polar graph help

Wheres the question?

im stuck with part a and also dont know how to do part b. for part A i have tried integrating with 9cos2theta with limits up to pi/2, but i get 0, im not to sure what limits i should be using? thanks
https://imgur.com/a/JCQguIC

I've only just noticed that you have included the mark scheme with the question, which is using exactly the method I've suggested. But do you see why it works and why your first go didn't?

Why are you integrating up to pi/2 ?

If theta = 0 and you begin increasing theta you have the initial line that then traverses the upper region of one segment of the shape up to theta =pi/4.

Plug some values in and you'll see this. When theta =pi/4 , r=0 and the direction traversed is anticlockwise. Thus integrating 1/2r^2 wrt theta from 0 to pi/4 gives the area of that sort of semi circular region. Theres four of them so then multiply by 4

is the 9/2 there because 3 squared and then they half it because of the equation with integrating polar curves?

Yes - you need to find 1/2 of the integral of r^2 wrt theta.

ah right that makes sense now, for part b however im still stuck. i get that PQ is 6, and i tried to calculate PS by using the fact that when theta = pi/4, the radius is 3(cospi/4)^1/2, using trig i can calculate the missing side which is sinpi/4 x 3(cospi/4)^1/2 and then multiply by 2 but for some reason that doesnt get me the answer.

doing it the differentiation way i can differentiate it fine but the simplifying bit im a bit lost, on line 4 part b not sure what they are doing?

You've forgotten to use 2 theta instead of just theta when substituting pi/4 into the original equation. If you do this with 2 theta, you'll see what that isn't going to work.

I think in the line you are referring to they are multiplying by (cos 2 theta)^1/2 and dividing by 3, as well as moving part of the equation to the other side. Is this the bit you mean?

yep just tried it and got a complex number...

yes i just multiplied by cos 2 theta ^1/2 and i got -3sin2theta sintheta + 3(cos2theta)^2 costheta
how is it still cos2theta ^1/2 and not to the power of 2?

Are you sure you should have that power 2 in your second term?

ah yes its 1 not 2, im at where they are at now after dividing by cos2theta ^1/2. i dont see at all what they are doing on the line after though.

Can't see a line where they divide by this, only one where they multiply by it. If you can be more specific about the line you're looking at, I'm sure we can help out.

on line 4 ive got to where they are, they just put the cos2theta on the denominator, i dont understand what they do to get sin2theta sintheta = cos2theta cos theta on the line after

This is what we were talking about before. From the line you agree with (the one that starts "At max/min;"

- multiply by (cos 2 theta)^1/2
- divide by 3
- move the negative piece to the other side of the equality sign

yep im there now thanks, which double angle rule are they using on the lhs where they get sin squared? and also how do they know that costheta(1-4sin^2theta) = 0 ?

They are using the standard identity for sin(2theta) in terms of sin(theta) and cos(theta), and combining it with the sin(theta) that is already there.

On the next line, they have brought everything onto one side and factorised out the cos(theta).

These are all bits from a regular maths A level - they shouldn't be giving you too much trouble if you're doing a question at this level.

we havent done these yet in normal maths and for some reason our mock exams for further maths is the entire syllabus so we have just had to kind of rush through it when revising. that being said i pretty much always lose marks on simplifying trig, can never figure out which identity to use