write sin(5θ)-sin(θ) as a product of two trigonometric functions

Question

whpalmer4 · Answer

You can use the identity

$$\sin u - \sin v = 2\cos(\frac{u+v}{2})\sin(\frac{u-v}{2})$$

anonymous · Answer

How would you write the equation using that formula?

jdoe0001 · Answer

well, that is the equation in full @tigers49 http://www.sosmath.com/trig/Trig5/trig5/img9.gif

jdoe0001 · Answer

just plug in your values

anonymous · Answer

oh okay thank you that's easy enough

whpalmer4 · Answer

yeah, the trick is remembering the formula :-)

jdoe0001 · Answer

the one whpalmer4 showed above is the so-called "sum to product" identity

anonymous · Answer

okay what about if i have something cos^24a-sin^4a as a single trigonometric function. would i use one of those formulas?

anonymous · Answer

cos^24a-sin^2 4a forgot the 2 square

whpalmer4 · Answer

$$\cos^24a-\sin^24a$$
Well, remember that $$\cos 2u = \cos^2u-\sin^2u$$
So you can just take $u=4a$ and write $$\cos^24a-\sin^24a=\cos 8a $$

anonymous · Answer

got it. okay so this would go also with sin6x cos2x - cos6x sin2x?

whpalmer4 · Answer

You have to do some pattern matching.  Here you have sin A cos B - cos A sin B.
That's one of the sum/difference formulas.

whpalmer4 · Answer

because each term has both a sin and cos, that's the sin sum/difference formula (for cos, the cosines are together and the sines are together).  Next you have to figure out if that's the sum or the difference.

whpalmer4 · Answer

With the sin sum/difference formulas, the sign on the left side matches the sign on the right side, so this is 
$$\sin(A-B) = \sin A \cos B - \cos A \sin B$$

whpalmer4 · Answer

A = 6x, B = 2x
so the final answer is?

anonymous · Answer

8x?

whpalmer4 · Answer

bzzzt, wrong!  but thanks for playing :-)

whpalmer4 · Answer

sin(A-B), A = 6x, B = 2x

whpalmer4 · Answer

A-B =

anonymous · Answer

sin4x

whpalmer4 · Answer

now we're talking!

whpalmer4 · Answer

do you have a good list of trig identities?

whpalmer4 · Answer

http://www.sosmath.com/trig/Trig5/trig5/trig5.html is the one I use

anonymous · Answer

No i don't.

whpalmer4 · Answer

make yourself a set of flashcards on 3x5 cards and drill on them for 5 minutes a day.  it'll be time well spent in the long run...

anonymous · Answer

Thank you I'm sure I'll be using this more real soon again.