# Complex Number finding the general Value of Theta

If $$(\cos\Theta+i\sin\Theta)(\cos2\Theta+i\sin2\Theta)(\cos3\Theta+i\sin3\Theta) \dots (\cos n\Theta+i\sin n\Theta)=i$$ then show that general Value of $$\Theta=\left[2r+\frac1{n(n+1)}\right]\pi$$ $$OR$$

$$\Theta = \frac{\pi+4\pi m}{n(n+1)}$$ How to get the value of theta

• Use $$(\cos A+i\sin A)(\cos B+i\sin B)=\cos(A+B)+i\sin(A+B)$$ – lab bhattacharjee Apr 7 '14 at 4:49
• It would be then $cos(\theta+2\theta+3\theta+..n\theta)+isin(\theta+2\theta+3\theta+..n\theta)$ where $$\theta+2\theta+3\theta+..n\theta=\frac{n\theta(n\theta+1)}2$$ – Abdullah Sorathia Apr 7 '14 at 5:00
• You have a mistake in $$\theta+2\theta+3\theta+..n\theta=\frac{n\theta(n\theta+1)}2$$ It should be $$\frac{1}{2} \theta n (n+1)$$ – Claude Leibovici Apr 7 '14 at 7:26
• Are you sure your formula is correct? Is $r$ a real number or a natural number? – john.abraham Apr 7 '14 at 7:58
• Using that $\cos(k\Theta)+i\sin(k\Theta)=e^{ik\Theta}$ we have $\begin{array}{ccc} e^{i\Theta}e^{i2\Theta}e^{i3\Theta}\dots e^{in\Theta} & = & i\\ e^{i\Theta(1+2+3+\dots+n)} & = & i\\ e^{i\Theta\frac{1}{2}n(n+1)} & = & i \end{array}$ Which is only true if (for $n,m\in\mathbb{N}$) $\begin{array}{ccc} \Theta\frac{1}{2}n(n+1) & = & \frac{1}{2}\pi+2\pi m\\ \Theta n(n+1) & = & \pi+4\pi m\\ \Theta & = & \frac{\pi+4\pi m}{n(n+1)} \end{array}$ I don't see how $\Theta$ could be anything else. Hopefully someone else can chime in on this. – john.abraham Apr 7 '14 at 11:17

As stated by john.abraham, the product can be written as an exponential raised to some sum which can then be equated to $\left({\frac{\pi}{2} + 2\pi m}\right)i$ (the exponent of $i = e^{\left(\frac{\pi}{2} + 2\pi m\right)i}$)--the $i$'s cancel:
$$\prod_{k = 1}^{n}\left(\cos\left(k\theta\right) + i\sin\left(k\theta\right)\right) = \prod_{k = 1}^n e^{k\theta i} = i = e^{\left(\frac{\pi}{2} + 2\pi m\right)i} \\ \prod_{k = 1}^n e^{k\theta i} = \exp\left(\sum\limits_{k = 1}^n \left(k\theta i\right)\right) \\ \sum\limits_{k = 1}^n \left(k\theta i\right) = \left(\theta \left(\sum\limits_{k = 1}^n k\right)\right)i = \left(\theta \frac{n(n + 1)}{2}\right)i = \left(\frac{\pi}{2} + 2\pi m\right)i \\ \theta \frac{n(n + 1)}{2} = \frac{\pi}{2} + 2\pi m \\ \theta = \frac{\pi + 4\pi m}{n(n+1)} \text{, q.e.d.}$$