Maths C3 - Trigonometry... Help??

The maximum value of $\sin(ax+b)$ and $\cos(ax+b) \forall a,b\in\mathbb{R}$ is 1 so when you have your equation in the R-alpha form, you can assume that the trigonometric function is 1 when finding the maximum.

As for the last bit, it's just the case of solving the trig equation and considering the smallest angle which will give the maximum.

Oh dear, I don't think I fully understand that first sentence I've never seen the symbol..$\forall$...before.

I can understand the bit where you say the maximum for $\sin(ax+b)$ and $\cos(ax+b)$ is 1 but I thought since I have to work out the smallest positive value of $\theta$ from the equation $13cos(\theta - 22.6)$ I wasn't sure how to go about doing so :/

Don't pay too much attention to the notation I used there - you don't need to be aware it. It simply means that the statement is true for any arbitrary values of a and b that you pick as long as they stay within the real numbers.

There's an infinite amount of solutions so you are simply looking for the smallest possible angle that will give the maximum, and we know that the maximum is given when the trig function is equal to 1, thus it's just the matter of solving a trig equation.

Oh right so I would do... $13cos (\theta-22.6) = 1$... and go from there?

No you would do $\cos(\theta - 22.6)=1$ because you want the cosine function to be maximised which in turn maximises $f(\theta)=13\cos(\theta - 22.6)$

Oh yeah I'm starting to understand now. It's because the angle $\theta$ would give a maximum at the same point along the x-axis for both of the equations.. $\cos(\theta - 22.6)$ and $13\cos(\theta - 22.6)$

It's almost like saying, because the maximum for...
$13\cos(\theta - 22.6)$... is 13

Then...
$13\cos(\theta - 22.6)=13$

$\cos(\theta - 22.6)=\frac{13}{13}$

$\cos(\theta - 22.6)=1$

So... $\theta = 22.6, 382.6,$ and so on (including minus figures)...

But since we want the smallest positive angle of $\theta$ then the answer would be...

$\theta = 22.6$

Exactly this.

Thank you so much @RDKGames you always manage to help me clear up my moments of confusion!!

I'm not quite sure how to answer part (d) of this question...

Can someone explain this to me. I'm not even sure I understand the question properly

The thing about squaring both sides of an equation is that it gives extra solutions. Consider the equation $x=2$. Squaring gives $x^2=4$ and so we conclude that $x=\pm 2$? No. Squaring has introduced a new solution.

Can someone please explain the second to last line? :/
Why does... A=(P+Q)/2 and B=(P-Q)/2 ??

I was also in the same position last year when I did my A-Levels. I used to use the website TutorRoom.co.uk (not sure if there is any relation to StudentRoom :/). The tutors on there would literally answer within 10-15 minutes and walk me through each question step-by-step. Only downside is that I had to beg my parents to pay for it. But the cost is actually less than some people pay for an hour of private tutoring and I could ask as many questions as I wanted, whenever I wanted. Totally recommend if you are self-studying like I was (or if you are struggling with Maths in general).

From $A+B = P$ and $A-B =Q$ we can add the two equations to get $(A+B) + (A-B) = P + Q \iff 2A = P + Q \iff A = \frac{P+Q}{2}$, we subtract the two to find B.

I was also in the same position last year when I did my A-Levels. I used to use the website TutorRoom.co.uk (not sure if there is any relation to StudentRoom :/). The tutors on there would literally answer within 10-15 minutes and walk me through each question step-by-step. Only downside is that I had to beg my parents to pay for it. But the cost is actually less than some people pay for an hour of private tutoring and I could ask as many questions as I wanted, whenever I wanted. Totally recommend if you are self-studying like I was (or if you are struggling with Maths in general).

Thank you Zacken!! With your help I've managed to get to the end of this exampole, but I'm still having trouble realising why

The things I'm still confused about...

Any wise words to help clear this up?

Also... are we expected to remember these identities off-by-heart?...

These identities are on the formula book. Even though this is Example 20, it's really just a proof of these identities.

Like any proof, it's useful to learn and understand it. You could be asked to prove it in the exam.

"I would never have known..." : That's fine since this is an individual proof and it isn't a technique that comes up in a variety of questions.