i was totally lost in what she was teaching, but something "clicked" in this chapter.
____________________________________________________________________

*polar is derived from the unit circle
*polar is in the form (r,theta) or r=__theta
*there are directional differences with r
*cos(theta)=x/r
*sin(theta)=y/r

**to convert from polar to rectangular>>> x=rcos(theta)
**to convert from rectangular to polar>>> y=rsin(theta)

***to convert from rectangular to polar>>> r= +- square root of x^2 + y^2
theta= tan^-1 (y/x)

____________________________________________________________________

EXAMPLE:

Give the polar coordinates for (3,4).

r= +- square root 3^2 + 4^2
r= +- 5

theta= tan^-1(4/3)
theta= 53.130degrees

*which quadrant is (3,4) in? <<<< I
THEREFOR---> the angle in quadrant I goes with the positive 5.

(5,53.130degrees) and (-5,233.130degrees)

__________________________________________________________________

To convert equations:

1. To go from rectangular to polar, Plug x=rcos(theta) and y=rsin(theta) into the equations, and solve for r.

2. To go from polar to rectangular, Use identites to get rid of the number in front of theta. Plug in y/r for sin(theta) and x/r for cos(theta). Plug in square root of x^2 + y^2 for r. Solve for y if possible.

_________________________________________________________________

EXAMPLE:

x^2 + y^2 = 1 <---- convert to polar


(rcos(theta))^2 + (rsin(theta))^2 = 1
r^2cos^2(theta) + r^2sin^2(theta) = 1
r^2(cos^2(theta)) + r^2(sin^2(theta)) = 1
r^2=1
r= +- 1
_________________________________________________________________

I really understood everything so far in this chapter, i just messed up the quiz we just took though. I thought i knew it, but obviously i messed up some of the problems. So if anyone could help me do the problems, where she gives you an equation and you have to find out what graph it is. It would really help me out. THANKSSS:)

Chapter 11 reflection

WHO DATTTTTTTTTTTT!!!!!!!!!!!!!!!!!!!!!! What a game, if you missed that u should go hang yourself. Anyways, enough with football and more with blogging about math, but first let me state how greatful i am to have theses blogs because they keep my math grades at a positive note when i'm not doing to well. In chapter 11 we learned about polars. A polar is derived from the unit circle. Polar is in the form of (r,theta) or r=theta. There are directional differences with r. costheta=x/r, this is to convert from polar to rectangula sintheta=y/r, this is to convert from polar to rectangular. Every point in polar has two names and they are given in cordinates.

To Convert equation:

Tp go from rectangular to polar plug x=rcos and y=rsintheta into the equations solve for r. To go from polar to rectangular use identities to get rid of # in front of theta plug in y/r for sin theta and x/r for costheta plug in xsquare + ysquare all square rooted for pie. Solve for y if possiable.

here are some formulas:

reitheta=rcistheta
eipie=1cispie=1cospie+1sinpie=-1
dc moivre's theorem zn=rncistheta

Here are some examples of graphs and how you name them
=2/3 (2^theta)_Logarithmic Spiral
r=Theta+3_Archimedes Spiral
r=4+2cos(theta)_Limacon
r=4 sin(6 theta)_Rose_4 pedals
r=3_Circle_Center at the Pole
r=3 sin (theta)_Circle_Intercects Poles


And once again who dat!!!!!!!!!!!!

Reflection 24!

WHO DAT?! :)
mkay, so that was the most exciting game EVERRRR!

and on to the math...

so this week we learned an entirely new way to graph: Polar.

Polar is in the form (r,theta) or r=_theta.


TO CONVERT POLAR TO RECTANGULAR:
cos(theta)= x/r sin(theta)=y/r
x=r cos(theta) y=rsin(theta)

TO CONVERT RECTANGULAR TO POLAR:
r= +/- sqrt (x^2+ y^2
theta= tan^-1(y/x)

Example:
Give the polar coordinates for(3,4)
r=+/- sqrt (9+16) = +/- 5
theta=tan^-1(4/3)= 53.130, 233.130

QUADRANT?
I & III


(5, 53.130degrees)
(-5,233.130degrees)



I still have trouble converting equations, anyone want to explain?
thanks!

