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# How many solutions does $$\sin(x) = 2$$ has?

I was studying roots of unity, de moivres theorem, Euler's formula while I was disturbed by a question that was exceedingly tough for me? It asked to calculate solution for sin x =2 ?Check out the wiki page for Euler's formula. In it the trigonometric applications section. I am getting four solution but the question shows only two of them. You can check a page to help yourself. Sin x =2 Please give an elaborate answer why can't there be four solutions?

Note by Puneet Pinku
1 year, 8 months ago

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It don't know why are you getting four answers. Better share your work.

Use $$\color{blue}{\text{Euler's Exponential Formula}}$$, $$\sin(\theta)\,=\,\dfrac{e^{i \theta} - e^{ - i \theta}}{2 i}$$.

Make a quadratic equation in $$e^{i \theta}$$ and then figure out its roots and express them in $$\color{red}{\text{Eulerian Form}}$$, i.e, $$|z| \cdot e^{i \cdot \text{arg(z)} }$$.

Taking natural logarithm on both sides and you're done :).

Feel free to ask if you've any doubt.

- 1 year, 8 months ago

sin(x) is bound by 1 and -1 , so it can't be equal to 2 without an additional component to change the bound range.

- 1 year, 8 months ago

Look into Euler's theorem. This bound only applies for real radial boundaries.

- 1 year, 8 months ago

Exactly..

- 1 year, 8 months ago