HL Maths Question

mp3guy

Off some weird mock paper we were given to do over the weekend, haven't the foggiest. From Q2 Paper 1



Thar' she blows, tough one, that power is n by the way, n+3 etc... I've tried subbing in the different n+ values into the sin and cos then messing around with the indices, didn't work, tried subbing in X for sin0 and Y for cos0 but to no avail. Anyone help?

Michael Collins

Looks pretty odd at first alright. Here's how you do it:

(For ease of notation I'm going to use computer notation i.e. U(n), U(n+3) etc for the indices, and leave out the theta, since it's the only variable)

U(n) = sin^n+cos^n

[U(n+3) - U(n+5)]/U(n+1) = [sin^(n+3) + cos^(n+3) - sin^(n+5) + cos^(n+5)] / [sin^(n+1) + cos^(n+1)]

RHS numerator can be factorised as:

sin^(n+3)[1-sin^2] + cos^(n+3)[1-cos^2]

= sin^(n+3)[cos^2] + cos^(n+3)[sin^2] ...since sin^2 + cos^2 = 1

= sin^(n+1)[sin^2.cos^2] + cos^(n+1)[cos^2.sin^2] ...just borrowing powers from the multiplicators

=[sin^(n+1) + cos^(n+1)][sin^2.cos^2]

and the first bracket cancels with the denominator above and you're left with the RHS given in the problem.

Yep, got the same as Mr. Collins.

(I'm not just saying that - I have it written in front of me, and just pressed refresh and saw he posted it up).

rjt

Using the definition of Un, we get:

U_(n+3)-U(n+5) = sin^(n+3) -sin^(n+5) + cos^(n+3) - cos^(n+5) {I'm omitting the theta's}

= sin^(n+3)x(1-cos^2) + cos^(n+3)x(1-sin^2)
= sin^(n+3)cos^2 + cos^(n+3)sin^2
= sin^2cos^2(sin^(n+1) + cos^(n+1))
= sin^2cos^2U_(n+1)

Therefore, ( U(n+3) - U_(n+5) ) / U_(n+1) = sin^2cos^2

If there's any problem with understanding this, I'll LaTeX it up for you; its hard to follow when written without proper formatting.

Edit: Agh, beaten to it

Michael Collins

rjt wrote:

...Edit: Agh, beaten to it

Yes! I win!

I was going to do it in LaTeX myself but realised it would take me about half an hour and I'd be beaten to it! I'm not the most proficient at LaTeX yet.

mp3guy

Thanks alot, was a tough one. It was kind of bastardish of them to put trig like that in question 2 paper 1 though.

md99

I HATE maths pre papers....

The real thing is USUALLY okay though