The Angle Addition Identities were derived using $\sin\theta \equiv \frac{opp}{hyp}$ and $\cos\theta \equiv \frac{adj}{hyp}$. In addition, $\theta$ was constrained to the interval of $[0, \frac{\pi}{2})$. The question addressed in this post is, do the Angle Addition Identities work if the arguments of the sine function and cosine functions are generalized to include negative numbers? In the following, $\theta$ will still be constrained to the interval of $[0, \frac{\pi}{2})$. The generalization of choice is $ \displaystyle \sin(-\theta) = -\sin\theta$ and $ \displaystyle \cos(-\theta) = \cos\theta$ This generalization is applicable to the unit circle, specifically in the 1st and 4th quadrants. The 2nd and 3rd quadrants are not currently needed for my purposes, but this may change in the future. The Angle Addition Identities, as derived previously, are

  • $\displaystyle \sin(x+y) = \sin x \cos y + \sin y \cos x$
  • $\displaystyle \cos(x+y) = \cos x \cos y – \sin x \sin y$.

Next use the above generalizations of the sine and cosine functions. There are three possible cases in which at least one of the input angles is negative.

  1. The first case to investigate is $y \rightarrow -y$ while maintaining $y \in [0,\frac{\pi}{2})$:
  • $\displaystyle \sin(x+(-y)) \stackrel{?}{=} \sin x \cos (-y) + \sin (-y) \cos x$
  • $\displaystyle \cos(x+(-y)) \stackrel{?}{=} \cos x \cos (-y) – \sin x \sin (-y)$.

2. The second case to investigate is $x \rightarrow -x$ while maintaining $x \in [0,\frac{\pi}{2})$:

  • $\displaystyle \sin((-x)+y) \stackrel{?}{=} \sin (-x) \cos y + \sin y \cos (-x) $
  • $\displaystyle \cos((-x)+y) \stackrel{?}{=} \cos (-x) \cos y – \sin (-x) \sin y $.

3. The third case and final case to investigate is $x \rightarrow -x$ while maintaining $x \in [0,\frac{\pi}{2})$ and $y \rightarrow -y$ while maintaining $y \in [0,\frac{\pi}{2})$:

  • $\displaystyle \sin((-x)+(-y)) \stackrel{?}{=} \sin (-x) \cos (-y) + \sin (-y) \cos (-x) $
  • $\displaystyle \cos((-x)+(-y)) \stackrel{?}{=} \cos (-x) \cos (-y) – \sin (-x) \sin (-y) $.

One realization to make is that the form of equations 1-3 is no different from the original Angle Addition Identities. So the question that remains is whether these identities make geometrical sense. This question can be further investigated with geometry. Three new pictures would be needed along with six separate proofs.

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