Have a good last week guys

And for your final papers you have probably seen this already but it is, , ,inspiring?
http://www.comedycorner.org/17.html

extra cred

where's pg 251 in the article, i cant find the page numbers!!!!

supplementary angles

cos(t)=sin(t+90)
To solve, it looks like it is written as a weird version as cos(x+y) With x equaling -10, because cos(-10) =cos(10) and sin(190)=sin(-10)

Pretest Questions

I sort of dont know how tp simplify for #10. I am also not sure how to do #16 or 17. For #16, b and c, I dont know how to express the equation. Thanks

supplementary angles

Im having trouble figuring out problems with supplementary angles. I know that if you have cos^2(x) + sin^2(y) where x+y = 180 then cos^2(x) + sin^2(y) = 1. But what is sin(t) in terms of cosine and vice versa? What I mean is sin(t) = cos(?). I was looking back on a problem, how would I solve cos(10)sin(70) + sin(20)sin(190)?

amp, period, vertical shift, formula

f(t)= Asin(bt+c)+d
A= Amplitude
2pi/b= Period
-c/b= Phase Shift
d= vertical shift

number 6b on the pretest

would another set of polar coordinates for the same point just be 4cos(9pi/4 + 2pi) , 4sin(9pi/4+2pi)?

I missed class on friday...

and there are a few things about the 12R book problems and the semester review problems that confused me.

for 12R #s 36 and 43, I think we need to use induction, and I was wondering if Mr. Karafiol said anything about whether we'll need to know this for the final.

as for the semester review problems... in # 60, we are to find dx/dt, but I don't understand what the "d" means. It's definitely not the difference quotient because there is no f(x).

For #65 I completely spazzed on how to find the limit of the series sum if you know the limit of terms within it. I know we just did this stuff in class so I feel really lame, but just because we know that the largest term approaches 4, how do we know what the series will converge to if we don't know how many terms we're adding?

Finally, for the last problem (the second #66) I determined that the common ratio of the series is sec(t)/2 and I know that a geometric series will only reach a finite number if the ratio is between 0 and 1. So I set sec(t) equal to both 0 and 1, and I found a maximum value for t (5pi/3), but the one equal to 0 has no solution. So how do you find the minimum value for t?

thanks.

Angular and Linear Velocity

Can someone give a quick explanation of how you do these calculations. For example, number 6 on the calculator section. For angular velocity i said that it was 108000000*pi/224.7*24 but i am not sure if that's right. Also, how do you convert that to linear velocity?
Thanks

gsp portion

is question 18 on the pretest supposed to relate to the gsp portion of the final?

Properties of Cosines and Sines

Can someone give an explanation or list of the properties of cos and sin because I am having a hard time remembering them. Thanks!

Happy Birthday!

Happy Birthday Abraham de Moivre, I'm so sorry that I didn't celebrate your birthday until this year!

A few snippets from Wikipedia, sorry that I didn't write them myself:

Born May 26, 1667 in Vitry-le-François, Champagne, France, Died November 27, 1754 in London, England

de Moivre was a Calvinist. He left France after the revocation of the Edict of Nantes (1685) and spent the remainder of his life in England.

It is reported in all seriousness that De Moivre correctly predicted the day of his own death. Noting that he was sleeping 15 minutes longer each day, De Moivre surmised that he would die on the day he would sleep for 24 hours. A simple mathematical calculation quickly yielded the date, November 27, 1754. He did indeed pass away on that day. (Yet another factoid you can use on next year's precalc class, Mr. Karafiol, and I'm sorry if you said this to us and I forgot).

He first discovered the "closed form" expression for Fibonacci numbers linking the nth power of phi to the nth Fibonacci number.

http://en.wikipedia.org/wiki/Abraham_de_Moivre

In honor of his 340th birthday, I'm going to recap all the fun stuff you can do with de Moivre's theorem, I'm so sorry if this isn't enough:

First of all, de Moivre's theorem states that (in complex form): r(cos(θ)+isin(θ))ⁿ=rⁿ(cos(nθ)+isin(nθ))
In polar form, it becomes
(r,θ)ⁿ=(rⁿ,nθ)

But remember, it also works backwards to find roots of polar coordinates!
So,
*nrt(r,θ)=(nrt(r),((θ+2kπ)/n)
Remember the +2kπ !

