change to the form Acos(x+w)

**JQ2009** · November 3rd 2009, 07:59 AM

How do you change F(x) = -cosx + sqrt3sint to the form:

Acos(x+w)

with A and w both exact values?

**Grandad** · November 3rd 2009, 10:07 AM

Hello JQ2009

Originally Posted by JQ2009

How do you change F(x) = -cosx + sqrt3sint to the form:

Acos(x+w)

with A and w both exact values?

I assume you mean

$F(x)= -\cos x + \sqrt3\sin x$

So, let $F(x) = A\cos(x+w)$ . Then:

$-\cos x + \sqrt3\sin x=A\cos(x+w)$

$=A\cos x \cos w - A\sin x\sin w$

Compare coefficients of $\cos x$ and $\sin x$ :

$-1 = A\cos w$

$\sqrt3 = -A\sin w$

Square and add:

$4 = A^2(\cos^2w+\sin^2w)$

$\Rightarrow A = -2$ (taking - sign)

Divide second equation by the first:

$\tan w = \sqrt3$

$\Rightarrow w = \pi/3$

$\Rightarrow F(x) = -2\cos(x+\pi/3)$

Grandad

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