Trig identity, tan(2A)

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March 10th 2010, 06:59 PM
monster

Trig identity, tan(2A)

Have a question i'm having some trouble with,

Verify the identity;

(sin A + sin 3A) / (cos A + cos 3A) = tan 2A

not sure best way to go about this one,
tried using sin3A = sin(2A + A) and expanding out but didn't seem to give me anything i wanted.

Any help would be greatly appreciated.

Cheers.
March 10th 2010, 07:18 PM
Black

Using the identities

$\sin(x \pm y) = \sin x \cos y \pm \sin y \sin x$

$\cos(x \pm y) = \cos x \cos y \mp \sin x \sin y$ ,

you can get

$\sin x + \sin y = 2\sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$

$\cos x + \cos y = 2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$ .

Set $x=3A$ and $y=A.$
March 11th 2010, 06:40 PM
monster

Ok, i can see how to finish the problem and this may seem like a stupid question but how did you get from the sin(x + y) formulae to the sinx + siny ones
March 12th 2010, 09:04 AM
Black

So you have

sin (a + b) = sin a cos b + sin b cos a

sin (a - b) = sin a cos b - sin b cos a

Adding the two equations gives you

sin (a + b) + sin (a - b) = 2 sin a cos b.

Now set x = a + b and y = a - b, find a and b in terms of x and y, and you'll get the identity for sin x + sin y
(similar for cos x + cos y).

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