równanie trygonometryczne

[airowin]

\(\displaystyle{ sin^{2}x+sin^{2}2x=sin^{2}3x}\)

z góry thx

[herfoo]

\(\displaystyle{ sin^{2}x+sin^{2}2x=sin^{2}3x}\)
\(\displaystyle{ sin^{2}x+4sin^{2}x(1-sin^{2}x)=sin^{2}x(3-4sin^{2}x)^{2}}\)
\(\displaystyle{ sin^{2}x+4sin^{2}x-4sin^{4}x=sin^{2}x(9-24sin^{2}x+16sin^{4}x)}\)
\(\displaystyle{ 5sin^{2}x-4sin^{4}x=9sin^{2}x-24sin^{4}x+16sin^{6}x}\)
\(\displaystyle{ 4sin^{6}x-5sin^{4}x+sin^{2}x=0}\)
\(\displaystyle{ sin^{2}x(4sin^{4}x-5sin^{2}x+1)}\)
\(\displaystyle{ sin^{2}x=0 (4sin^{4}x-5sin^{2}x+1)=0}\)
\(\displaystyle{ x=k\cdot \pi k\in C}\)
\(\displaystyle{ sin^{2}x=t t\in}\)
\(\displaystyle{ 4t^{2}-5t+1=0}\)
\(\displaystyle{ \Delta_{t}=9}\)
\(\displaystyle{ \sqrt{\Delta}=3}\)
\(\displaystyle{ t_{1}=\frac{1}{4}}\)
\(\displaystyle{ t_{2}=1}\)
\(\displaystyle{ t_{1}=sin^{2}x}\)
\(\displaystyle{ sin^{2}x=\frac{1}{4} sinx=\frac{1}{2} sinx=-\frac{1}{2}}\)
\(\displaystyle{ x=\frac{\pi}{6}+2k\pi x=\frac{5}{6}\pi + 2k\pi x=\frac{7}{6}\pi + 2k\pi x=\frac{11}{6} \pi + 2k\pi}\)
\(\displaystyle{ sin^{2}x=1 sinx=1 sinx=-1}\)
\(\displaystyle{ x=\frac{\pi}{2}+k\pi}\)

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