Question

If $\sin \left(A\right)=\frac{3}{5}$, find all other trigonometric ratios of angle $A$ (using trigonometric identities).

Answer 1

Compute all the trigonometric ratios:

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

Let's plot the figure.

So, in $△ABC$ we have $\angle B=90°$.

And,

$AC=5$ and $BC=3$

By Pythagoras theorem,

$$\begin{gathered} AB=\sqrt{A{C}^2-B{C}^2} \\ =\sqrt{5^2-3^2} \\ =\sqrt{25-9} \\ =\sqrt{16} \\ =4 \end{gathered}$$

Now, we know that,

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

Hence, the required values are obtained.