Reflection 1/24

In this chapter we are learning how to do polar graphs instead of regular ones. There are limacons, roses, archimedes spirals, logarithmic spirals, cardiods. You have to know the formula for each one.


limacon...r=a+b sin(theta) OR r= a+b cos(theta)
cardiod...which is the same as limacon except a and b or equal
rose...r= a sin(n theta) OR r= a cos(n theta) *n=how many petals
Archimedes spiral...r= a theta+b
logarithmic spiral...r= b^theta
circle with its center at the pole...r=c
circle that intersects with the pole...r= a sin(theta) OR r= a cos(theta)

2 sin(4theta)-would be a rose with 4 petals

3+3 Cos (theta)- would be a cardiod

2theta+4- would be an Archimedes spiral

2+5 sin (theta)- would be a limacon

__________________________________________________________________________

I need help with converting rectangular to polar and polar to rectangular

reflection # something, i lost count

Ok, first off, i'd like to say, WHO HAS TIME TO DO BLOGS WHEN THE SAINTS ARE GOING TO THE SUPER BOWL....me i guess.........................

Ok so, here are some formulas for finding the type of graphs:

Limacon_r=a+b sin(theta) OR r=a+b cos(theta)
Rose_r=a sin(N theta) OR r=a cos (N theta)
N stands for number of petals
Archimedes spiral_r=a theta+b
Logarithmic spiral_r=a b^theta
Circle with its center at the pole_r=c
Circle that intersects with the pole_r=a sin(theta) OR r=a cos(theta)

Examples:

r=2/3 (2^theta)_Logarithmic Spiral
r=Theta+3_Archimedes Spiral
r=4+2cos(theta)_Limacon
r=4 sin(6 theta)_Rose_4 pedals
r=3_Circle_Center at the Pole
r=3 sin (theta)_Circle_Intercects Poles

____________________________________________________

Now, i dont understand how to change into polar and to rectangular....

Help?? :)

thanks


WHO DAT?

COLTS vs. SAINTS = WIN WIN SITUATION!!!!!!!!!!!!!!

reflection

Well from what I know, chapter eleven is basically so far just knowing the formulas of graphs. I wasn't there friday so im not sure if we did anything new or not but this is all i know so far.

limacon...r=a+b sin(theta) OR r=a+b cos(theta)
rose...r=a sin(n theta) OR r=a cos (n theta)
**n=how many petals
Archimedes spiral...r=a theta+b
logarithmic spiral...r=a b^theta
circle with its center at the pole...r=c
circle that intersects with the pole...r=a sin(theta) OR r=a cos(theta)

examples:
1) r=theta+2
2) r=2+3cos(theta)
3) r=5
4) r=3sin(4 theta)
5) r=1/2(3^theta)
6) r=2sin(theta)

1) archimedes spiral
2) limacon
3) circle with its center at the pole
4) rose with 4 petals
5) logarithmic spiral
6) circle that intersects with the pole


I dont understand the homework. I tried to do it and it just wasn't happening. haha. help?

reflection 23

Yeaaaahhhh, the saints just won!!! This week wasn' t too hard..I actually understood a lot of it. I think the easiest is chapter 11-3 and 11-4, the stuff we did friday. Which was De Moivre's Theorem...

z^n=r^ncisntheta

EXAMPLES..

1) z=2cis20 Find z^2
z^2=2^2cis2(20)
z^2=4cis(40)

2) z=4cis15 Find z^4
z^4=4^4cis4(15)
z^4=256cis60


The stuff that I didn't really understand, was from the beginning of the week..in chapter 11-1. From the homework problems, I didn't get 13-21 odd and 25-34. But when we went over them in class, i realized that i had to type the whole equation into my calculator. But you have to make sure that your in radians and that you are in polar for the graphs to come up correctly.

Reflection 23

This week was pretty much packed full of information and new formulas and stuff from chapter 11. Even though we had a 4 day week, it still seemed to be a very slow and boring week, but at least it is over. As I said, we learned a lot of stuff in chapter 11, and I'm going to put up some formulas and examples of what we learned this week.

