- #1

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- TL;DR Summary
- A general rule for adding two trigonometric functions that have unidentical amplitudes

The trig identities for adding trig functions can be seen:

But here the amplitudes are identical (i.e. A = 1). However, what do I do if I have two arbitrary, real amplitudes for each term? How would the identity change?

Analysis: If the amplitudes do show up on the RHS, we would

I don't know the derivations of the above identities, so I'm wondering if there is a way to generalize something like Asin(a) + Bsin(b) such that these terms are accounted for on the RHS of an equivalent expression.

But here the amplitudes are identical (i.e. A = 1). However, what do I do if I have two arbitrary, real amplitudes for each term? How would the identity change?

Analysis: If the amplitudes do show up on the RHS, we would

*expect*them to either be a product or sum of these, possibly signed, or simply explicitly states. For A = 1, it may be difficult to see where they appear if explicitly stated, as they disappear as a factor. However, what we do see is that 2 appears in front of all of the RHS identities, which is a hint that for A_1 = A_2 = 1, we simply add them together and place them as a factor in front of the expression. However, this isn't necessarily the case, and simply and intuition, and considering 2 also appears in the denominators of the inner terms, it isn't a given that the number 2 shows up for this reason.I don't know the derivations of the above identities, so I'm wondering if there is a way to generalize something like Asin(a) + Bsin(b) such that these terms are accounted for on the RHS of an equivalent expression.