Question

If $\sin \left(A\right)=\frac{3}{4}$, Calculate $\cos \left(A\right)$ and $\tan \left(A\right)$?

Answer 1

Step:1 Find the base value.

Given: $\sin \left(A\right)=\frac{3}{4}\to \left(1\right)$

We know, by $\sin$ definition;

$\sin \left(A\right)=\frac{P}{H}=\frac{3}{4}\to \left(2\right)$

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

Perpendicular $P=3$, and Hypotenuse $H=4$

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

$$\begin{gathered} H^2=P^2+B^2 \\ \Rightarrow B^2=4^2-3^2 \\ \Rightarrow B=\sqrt{7} \end{gathered}$$

Hence, Base $B=\sqrt{7}$.

Step:2 Computing the required value

By definition,

$$\begin{gathered} \cos \left(A\right)=\frac{B}{H} \\ \Rightarrow \cos \left(A\right)=\frac{\sqrt{7}}{4} \end{gathered}$$

Also, $\tan \left(A\right)=\frac{P}{B}$.

$\Rightarrow \mathrm{ta}n\left(A\right)=\frac{3}{\sqrt{7}}$

Hence, the required values are $\cos \left(A\right)=\frac{\sqrt{7}}{4}$ and $\tan \left(A\right)=\frac{3}{\sqrt{7}}$.