Rownanie trygonometryczne

[ptasiek123]

a)
\(\displaystyle{ \left( \frac{4}{9} \right) ^{2\sin ^{2}x }+ \left( \frac{2}{3} \right) ^{4\cos ^{2}x } = \frac{26}{27}}\)
b)
\(\displaystyle{ 2 ^{1+2\log _{2}\cos x } - \frac{3}{4}=9 ^{0,5+\log _{3}\sin x }}\)

[chlorofil]

a)
\(\displaystyle{ \left( \frac{4}{9}\right)^{2\sin ^2 x} + \left( \frac{2}{3}\right)^{4\cos ^2 x} = \frac{26}{27}\\
\left( \frac{2}{3}\right)^{4\sin ^2 x} + \left( \frac{2}{3}\right)^{4-4\sin ^2 x} = \frac{26}{27}\\
\left( \frac{2}{3}\right)^{4\sin ^2 x} + \frac{\left(\frac{2}{3}\right)^4}{\left( \frac{2}{3}}\right)^{4\sin ^2 x}} = \frac{26}{27}}\)

I teraz wystarczy podstawić: \(\displaystyle{ t=\left( \frac{2}{3}\right)^{4\sin ^2 x}}\)

b) Wsk.:
\(\displaystyle{ a^{\log _a x} = x\\
k \cdot \log _a x = \log _a x^k}\)