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?

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

Use $\color{#3D99F6}{\text{Euler's Exponential Formula}}$ , $\sin(\theta)\,=\,\dfrac{e^{i \theta} - e^{ - i \theta}}{2 i}$ .

Make a quadratic equation in $e^{i \theta}$ and then figure out its roots and express them in $\color{#D61F06}{\text{Eulerian Form}}$ , 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.

Aditya Sky - 5 years 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.

Robert VanMatre V - 5 years ago

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

Sal Gard - 5 years ago

Exactly..

Aditya Sky - 4 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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

Comments

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