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In Stitz' and Zeager's 'Precalculus' textbook, v.3.0, pg. 784, we find the following exercise (#65): >[V]erify the identity. Assume all quantities are defined: > >$$\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)$$ Now, I managed to verify the identity. I wanted to ask about something else I found while trying to solve this problem. Working with the left-side of the equation we have: $ = \sin(2\theta + \theta) $ $ = {\color{red}\sin(2\theta)\cos(\theta)} + \cos(2\theta)\sin(\theta) $ --- via sum identity $ = {\color{red}\frac{1}{2}[\,\sin(\theta) + \sin(3\theta)]} + \cos(2\theta)\sin(\theta) $ --- via product-to-sum identity The recovery of $\sin(3\theta)$ in the derivation seems to suggest that $\sin(3\theta)$ itself can be written as an infinite series, but I don't know much about infinite series of trigonometric functions (yet). Would be interested to see if this is the case.
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Sat, 31 Dec 2022 04:48 GMT