Czy jest to poprawnie rozwiązane??
1.Wszystkie wartości z dla \(\displaystyle{ sinz=2}\)
\(\displaystyle{ sinz = \frac{e ^{iz}- e^{-iz} }{2i}}\)
\(\displaystyle{ 2 = \frac{e ^{iz}- e^{-iz} }{2i}}\)
\(\displaystyle{ 4i = e ^{iz} - e ^{-iz}}\)
\(\displaystyle{ e ^{iz} =t}\)
\(\displaystyle{ t^2-4it-1=0}\)
\(\displaystyle{ t _{1}, t_{2} = 2i \mp \sqrt{3}i}\)
\(\displaystyle{ e ^{iz} = 2i \mp \sqrt{3}}\)
\(\displaystyle{ i = e ^{i \frac{ \pi }{2} }}\)
\(\displaystyle{ e ^{iz} = e ^{i \frac{ \pi }{2} }(2 \mp \sqrt{3})}\)
\(\displaystyle{ e ^{iz} = e ^{i(x+iy)}=e ^{ix}e ^{-y}}\)
\(\displaystyle{ e ^{-y} = (2 \mp \sqrt{3}) \Rightarrow y = ln (2 \mp \sqrt{3})}\)
\(\displaystyle{ x= \frac{ \pi }{2}}\)
2.Rozwiąż:
\(\displaystyle{ |z+i|+|z-i|=4}\)
\(\displaystyle{ |x+iy+i|+|x+iy-i|=4}\)
\(\displaystyle{ \sqrt{x^2+(y+1)^2} + \sqrt{x^2+(y-1)^2}=4}\)
\(\displaystyle{ x^2+(y+1)^2 = 16 -x^2-(y-1)^2}\)
\(\displaystyle{ 2x^2+2y^2=14}\)
\(\displaystyle{ ( \frac{x}{ \sqrt{7} } )^2+( \frac{y}{ \sqrt{7} } )^2=1}\) - jest to elipsa