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Question Number 198197 by necx122 last updated on 13/Oct/23

$$pleasehelpe$$$$sinz=2.Findz$$

Commented by mokys last updated on 13/Oct/23

Answered by mr W last updated on 13/Oct/23

$$e^{iz}=\cos z+i\sin z$$$$e^{-iz}=\cos z-i\sin z$$$$\Rightarrow e^{iz}-e^{-iz}=2i\sin z$$$$\Rightarrow \sin z=\frac{e^{iz}-e^{-iz}}{2i}$$$$\sin z=2$$$$\Rightarrow \frac{e^{iz}-e^{-iz}}{2i}=2$$$$\Rightarrow {\left(e^{iz}\right)}^2-4i\left(e^{iz}\right)-1=0$$$$\Rightarrow e^{iz}=(2\pm \sqrt{3})i=(2\pm \sqrt{3})e^{(2k\pi +\frac{\pi }{2})i}$$$$\Rightarrow iz=\ln (2\pm \sqrt{3})+(2k\pi +\frac{\pi }{2})i$$$$\Rightarrow z=2k\pi +\frac{\pi }{2}\pm i\ln (2+\sqrt{3})$$

Commented by necx122 last updated on 14/Oct/23

$$Thankyousomuch$$

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