Question

If $\text{cosec}\theta =\frac{5}{4}$, then find the values of $\cot \theta ,\sin \theta ,\cos \theta$.

Answer 1

Given $\text{cosec}\theta =\frac{5}{4}$
Since, $\sin \theta =\frac{1}{\text{cosec}\theta }$
$\Rightarrow \sin \theta =\frac{1}{\frac{5}{4}}$
$\therefore \sin \theta =\frac{4}{5}$
As we know that ${\sin }^2\theta +{\cos }^2\theta =1$
$\Rightarrow \cos \theta =\sqrt{1-{\sin }^2\theta }$
$=\sqrt{1-(\frac{4}{5}{)}^2}$
$=\sqrt{1-\frac{16}{25}}$
$=\sqrt{\frac{25-16}{25}}$
$=\sqrt{\frac{9}{25}}$
$\therefore \cos \theta =\frac{3}{5}$
Thus, $\cot \theta =\frac{\cos \theta }{\sin \theta }$

$\Rightarrow \cot \theta =\frac{\frac{3}{5}}{\frac{4}{5}}$
$=\frac{3}{5}\times \frac{5}{4}$

$\therefore \cot \theta =\frac{3}{4}$