Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If $\alpha+\beta+\gamma=\pi$ then $\tan\alpha+\tan\beta+\tan\gamma=\tan\alpha\tan\beta\tan\gamma$.

If $\alpha+\beta+\gamma=\pi$ then $\sin(2\alpha)+\sin(2\beta)+\sin(2\gamma) = 4\sin\alpha\sin\beta\sin\gamma$.

These get referred to as "conditional identities" in books on trigonometry.

Are there any out there in published sources where the "condition" is something other than that the sum of three angles or (to be moderately daring?) the sum of several angles, is $\pi$? How many relatively basic ones (as opposed to complicated ones derived from these and others like them) are in somewhat standard sources besides these two? Any at all?

share|improve this question
add comment

1 Answer 1

Here is an identity I made up. Suppose that $$a+b+c+d=2\pi.$$ Then $$4\cos(a)\cos(b)\cos(c)\cos(d)-4\sin(a)\sin(b)\sin(c)\sin(d)$$ $$=\cos(2a)+\cos(2b)+\cos(2c)+\cos(2d).$$

Are these the type of things that you are looking for?

share|improve this answer
    
I'm not sure this one isn't merely one of the "derived" sort I mentioned. I know lots of those. For example, if $\alpha+\beta+\gamma+\delta+\varepsilon=\pi$ then $\sin(2\alpha)+\cdots+\sin(2\varepsilon)$ $=4\underbrace{\sin\alpha\sin\beta\sin\gamma}_{\text{sines}}\ \underbrace{\cos\delta\cos\varepsilon}_{\text{cosines}}+\{\text{9 other such terms}\}$ ${}-8\sin\alpha\sin\beta\sin\gamma\sin\delta\sin\varepsilon$.${}\qquad{}$ –  Michael Hardy Dec 26 '12 at 1:20
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.