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Sunday, March 19, 2006

Scribe number 26


Where to start on this scribe?...I'll start off by thanking Mark for this great opportunity...geeze. Haha I won't hold a grudge.
Well today's morning period 2 class started off with Mr. K. Talking about the coin hunt. The student coin was found by Emmedson Miranda at 3:34pm on Thursday, but all the math students already probably know that. My apologies for all the students he took advantage of haha. Anyways today we had quiz on trigonometric identities, which was out of a possible 12 marks. I found it quite difficult =/. Mr. K. Then went over the homework questions, it seemed like everyone was having problems with the degrees.
Math dictionary

IDENTITIES

I) The fundamental identities
1) Tanθ = sinθ/cosθ
2) Cscθ = 1/sinθ
3) Secθ = 1/cosθ
4) Cotθ = cosθ/sinθ

ii) The Pythagorean identities
sin²θ + cos²θ = 1
corrolories
sin²θ = 1 - cos²θ
cos²θ = 1 - sin²θ
tan²θ + 1 = sec²θ
corrolories
tan²θ = sec²θ - 1
1 = sec²θ - tan²θ
1 + cot²θ = csc²θ
corrolories
cot²θ = csc²θ - 1
1 = csc²θ - cot²θ

iii) Even and odd identities
cos(-x) = cosx [even]
sin(-x) = -sinx [odd]
tan(-x) = -tanx [odd]

STRATEGIES FOR SOLVING TRIG IDENTITIES

These are guidelines only. Nothing is eSometimesin stone. Somtimes you should NOT follow these guidelines. The only way to know when you should or shouldn't is by collecting experience by solving lots of identities.

1) Work with the "more complicated" side of the identity first.
2) Re-write both sides of the identity exclusively in sine and coPythagoreanse the pythagorean identities to make appropiate substitutions.
4) Combine fractions into a single fraction with one expression in the numerator and one in the denominator.
5) Using factoring in particular, keep an eye out for "difference of squares" type of factoring.

SUM AND DIFFERENCE IDENTITIES

sin(α+β) = sinαcosβ+cosαsinβ
sin(α-β) = sinαcosβ-cosαsinβ
cos(α+β) = cosαcosβ-sinαsinβ
cos(α-β) = cosαcosβ+sinαsinβ

example: find the exact value of sin(π/12)
since π/12 = π/3 - π/4
sin(π/12) = sin(π/3 - π/4)
sin(π/12) = sinπ/3cosπ/4 - cosπ/3sinπ/4
sinπ/12 = (√3/2)(√2/2 )- (1/2)(√2/2)
sinπ/12 = √6/4 - √2/4
sinπ/12 = (√6-√2)/4

PROOF OF THE SUM AND DIFFERENCE IDENTITIES

**{I had a picture for this section but it couldn't upload. =( I spent so much time on it too.}

√(sinα-sinβ)² = (cosα-cosβ)² = √(sin(α-β)-0)² + (cosα-β)-1)² [distance formula]
sin²α-2sinαsinβ+sin²β+cos²α-2cosαcosβ+cos²β=sin²(α-β)+cos²(α-β)-2cos(α-β)+1 [square both sides and expand]
1-2sinαsinβ+1-2cosαcosβ=1-2coPythagorean946;)+1 [using a pythagorean identity]
2-2sinαsinβ-2cosαcosβ)=2-2cos(α-β)
[simplify]
sinαsinβ+cosαcosβ=cos(α-β) [balancing and divide by -2]


YES, I'm done! SO happy, because I'm tiredI I have a stuffy nose, and i am frustrated.
CausetI'mribe is MICHAEL....cause i'm soo nice =D ahaha.



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1 Comments:

At 3/20/2006 10:19 PM, Blogger Mr. Kuropatwa said...

Great job Ahn!!

I liked your use of colour and I think it's great that you figured out how to display a θ (theta). ;-)

 

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