What Trig Identity are these?

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January 10th 2013, 07:04 AM
AZach

What Trig Identity are these?

I saw these trig identities at the beginning of this video on PatrickJMT » Trigonometric Integrals – Part 5 of 6 and I don't remember using these identities before. They weren't in the back of my textbook either. Does anyone know what they're called, and maybe a simple method for deriving them? And by simple I mean along the lines of deriving a half-angle formula using Pythagorean theorem and double-angle formulas. It doesn't have to be simple, I'm just hoping there's a simple way to memorize them!

Here's the identities if the link doesn't work:

$sin(Ax)cos(Bx) = \frac{1}{2}[sin(Ax-Bx)+sin(Ax+Bx)]$
$sin(Ax)sin(Bx) = \frac{1}{2}[cos(Ax-Bx)-cos(Ax+Bx)]$
$cos(Ax)cos(Bx) = \frac{1}{2}[cos(Ax-Bx)+cos(Ax+Bx)]$

Edit: I just realize this might be better of in the Trig forum and not calculus. I posted here just out of habit! And fix't the typos
January 10th 2013, 07:49 AM
ebaines

Re: What Trig Identity are these?

I believe you have mis-written these identities; they should be:

$\sin(Ax)\cos(Bx) = \frac 1 2 [ \sin(Ax -Bx) + \sin(Ax + Bx)]$

$\sin(Ax) \sin(Bx) = \frac 1 2[ \cos(Ax-Bx) - \cos(Ax + Bx) ]$

$\cos(Ax) \cos(Bx) = \frac 1 2[ \cos(Ax-Bx) + \cos(Ax + Bx) ]$

These can all be derived from the basic identities:

$\sin (Ax + Bx) = \sin (Ax) \cos(Bx) + \cos(Ax)\sin(Bx)$ ,
$\cos(Ax + Bx) = \cos(Ax) \cos(Bx) - \sin(Ax) \sin (Bx)$

For example:

$\frac 1 2 [\sin(Ax -Bx) + \sin (Ax + Bx)] = \frac 1 2 [\sin(Ax) \cos(Bx) - \cos(Ax) \sin(Bx) + \sin(Ax)\cos(Bx) + \cos(Ax)\sin(Bx) ] = \sin(Ax)\cos(Bx)$

The other two can be derived in similar fashion.
January 10th 2013, 08:08 AM
Soroban

Re: What Trig Identity are these?

Hello, AZach!

Careful! . . . Your identities have terrible typos.

Product-to-Sum Indentities

. . $\begin{array}{ccc}\sin A\sin B &=& \frac{1}{2}\left[\cos(A-B) - \cos(A+B)\right] \\ \\[-3mm] \cos A\cos B &=& \frac{1}{2}\left[\cos(A-B) + \cos(A+B)\right] \\ \\[-3mm] \sin A\cos B &=& \frac{1}{2}\big[\sin(A-B) + \sin(A+B)\big] \end{array}$

Sum-to-Product Identities

. . $\begin{array}{ccc}\sin A + \sin B &=& 2\sin(\frac{A+B}{2})\cos(\frac{A-B}{2}) \\ \\[-3mm] \sin A - \sin B &=& 2\cos(\frac{A+B}{2})\sin(\frac{A-B}{2}) \\ \\[-3mm] \cos A + \cos B &=& 2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2}) \\ \\[-3mm] \cos A - \cos B &=& \text{-}2\sin(\frac{A+B}{2})\sin(\frac{A-B}{2}) \end{array}$

These are lesser-known identities,
. . but become more important later on.