Question

If $\sin \left(A\right)=\frac{1}{3}$ find the value of $\tan \left(A\right)$.

Answer 1

Step:Find the base value.

Given: $\sin \left(A\right)=\frac{1}{3}\to \left(1\right)$

As we know, by $\sin$ definition;

$\sin \left(A\right)=\frac{P}{H}=\frac{1}{3}\to \left(2\right)$

By comparing equation $\left(1\right)$ and $\left(2\right)$, we have,

Perpendicular, $P=1$ and Hypotenuse , $H=3$

On using Pythagoras theorem in $△ABC$, we get

$$\begin{gathered} H^2=P^2+B^2 \\ \Rightarrow B^2=3^2-1^2 \\ \Rightarrow B=\sqrt{8} \\ \Rightarrow B=2\sqrt{2} \end{gathered}$$

Hence, Base is $2\sqrt{2}$.

Step:Computing the required value.

By definition,

$$\begin{gathered} \tan \left(A\right)=\frac{P}{B} \\ \Rightarrow \tan \left(A\right)=\frac{1}{2\sqrt{2}} \end{gathered}$$

Hence, the required value is $\tan \left(A\right)=\frac{1}{2\sqrt{2}}$.