A question about harmonic form of trigonometric functions. The question is:
i) Find the maximum and minimum values.
ii) the smallest non-negative value of x for which this occurs.
12cos(a)-9sin(a)
I think it should be changed into the form of Rcos(a+x) and it should be 15cos(a+36.87), and I get the answer i)+15 / -15 ii)323.13 (360-36.87) / 143.13 (180-36.87).
But the answer given by the book is " i)15, -15 ii)306.87, 143.13 "
I'm really confused by that answer..Am I wrong?
BTW, I'm self studying A-level further pure mathematics, but the book(written by BRIAN and MARK GAULTER published by Oxford university press) I get seems not very helpful.
so I truly hope someone can recommend some books/websites for self learning.
 A: Using the formula $\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$, we get
$$
12\cos(a)-9\sin(a)=15\sin(a+\pi+\arctan(-4/3))
$$
So the maximum and minimum are $+15$ and $-15$.
The smallest non-negative value for the maximum would be when $a+\pi-\arctan(4/3)=5\pi/2$; that is, $a=3\pi/2+\arctan(4/3)$.
The smallest non-negative value for the minimum would be when $a+\pi-\arctan(4/3)=3\pi/2$; that is, $a=\pi/2+\arctan(4/3)$.
Problem with Book Answer:
Converting to degrees, my answers are
maximum at $323.1301^\circ$ and minimum at $143.1301^\circ$.
It appears the first book answer is wrong. The answers should be $180^\circ$ apart.
A: We review the (correct) procedure that you went through. We have $12^2+9^2=15^2$, so we rewrite our expression as 
$$15\left(\frac{12}{15}\cos a -\frac{9}{15}\sin a\right).$$
Now if $b$ is any angle whose cosine is $\frac{12}{15}$ and whose sine is $\frac{9}{15}$, we can rewrite our expression as 
$$15\left(\cos a \cos b -\sin a \sin b\right),$$
that is, as $15\cos(a+b)$. 
The maximum value of the cosine function is $1$, and the minimum is $-1$. So the maximum and minimum of our expression are $15$ and $-15$ respectively. The only remaining problem is to decide on the appropriate values of $a$.
For the maximum, $a+b$ should be (in degrees) one of $0$, $360$, $-360$, $720$, $-720$, and so on. The angle $b$ is about $36.87$ plus or minus a multiple of $360$. So we can get the desired kind of sum $a+b$ by choosing $a\approx 360-36.87$, about $323.13$. 
It is not hard to do a partial verification our answer by calculator. If you compute $12\cos a -9\sin a$ for the above value of $a$, you will get something quite close to $15$. The book's value gives something smaller, roughly $14.4$. The book's value is mistaken. It was obtained by pressing the wrong button on the calculator, $\sin^{-1}$ instead of $\cos^{-1}$.  
For the minimum, we want $a+b$ to be $180$ plus or minus a multiple of $360$. Thus $a$ is approximately $180-36.87$.
