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March 29, 2024, 08:17:24 am

Author Topic: Small question about Products to Sums and Sums to Products  (Read 1316 times)  Share 

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decenthumanbeing

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Small question about Products to Sums and Sums to Products
« on: February 01, 2019, 09:01:39 pm »
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Hi :))

This isn't really a major question or anything its just me trying to cut down on the amount of formulas i need to remember because im very very lazy.
With the Products to Sums and Sums to Products formulae:

2sinAcosB=sin(A+B)+sin(A-B) (1)
2sinBcosA=sin(A+B)-sin(A-B) (2)
2cosAcosB=cos(A+B)+cos(A-B) (3)
-2sinAsinB=cos(A+B)-cos(A-B) (4)

sinS+sinT=2sin((S+T)*1/2)cos((S-T)*1/2) (5)
sinS-sinT=2sin((S+T)*1/2)sin((S-T)*1/2) (6)
cosS+cosT=2cos((S+T)*1/2)cos((S-T)*1/2) (7)
cosS-cosT=-2cos((S+T)*1/2)cos((S-T)*1/2) (8)

I know that (1) and (3) can be found by the compound angle formula which is pretty easy and I know that (2) is just where A and B has been swapped over so its easy to remember. (5), (6), (7) and (8) too are also easy to remember with the corresponding products to sums formula.

Anyway... what I'm asking is if there's a way to derive equation (4) using equations (1) and (3). My teacher told us there was a way but I can't figure it out and I've been trying for a while.

Thank you very much :"D

RuiAce

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Re: Small question about Products to Sums and Sums to Products
« Reply #1 on: February 01, 2019, 09:09:01 pm »
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Not too sure why you would bother actually. It's not hard to expand \( \cos(A+B) - \cos(A-B) = (\cos A \cos B - \sin A \sin B) - (\cos A \cos B + \sin A \sin B) \) either. I've always just expanded the compound angles in all 4 cases for the product to sums.

decenthumanbeing

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Re: Small question about Products to Sums and Sums to Products
« Reply #2 on: February 01, 2019, 09:17:18 pm »
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Not too sure why you would bother actually. It's not hard to expand \( \cos(A+B) - \cos(A-B) = (\cos A \cos B - \sin A \sin B) - (\cos A \cos B + \sin A \sin B) \) either. I've always just expanded the compound angles in all 4 cases for the product to sums.

I know that's true (and I'll try expanding when problems like that come up) but I would feel confident to have someone explain relationship between sin and cos that would allow me to convert (3) to (4).

RuiAce

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Re: Small question about Products to Sums and Sums to Products
« Reply #3 on: February 01, 2019, 10:24:09 pm »
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I know that's true (and I'll try expanding when problems like that come up) but I would feel confident to have someone explain relationship between sin and cos that would allow me to convert (3) to (4).
Unless you're talking about some substitution like \(A = \frac\pi2 - x\) and \(B = \frac\pi2 - y\), I personally can't see any relationship between the identities

(When you use these, you end up with some complementary angle identities and a bit of ASTC identities)
« Last Edit: February 01, 2019, 10:28:07 pm by RuiAce »