Question

If $0^o<x^o<{90}^o$ and $\cos x=\frac{3}{\sqrt{10}}$, then compute $\log \sin x+\log \cos x+\log \tan x$.

Answer 1

$\cos x=\frac{3}{\sqrt{10}}$

$\sin x=\sqrt{1-{\cos }^{2}x}$

$=\sqrt{1-\frac{9}{10}}$

$=\sqrt{\frac{1}{10}}$

$=\frac{1}{\sqrt{10}}$

$\tan x=\frac{\sin x}{\cos x}=\frac{\frac{1}{\sqrt{10}}}{\frac{3}{\sqrt{10}}}=\frac{1}{3}$

$\log \sin x+\log \cos x+\log \tan x$

$=\log \left(\frac{1}{\sqrt{10}}\right)+\log \left(\frac{3}{\sqrt{10}}\right)+\log \left(\frac{1}{3}\right)$

$=\log \left(\frac{1}{\sqrt{10}}\times \frac{3}{\sqrt{10}}\times \frac{1}{3}\right)$ $\left(\log a+\log b=\log ab\right)$

$=\log \left(\frac{1}{10}\right)$

$=\log 1-\log 10$

$=-\log 10$ $(\log \frac{a}{b}=\log a-\log b)$