MHB What is the Simplified Form of This Trigonometric Identity?

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The discussion centers on simplifying the trigonometric identity $\sin(A) + \sin(A + \frac{2\pi}{3}) + \sin(A + \frac{4\pi}{3}) = 0$. Participants focus on breaking down the left-hand side using angle addition formulas and identifying the contributions of the sine and cosine terms. Key calculations involve evaluating $1 + \cos(\frac{2\pi}{3}) + \cos(\frac{4\pi}{3})$ and $\sin(\frac{2\pi}{3}) + \sin(\frac{4\pi}{3})$. The goal is to verify the identity through these simplifications. Ultimately, the identity holds true, confirming the original equation.
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$\sin\left({A}\right)+\sin\left({A+\frac{2\pi}{3}}\right)+\sin\left({A+\frac{4\pi}{3}}\right)=0$

$L.H.S=\sin\left({A}\right)+\left(\sin\left({A}\right)\cos\left({\frac{2\pi}{3}}\right)+\cos\left({A}\right)\sin\left({\frac{2\pi}{3}}\right)\right)+\left(\sin\left({A}\right)\cos\left({\frac{4\pi}{3}}\right)+\cos\left({A}\right)\sin\left({\frac{4\pi}{3}}\right)\right) $

From there?
 
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You've only given an expression...what is the actual identity to be verified?
 
Corrected now
 
Silver Bolt said:
$\sin\left({A}\right)+\sin\left({A+\frac{2\pi}{3}}\right)+\sin\left({A+\frac{4\pi}{3}}\right)=0$

$L.H.S=\sin\left({A}\right)+\left(\sin\left({A}\right)\cos\left({\frac{2\pi}{3}}\right)+\cos\left({A}\right)\sin\left({\frac{2\pi}{3}}\right)\right)+\left(\sin\left({A}\right)\cos\left({\frac{4\pi}{3}}\right)+\cos\left({A}\right)\sin\left({\frac{4\pi}{3}}\right)\right) $

From there?

I would write the LHS as:

$$\sin(A)\left(1+\cos\left(\frac{2\pi}{3}\right)+\cos\left(\frac{4\pi}{3}\right)\right)+\cos(A)\left(\sin\left(\frac{2\pi}{3}\right)+\sin\left(\frac{4\pi}{3}\right)\right)$$

Now, what are:

$$1+\cos\left(\frac{2\pi}{3}\right)+\cos\left(\frac{4\pi}{3}\right)=?$$

$$\sin\left(\frac{2\pi}{3}\right)+\sin\left(\frac{4\pi}{3}\right)=?$$
 
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