$\sin 5 x=\cos x$

Step 1 ：Rewrite the equation as \(\sin 5x = \sin (90 - x)\)

Step 2 ：Find the solutions for the equation: \(x = -i \log(-(1 - \sqrt{3}i)e^{15i}/2)\), \(x = -i \log((1 - \sqrt{3}i)e^{15i}/2)\), \(x = -i \log(-(1 + \sqrt{3}i)e^{15i}/2)\), \(x = -i \log((1 + \sqrt{3}i)e^{15i}/2)\), \(x = -i \log(-\sin(\pi/4 + 45/2) - i\cos(\pi/4 + 45/2))\), \(x = -i \log(\sin(\pi/4 + 45/2) + i\cos(\pi/4 + 45/2))\), \(x = -i \log(-e^{15i})\), \(x = -i \log(-e^{-i(\pi + 90)/4})\), \(x = -45/2 + 31\pi/4\), and \(x = 15 - 4\pi\)

Step 3 ：\(\boxed{x = -i \log(-(1 - \sqrt{3}i)e^{15i}/2), x = -i \log((1 - \sqrt{3}i)e^{15i}/2), x = -i \log(-(1 + \sqrt{3}i)e^{15i}/2), x = -i \log((1 + \sqrt{3}i)e^{15i}/2), x = -i \log(-\sin(\pi/4 + 45/2) - i\cos(\pi/4 + 45/2)), x = -i \log(\sin(\pi/4 + 45/2) + i\cos(\pi/4 + 45/2)), x = -i \log(-e^{15i}), x = -i \log(-e^{-i(\pi + 90)/4}), x = -45/2 + 31\pi/4, x = 15 - 4\pi}\)