Question

If $\cot \theta =\frac{3}{4},$ and $\theta$ is acute angle, then find $\csc \theta$

Answer 1

$\cot \theta =\frac{3}{4}$ is positive.

$\therefore \theta$ lies in first or third quadrant.

We know that, $1+{\cot }^2\theta ={\csc }^2\theta$

$\Rightarrow {\cot }^2\theta =\frac{9}{16}$

$\Rightarrow {\csc }^2\theta -1=\frac{9}{16}$

$\Rightarrow {\csc }^2\theta =\frac{9}{16}+1=\frac{9+16}{16}=\frac{25}{16}$

since $\theta$ lies in the first or third quadrant and $\csc \theta$ is negative in the third quadrant and positive in the first quadrant.

$\Rightarrow \csc \theta =\pm \frac{5}{4}$