Question

If $\cos x=-\frac{3}{5},x$ lies in the third quadrant, find the values of other five trigonometric functions.

Answer 1

Step 1: Solve for value of $\sin x$
$\cos x=-\frac{3}{5}$ and $x$ lies in $3^{rd}$ quadrant.
$\Rightarrow {\sin }^{2}x+{\left(\frac{-3}{5}\right)}^2=1$
$\Rightarrow {\sin }^{2}x+\frac{9}{25}=1$
$\Rightarrow {\sin }^{2}x=1-\frac{9}{25}$
$\Rightarrow {\sin }^{2}x=\frac{16}{25}$
$\Rightarrow \sin x=\pm \frac{4}{5}$
Since, $x$ is in $3^{rd}$ quadrant and $\sin x$ is negative $3^{rd}$ quadrant.
$\therefore \sin x=\frac{-4}{5}$

Step 2: Solve for value of $\tan x$
We know that $\tan x=\frac{\sin x}{\cos x}$
$\therefore \tan x=\frac{\left(\frac{-4}{5}\right)}{\left(\frac{-3}{5}\right)}=\frac{4}{3}$

Step 3: Solve for value of $\text{cosec}x$
We know that $\text{cosec}x=\frac{1}{\sin x}$
$\therefore \text{cosec}x=\frac{1}{\frac{-4}{5}}=\frac{-5}{4}$

Step 4: Solve for value of $\sec x$
We know that $\sec x=\frac{1}{\cos x}$
$\therefore \sec x=\frac{1}{\frac{-3}{5}}=\frac{-5}{3}$
$\tan x=\frac{4}{3}$

Step 5: Solve for value of $\cot x$
We know that $\cot x=\frac{1}{\tan x}$
$\therefore \cot x=\frac{1}{\frac{4}{3}}=\frac{3}{4}$