Question

The equation $sin\left(z\right)=10$ has

• A. no real (or) complex solution
• B. exactly two distinct complex solution
• C. a unique solution
• D. an infinite number of complex solutions

Answer 1

The correct option is D an infinite number of complex solutions

$sinz=10$
$\Rightarrow \frac{e^{iz}-e^{-iz}}{2i}=10\Rightarrow e^{iz}-\frac{1}{e^{iz}}=20i$
$\Rightarrow (e^{iz}{)}^2-20i(e^{iz})-1=0$
$\Rightarrow {\alpha }^2-20i\alpha -1=0$, where $\alpha =e^{iz}$
$\Rightarrow \alpha =\frac{-(-20i)\pm \sqrt{-400+4}}{2}$
$=\frac{20i\pm \sqrt{-396}}{2}=\frac{20i\pm 6i\sqrt{11}}{2}$
or $e^{iz}=10i\pm 3\sqrt{11}.i$
$\Rightarrow lo{g}_e\left(e^{iz}\right)=log(10i\pm 3\sqrt{11}i)$
$\Rightarrow iz=log\left[i\right(10\pm 3\sqrt{11}\left)\right]$
$=logi+log(10\pm 3\sqrt{11})$
$[\because [lo{g}_e{e}^{\frac{\pi }{2}i}=log1+lo{g}_e{e}^{(\frac{\pi }{2}\pm 2n\pi )i}]$
$\Rightarrow iz=log1+i(\frac{\pi }{2}\pm 2n\pi )+log(10\pm 3\sqrt{11})$
$\Rightarrow iz=i(\frac{\pi }{2}\pm 2n\pi )+log(1-\pm 2\sqrt{11})$
$\Rightarrow z=(\frac{\pi }{2}\pm 2n\pi )-ilog\left(10\pi \sqrt{11}\right)$
Wherer $n=0,1,\mathrm{2...}$
$\therefore sinz=10$ has infinite number of complex solutions.