11-2 Imaginary Numbers are no longer "imaginary"

Rectangular form is defined as
a + bi

Polar form is defined as
z = r cos theta + r sin theta i
abbreviated z = r cis theta

Example: Express 2 cis 50degrees in rectangular form

2 cos 50 + 2 sin 50 i


-1-2i in polar form
radius = +- sqrt of ((-1)^2 + (-2)^2)) = +- sqrt of (5)
theta = tan^-1(-2/-1)
theta = tan^-1(1)

if you do this, tangent is positive in the first and third quadrants, so it comes out to be 63.435 and 243.435

Since the 63 is positive for cosine, we can put it with the positive sqrt of 5.
And the 243 is negative for cosine, we can put it with the negative sqrt of 5.

z= sqrt of 5 cis 63.435
z= sqrt of 5 cos 63.435 + sqrt of 5 sin 63.435 i
z= negative sqrt of 5 cis 243.435
z- negative sqrt of 5 cos 243.435 + negative sqrt of 5 sin 243.435 i

It is now in polar form.

Reflection #23

Okay, ch. 11 is really confusing. I'm so lost and it doesn't help that I wasn't there Thursday. Anyway, one thing I did understand was how to tell what formula went with what graph.

limacon...r=a+b sin(theta) OR r=a+b cos(theta)
rose...r=a sin(n theta) OR r=a cos (n theta)
**n=how many petals
Archimedes spiral...r=a theta+b
logarithmic spiral...r=a b^theta
circle with its center at the pole...r=c
circle that intersects with the pole...r=a sin(theta) OR r=a cos(theta)

examples:
1) r=theta+2
2) r=2+3cos(theta)
3) r=5
4) r=3sin(4 theta)
5) r=1/2(3^theta)
6) r=2sin(theta)

1) archimedes spiral
2) limacon
3) circle with its center at the pole
4) rose with 4 petals
5) logarithmic spiral
6) circle that intersects with the pole


Now, what I don't understand...everything we did Thursday. Like I said, I wasn't there and I tried to do the homework, and I'm so lost. If anyone could explain, that be great. Maybe even an example from the homework on page 406. ^^

reflection 24

this week we started chapter 11 which was all about polar...

*polar is derived from the unit circle
*it's in the form (r,theta) or r = ___theta
*there are directional differences with r

cos theta = x/r sin theta = y/r

(converted to rectangular) x = rcos theta y = rsin theta
r = +/- sqrt. x^2 + y^2 theta = tan^-1 (y/x)

to go from rectangular to polar, plug x - rcos theta and y = rsin theta into the equations. solve for r

to go from polar to rectangular, use identities to get rid of # in front of theta. plug in y/r for sin theta and x/r for cos theta. plug in sqrt. x^2 + y^2 for r. solve for y if possible

reflection 24

This week was longer than normal, even though it was 4 days. we started a new chapter about polar and rectangular, and how to convert

cosA=x/r sinA=y/r

polar to rectangular
r=+/- {x^2+y^2 theta=tan^-1 (y/x)


rectangular to polar

(4,30*)
first, draw four on a number line, then place the point at the approximate angle

remember that every point in polar has 2 names

(4,30*)=(-4,210*)

give the polar coordinates for (3,4) Quandrant I

r=+/- {3^2 + 4^2 = +/-5 tan^-1(4/3)=53.13
(5,53.13)
(-5,233.13)

x^2 + y^2 = 1 convert to polar 0=theta

Rcos0^2+Rsin0^2=1
R^2cos^20+R^2sin^20
r^2(cos^20 +sin^20)=1
r^2=1
r=+/-1

REFLECTION #23

Okay this week went by really slow and I'm going to come right out and say that I don't like what we're learning in this chapter. I don't know why, but this stuff is annoying me a lot. And I'm barely comprehending it right now so I'm a little worried about that test on Tuesday. The only thing that I'm positive of how to do is identifying the polar graphs (the first part of the quiz we took on Friday). Soooooo..I'll attempt to explain how to do something but I probably won't understand what I'm doing myself. but i tried.

