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The following diagram justifies the cosine angle sum rule quite nicely when $0 \alpha, \beta, \alpha + \beta < \pi / 2$.

cosine sum rule

Question: When extending this diagram to the other quadrants (i.e., when $\alpha + \beta > \pi/2$), the argument doesn't seem work anymore, at least not naturally? Am I mistaken? Does there exist a geometric argument for the cosine angle-sum rule for the other quadrants?

user1770201
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  • What’s your geometric definition of cosine for the other quadrants? – amd Sep 04 '16 at 16:40
  • I suppose "adjacent over hypotenuse", or the usual unit circle definition. – user1770201 Sep 04 '16 at 16:41
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    Woo-hoo! That's my diagram! :) (Nice to know it's appreciated.) Adapting it to other quadrants isn't too much of an ordeal. In this answer, I show a couple of obtuse cases that give a sense of how things work. – Blue Sep 04 '16 at 16:53
  • This rule is the real part of the complex exponential rule $e^{j(\alpha+\beta)} = e^{j\alpha}.e^{j\beta}$, which represents the product of complex numbers of length 1 also being a complex number of length 1. That is a good geometric representation. – Paul Sep 04 '16 at 17:00
  • By the way: This answer, which discusses the development of trig values for angles beyond the first quadrant, may be helpful. – Blue Sep 04 '16 at 17:16

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