Can Sin(x) Be Greater Than One ?

The simple answer for this problem is YES !

We already know that \(-1\leq sin\left ( x \right ) \leq1\) , \(\forall x \in \mathbb{R}\) . Therefore, any solution to  

\( \left | sin \left ( x \right ) \right | > 1 \) would implies values of \(x\) such that \(x \notin \mathbb{R}\).

Example:

Consider the problem \(sin\left ( \theta \right )\) = 2 . What is the value of \(\theta\) ?

Start by utilizing the Euler's Formula:

 \(e^{i\theta} = cos\left ( \theta \right ) + isin\left ( \theta \right )\)

then apply the Euler's Formula for \(-\theta\) to have :

\(e^{i\left ( -\theta \right )} = cos\left ( \theta \right ) - isin\left ( \theta \right ) \)

Now, subtract these two formulas like follows:

\( \begin{matrix} & e^{i\theta} = cos\left ( \theta \right ) + isin\left ( \theta \right ) \\ - & \\ & e^{i\left ( -\theta \right )} = cos\left ( \theta \right ) - isin\left ( \theta \right ) \\ & ----------- \\ \Rightarrow & e^{i\theta} - e^{i\left ( -\theta \right )} = 2isin\left ( \theta \right ) \\ \end{matrix} \)

Rewrite the last equation in term of \(sin\left (\theta \right )\)

\( \Rightarrow sin\left (\theta \right ) = \frac{e^{i\theta}-e^{-i\theta}}{2i}\)

Now, Since the original problem is \(sin\left ( \theta \right )\) = 2, we may substitute the Left Hand Side with

\( \frac{e^{i\theta}-e^{-i\theta}}{2i}\) to have:

\(\frac{e^{i\theta}-e^{-i\theta}}{2i} =2\) \(\Rightarrow e^{i\theta}-e^{-i\theta} = 4i\)

To find the value of \(e^{i\theta}\) ; multiply the both sides with \(e^{i\theta}\) like follows:

\(e^{i\theta} \left [e^{i\theta}-e^{-i\theta} \right ]\) \(\Rightarrow e^{2i\theta} - 1 = 4ie^{i\theta}\)

\(\Rightarrow e^{2i\theta} - 4ie^{i\theta} - 1\)

Apply the Quadratic Formula \( \frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\) on the last equation to evaluate \(e^{i\theta}\)

\(e^{i\theta} = \frac{4i\pm \sqrt{-16-4\left (1 \right )\left (-1 \right )}}{2}\)

\(e^{i\theta} = \frac{4i\pm 2i\sqrt{3}}{2}\)

\(e^{i\theta} = 2i\pm i\sqrt{3} \) \(\Rightarrow e^{i\theta} = i(2 \pm \sqrt{3} ) \)

Now, multiply both sides with \(ln\) to exclude \(\theta\) like follows:

\( ln\left ( e^{i\theta} \right ) = ln\left ( i(2 \pm \sqrt{3} ) \right ) \)

\( i \theta = ln \left ( i \right ) \pm ln \left ( 2 \pm \sqrt{3} \right ) \)

\( i \theta = i\frac{\pi}{2} + ln \left ( 2 \pm \sqrt{3} \right ) \)

\( \theta = \frac{\pi}{2} - iln \left ( 2 \pm \sqrt{3} \right ) \)

Hence \( \theta = \frac{\pi}{2} \pm iln \left ( 2 + \sqrt{3} \right ) + 2 \pi n\) where, \(n \in \mathbb{N}\)