Okay so looking over my notes and whatnot, I think converting to polar is kinda easy so I'll explain that.
Ex. 1.) Convert (-12,5) to polar
*The first thing you have to do is plug -12 and 5 into this formula > r = +/- sqrt of x^2 + y^2
*So you get r = +/- sqrt of 144 + 25
*And simplifying that you get +/- 13 (*13 and -13 will be the x coordinate for your two polar coordinates)
*Then you use tangent inverse to find the y coordinates
*So you get x = tan^-1(-5/12)
*Your reference angle is 22.620 and tangent is negative in the 2nd and 4th quadrants.
*So your y coordinates are 157.380 degrees and 337.380 degrees.
*Your final answers are (13 , 157.380) (-13 , 337.380)
(*I don't know how to determine which y coordinate goes with which x coordinate. help?).

Here's an example of converting to rectangular. it's pretty easy actually.

Ex. 2.) Convert (2,45) to rectangular
*Okay for this you're going to use x = rcostheta and y = rsintheta
*Then you just plug in 2 for r in both equations, and plug in 45 for theta in both equations
*Then you simplify them
*So you get x = 2cos45 and y = 2sin45
*Then you get x = 2(sqrt of 2/2) and y= 2(sqrt of 2/2)
*And simplifying that you get square root of 2 and square root of 2
*Then you put them in point form for your final answer: (sqrt of 2, sqrt of 2)

*Alright well that's basically all I understand so far (which isn't really good at all). I have to say that not being at school on Thursday made me more confused than ever. I'm never missing this class again. hah. Well if anyone would like to help me out and explain a few things that we learned on Thursday that would be great (:

Reflection 24

So this week intimidated me with a new chapter but it wasn't all that hard, definately manageable...I think the easiest thing to do though was converting retangular points for polar:

Give the polar coordinates for (3,4)
first you have to use the formula r = square root of (x^2 + y^2)
so for this problem r = square root of (3^2 + 4^2) which gives you square root of (9 + 16) = (25)

so now you have your x = + or - 5

Now pllug your y/x into the inverse of tangent:
a = tan^-1(4/3) = 53.130 degrees. Now find where tangent is positive, quadrants I and III, so your other angle is going to be 233.130 because you ad 180 to 53.130 to move it to the third quadrant. Now you have to graph the two to figure out which goes with positive 5 and which goes with negative 5.

In this case, it's: (5 , 53.130 degrees) and (-5 , 233.130 degrees)

Also, changing from polar to rectangular is easy:

Give the rectangular coordinates for (3,30 degrees)
use your formulas x = rcosa and y = rsina
x = 3(square root of (3)/2) and y = 3(1/2)

solve and you end up with: (3 square root of 3/2, 3/2)

_________________________________________________________________
I still don't really grasp the whole imaginary numbers not be imaginary anymore, but real....

Reflection

This week we learned all kind of new stuff from chapter 11, but i find chapter 11 is pretty simple. I understood 11-3 and 11-4. We learned stuff about cardioids, limacons, rose, petals, archimedes spirals, and logmarithmisc spirals. The quiz on that was easy because its simple to understand the formulas with the graphs.

EXAMPLES FROM 11-4 AND 11-3

1)re^i(theta)=rcis(theta)
3e^i(2pi)=3cis2pi
=3cos2pi+3sin2pi(i)
=3(1)+3(0)i
=3

2)De Moivre's Thereom
z^n=r^ncis(n)(theta)
z=2cis20degrees Find z^2
z^2=2^2cis2(20degrees)
z^2=4cis40degrees

3)De Moivre's Thereom
4cis15degrees Find z^4
z^4=4^4cis4(15degrees)
z^4=256cis60degrees

4)Find z^1/4, z=16cis180degrees
z^1/4=16^1/4cis(180*/4+0X360*/4)= 2cis45*
z^1/4=16^1/4cis(180*/4+1X360*/4)= 2cis135*
z^1/4=16^1/4cis(180*/4+2X360*/4)= 2cis225*
z^1/4=16^1/4cis(180*/4+3X360*/4)= 2cis315*

5)Find z^1/2, z=9cis60*
z^1/2=9^1/2cis(60*/2+0X360*)= 3cis30*
z^1/2=9^1/2cis(60*/2+1X360*)= 3cis210*

I'm still having trouble with stuff from section 11-1, so if anyone wants help me, that would be great.