Question

If $\sin \left(A\right)=\frac{3}{5}$, what is $\cos \left(A\right),\tan \left(A\right),cosec\left(A\right),\sec \left(A\right),\cot \left(A\right)$?

Answer 1

Compute all other trigonometric ratios:

Given:

$\sin \left(A\right)=\frac{3}{5}$ and $A$ is an acute angle.

So, in $△ABC$ we have $\angle B=90°$. And,

$AC=5$ and $BC=3$.

By Pythagoras theorem,

$\begin{array}{rcl}AB& =& \sqrt{A{C}^2-B{C}^2}\\ & =& \sqrt{5^2-3^2}\\ & =& \sqrt{16}\\ & =& 4\end{array}$

Now, we know

$\begin{array}{rcl}\cos \left(A\right)& =& \frac{AB}{AC}=\frac{4}{5}\\ \tan \left(A\right)& =& \frac{BC}{AB}=\frac{3}{4}\\ \cot \left(A\right)& =& \frac{1}{\tan \left(A\right)}=\frac{4}{3}\\ \sec \left(A\right)& =& \frac{1}{\cos \left(A\right)}=\frac{5}{4}\\ cosec\left(A\right)& =& \frac{1}{\sin \left(A\right)}=\frac{5}{3}\end{array}$

Hence, the required values are obtained.