*nrt=nth root, unfortunately I can't write it in pretty print, sorry!

I'm sorry if I forgot anything else you can do with de Moivre's theorem.

Oh, it's also National sorry day in Australia, sorry I didn't mention that at the beginning.

Vector Physics

Hey, does anyone remember how to to the comp of a vector and the proj of a vector onto another? And what exactly does "comp" tell you?

12.5 #19

I understand why it works, just don't understan how i would prove it.

Eccentricity

I forgot how to find the eccentricities of narrow vs. wide ellipses vs. circles. I know wide ellipses have an eccentricity close to 1, circles have an eccentricity of 0 (since focal length is 0), but then what are narrow ellipses? I seem to remember eccentricity must be greater than 0, so is it also between 0 and 1 for narrow ellipses?

Also, for number 8 on the pretest, I'm coming up with y=-15/2, since x^2+16=32 is the same as x^2=32(y-1/2), so 4p=32, and p=8, so the directrix of the parabola would be y=1/2-8=-15/2, which isn't an option. Where did I go wrong, or is the 16 a mistake on the test?

Problem on pretest

when do you use B^2 - 4AC is less than 0, greater than 0, or equal to 0, and what do A, B, and C stand for?

12.4.42 b

I think i am confused by the wording, but also for part a, don't you get (1^4, 4*theta) for an answer? This is not what I got for 41. Any approaches?

A Little About Parametrics for Conics

Ellipses

We know that the standard form for this conic section is (x^2)/a +(y^2)/b = 1

The parametric equation for an ellipse is x = a * cos (t)
y = b * sin (t)

Hyberbolas

We know that the standard form for this conic section is (x^2)/a - (y^2)/b = 1

The parametric equation for an hyperbola is

y = a * sec (t)

x = b * tan (t)

*the y values and x values will vary according to what axis the hyperbola is on

Parabola

The parametric equation for the parabola is

x = t^2/4p

y = t

The standard equation for the parabola is x^2 = 4py and y^2 = 4px

* Make sure to set the mode of the TI-89 to Parametric

parametric equations of conics?

I'm not sure if this will be on the test at all, but to add to everyone's conic charts, here are the parametric equations for all the conic sections:
Ellipse: x = a*cos(t), y=b*sin(t)
Hyperbola: x=a*sec(t) (or a/cos(t)), y=b*tan(t) (or the other way around if the vertices lie on the y-axis)
Parabola: x=t^2/4p, y=t

i hope this is actually helpful...
and speaking of parametrics, does anyone have an idea of the other main concepts that are likely to be on the test regarding parametrics other than the projectile motion equations or expressing parametrics in terms of x and y? (do we need to know about cycloids?)

Degrees of Rotation

I don't believe we're covering degrees of rotation.

Pretest Number 11

3(-)= pi/2 +2k*pi ... so (-)=pi/6 +2/3k*pi. The values are pi/6, 5pi/6, and 9pi/ 6. You can also think of it geometrically, it is going to make an equilateral triangle, so split the points up 3 ways.

Pretest number 11

So I know that sin can get to one and this means that the graph is furthest from pole when 3(-)
=pi/2. So then I know that (-) must be pi/6 and inside 0 to 2pi isnt that the only value for theta? I know it is not so please help me out thank you.

ps (-) is a theta symbol.

Degrees of Rotation

I think I missed the day Mr. Karafiol explained degrees of rotation of conic sections? I read the part in the book, but I'm still kinda confused about it. Can anyone try to explain it to me?

thanks.

Conics Review

Here's a link to a spreadsheet I made to help remember the properties of different conics.

https://www.cjhghs.com/051307 Conics Chart.xls

Pretest #12

Based on the picture i said that the focus was (0,p) and the latus rectum intersected the parabola at (x,p) and (-x,p). I then said the length was 2x and solved x^2=4py for 2x in terms of p. Is that what is supposed to be done? It seems like it should be more complicated.

Semester Review Packet # 54

I know we've done multiple problems like this one but could someone please remind me how to do it?

sin a = 4/5 and sin b = 5/13

find sin (a-b)