Prelim 2016 Maths Help Thread (3 Viewers)

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InteGrand

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If sin (theta)=3/4, 90 degrees is smaller than (theta) is smaller than 180 degrees, evaluate in surd form,

a. Sin2(theta)
b. cos2(theta)
c.tan2(theta)
In which quadrant does 2(theta) lie in?


The others are quite similar in nature.
 

eyeseeyou

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1. sin(45-x)cos(45-x)
2. (1-cos 2theta)/ (1+cos2theta)
3. 2cos squared 3x -1
4. (1-t^2)/(1+t^2) where t=tan theta/2
 

eyeseeyou

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1. Using double angle formula for tan (theta), find tan 22.5 in simplest surd form
2. If tan (alpha)=k tan (beta), show that
(k-1)sin(aplha +beta) = (k+1) sin (alpha - beta)
 

InteGrand

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1. Using double angle formula for tan (theta), find tan 22.5 in simplest surd form
2. If tan (alpha)=k tan (beta), show that
(k-1)sin(aplha +beta) = (k+1) sin (alpha - beta)
1) Let t = tan(22.5 deg), then 2t/(1 – t^2) = tan(45 deg) = 1, by the double angle formula.

So 2t = 1-t^2 ==> t^2 +2t = 1.

So (t+1)^2 = 2

i.e. t = 1 + √(2) (taking the positive solution; the negative solution is 1 – √(2), which we know isn't the right one, since t > 0 as 22.5 deg is acute).
 

leehuan

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Oops I got beaten :p
Tip: If you have cos(2x) by itself then it might be hard to figure out which to use.

But if you have 1-cos(2x) try to get used to rewriting as 2sin^2(x)
And 1+cos(2x) as 2cos^2(x)
 

eyeseeyou

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Tip: If you have cos(2x) by itself then it might be hard to figure out which to use.

But if you have 1-cos(2x) try to get used to rewriting as 2sin^2(x)
And 1+cos(2x) as 2cos^2(x)
I was stuck at first and then thought about double angle then when I tried double angle I got some weird answer

Anyways thanks leehuan
 

leehuan

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I was stuck at first and then thought about double angle then when I tried double angle I got some weird answer

Anyways thanks leehuan
Just don't forget with cos you have three choices for the double angles. Reason is because of the Pythagorean identity giving you three versions
 

eyeseeyou

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Just don't forget with cos you have three choices for the double angles. Reason is because of the Pythagorean identity giving you three versions
yeah I know but which is the best to take?
 

eyeseeyou

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(sin2theta+1)/(cos2theta)=(costheta-sintheta)/(costheta+sintheta)=2sec2theta
 

leehuan

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yeah I know but which is the best to take?
The rule of thumb is when you have 1-cos(2x) you end up with 2sin^2(x)
And 1+cos(2x) gives 2cos^2(x)

You'll nee those identities when you learn to integrate sin^2(x) as well in the future.

Otherwise when you have cos(2x) by itself there is no rule of thumb. There's more intuition required for those. e.g. difference of two squares may mean I leave it in the original form cos^2(x)-sin^2(x)
 
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