Question

Find the value of other five trigonometric functions if cot x = $\frac{3}{4}$, and x lies in third quadrant.

Answer 1

Here cot x = $\frac{3}{4}$
tan x = $\frac{1}{cotx}=\frac{4}{3}$

Now ${\text{sec}}^{2}x=1+{\text{tan}}^2$ x

$\Rightarrow {\text{sec}}^{2}x=1+{\left(\frac{4}{3}\right)}^2$

$\Rightarrow {\text{sec}}^{2}x=1+\frac{16}{9}$

$\Rightarrow {\text{sec}}^{2}x=\frac{25}{9}\Rightarrow$ sec x = $\pm \frac{5}{3}$

But x lies third quadrant.

$\therefore$ sec x = $\frac{-5}{3}$

cos x = $\frac{1}{secx}=\frac{-3}{5}$

Also $si{n}^{2}x+co{s}^2$ x =1

$\Rightarrow si{n}^{2}x+{\left(\frac{-3}{5}\right)}^2=1$

$\Rightarrow si{n}^{2}x=1-\frac{9}{25}$

$\Rightarrow si{n}^{2}x=\frac{16}{25}\Rightarrow sinx=\pm \frac{4}{5}$

But x lies in third quadrant.

$\therefore sinx=\frac{-4}{5}$

$\therefore$ cosec x = $\frac{1}{sinx}=\frac{-5}{4}$