Question

Решите уравнение $81^{sin^{2} x} +81^{cos^{2}x } =30$

Answer 1

Ответ:

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Объяснение:

пусть

$81^{sin^{2} x} =t \\81^{1-sin^{2} x} = \frac{81}{81^{sin^{2} x}} =\frac{81}{t}$

t+81\t=30 ,t²-30t+81 =0 ,D=900-324=576=24²

t1=27 , t2=3

1.

$27=3^{3} , 81^{sin^{2} x} =t => 3^{4*sin^{2} x} =3^{3} \\4*sin^{2} x=3 , sin^{2} x=\frac{3}{4} , \\sin x=\frac{\sqrt{3} }{2} =>x=\frac{\pi }{3} +2\pi n , x=\frac{2\pi }{3} +2\pi n\\,\\\\sin x=-\frac{\sqrt{3} }{2} =>x=-\frac{\pi }{3} +2\pi m , x=-\frac{2\pi }{3} +2\pi m$

2.

$81^{sin^{2} x} =t => 3^{4*sin^{2} x} =3^{1} \\4*sin^{2} x=1 , sin^{2} x=\frac{1}{4} , \\sin x=\frac{1 }{2} =>x=\frac{\pi }{6} +2\pi n , x=\frac{5\pi }{6} +2\pi n\\,\\\\sin x=-\frac{1}{2} =>x=-\frac{\pi }{6} +2\pi m , x=-\frac{5\pi }{6} +2\pi m$