# Sum and Difference of 2 cosine functions

This question has been troubling me for days, I really haven't got a clue how to handle it:

$f(x) = -3+2\cos(x)$

$g(x) = \cos(x-\dfrac{1}{4}\pi)-2$

Get the sum ($s(x)=f(x)+g(x)$) and difference ($d(x)=f(x)-g(x)$) of these functions.

Can you guys please explain how to tackle these problems IN GENERAL, because I don't know the action scheme for solving a question like this one.

I have a TI-84+ with graphing abilities (calc intersect, min/max, dy/dx etc.) which I'm allowed to use. Please help me, I need urgent help with this question!

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Why isn't this question appearing on the front page for the love of god? – JohnPhteven Sep 15 '12 at 19:23
The constant terms just add directly, of course. For the sum of two cosines, does the intuitive explanation of a similar question here help? – Henning Makholm Sep 16 '12 at 14:02

$g(x)=\cos(x-\frac{\pi}{4})-2=\cos x\cos \frac{\pi}{4} + \sin x\sin \frac{\pi}{4} -2$ $=\frac{1}{\sqrt2}\cos x+\frac{1}{\sqrt2}\sin x -2$

So, $s(x)=f(x)+g(x)=3+2\cos x + \frac{1}{\sqrt2}\cos x+\frac{1}{\sqrt2}\sin x -2$ $=-5+\frac{2\sqrt2+1}{\sqrt2}\cos x+\frac{1}{\sqrt2}\sin x$

Let $r\cos y=\frac{2\sqrt2+1}{\sqrt2}$ and $r\sin y = \frac{1}{\sqrt2}$

$\frac{2\sqrt2+1}{\sqrt2}\cos x+\frac{1}{\sqrt2}\sin x=r(\cos y\cos x+\sin y\sin x)$ $=r\cos(x-y)$

Squaring and adding we get, $r^2=\frac{(2\sqrt2+1)^2+1}{2}=5+2\sqrt2$

On division, $\tan y=\frac{1}{2\sqrt2+1}\implies y=\tan^{-1}(\frac{1}{2\sqrt2+1})$

So, $s(x)=f(x)+g(x)=-5+\sqrt{5+2\sqrt2}\cos(x-\tan^{-1}(\frac{1}{2\sqrt2+1}))$

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Your answer seems to be of a higher mathematical level than I have and I don't think your approach is the way I should approach it, because it seems to be way tougher. I will however give you props for taking the time to answer it! – JohnPhteven Sep 16 '12 at 9:46
Would you mind sharing your approach, also where you faced difficulties in my method. – lab bhattacharjee Sep 16 '12 at 9:50
I haven't faced difficulties, but, at my school, you have to use the methods used in the book. I once had an extremely bad experience where I aced a test using methods of a class 1 year higher (thus different than methods in my book) and instead of getting full marks I got half marks. Method in the next comment (running out of characters. – JohnPhteven Sep 16 '12 at 9:54
Correction model: college.iwebs.ws/VWO%20B%20deel%202/VWO%20B%20deel%202%20H6.pdf Scroll down to page 19, question 55. 55a is the sum, 55b is the difference. It is in a different language, but mathematics is universal so I don't think you'll have trouble understanding it, if you do, ask me! Of course, we are allowed to use the TI84, that's where they get the min and max, however, the other things like the amplitude, period and the the 'evenwichtsstand' (which is in English I believe equilibrium value) – JohnPhteven Sep 16 '12 at 9:56
@ZafarS: if you need to understand that method specifically, you should probably add a comment along those lines to the question. – Ben Millwood Sep 17 '12 at 17:29