Can you please give me a hint for the following exercise: $$(\cos x+i\sin x)(\cos 2x+i\sin 2x)(\cos 5x+i\sin 5x)={i+1\over \sqrt 2 }$$
Thank you!
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3Use polar forms on both sides... – Macavity May 11 '14 at 15:03
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@Macavity But can you please explain me how can I transform the left side into a polar form? – wonderingdev May 11 '14 at 15:25
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1You have the formula in the answers posted - use Euler's formula and you have $\cos kx + i \sin kx = e^{ikx}$. Multiply three such terms for the LHS. It is a more easy form to work with when you are multiplying such things.. – Macavity May 11 '14 at 15:29
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So it's the only and best possibility, is it? – wonderingdev May 11 '14 at 15:29
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Best probably yes, certainly not the only one - for e.g. you could multiply out the LHS to get a monster, simplify using trigonometry and equate real and imaginary parts. Not something you want to do if you have other choices. – Macavity May 11 '14 at 15:31
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@Macavity Thank you very much – wonderingdev May 11 '14 at 15:32
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Note that the top is given by $e^{ix}\cdot e^{2ix}\cdot e^{5ix} = e^{8ix} = \frac{1+i}{\sqrt{2}}$ and so $\cos(8x) = \sin(8x) = 1/\sqrt{2}$ implies $8x = \pi/4$ implies $x = \pi/32 + \pi\cdot n/4$ where $n \in \mathbb{Z}$. This was edited to consider less trivial solutions - if your goal is a complete solution set.
Christopher K
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2"implies 8x=π/4" No, one could also have 8x=9π/4 (and a few others as well). – Did May 11 '14 at 15:08
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