Jika 0<x<(pi)/(2) dan 2 sin ^(2) x+cos ^(2) x=(34)/(25)

\(\tan x = \frac{3}{4}\)

Pembahasan

Kita diberikan persamaan trigonometri \(2 \sin^2 x + \cos^2 x = \frac{34}{25}\) dengan rentang \(0 < x < \frac{\pi}{2}\). Kita perlu mencari nilai \(\tan x\).

Menggunakan identitas \(\sin^2 x + \cos^2 x = 1\), kita bisa mengganti \(\cos^2 x = 1 - \sin^2 x\) ke dalam persamaan:
\(2 \sin^2 x + (1 - \sin^2 x) = \frac{34}{25}\)
\(\sin^2 x + 1 = \frac{34}{25}\)
\(\sin^2 x = \frac{34}{25} - 1\)
\(\sin^2 x = \frac{34 - 25}{25}\)
\(\sin^2 x = \frac{9}{25}\)

Karena \(0 < x < \frac{\pi}{2}\), \(\sin x\) bernilai positif. Maka:
\(\sin x = \sqrt{\frac{9}{25}} = \frac{3}{5}\)

Selanjutnya, kita bisa mencari \(\cos x\) menggunakan \(\sin^2 x + \cos^2 x = 1\):
\((\frac{3}{5})^2 + \cos^2 x = 1\)
\(\frac{9}{25} + \cos^2 x = 1\)
\(\cos^2 x = 1 - \frac{9}{25}\)
\(\cos^2 x = \frac{16}{25}\)

Karena \(0 < x < \frac{\pi}{2}\), \(\cos x\) juga bernilai positif. Maka:
\(\cos x = \sqrt{\frac{16}{25}} = \frac{4}{5}\)

Terakhir, kita hitung \(\tan x\):
\(\tan x = \frac{\sin x}{\cos x} = \frac{3/5}{4/5} = \frac{3}{4}\)