Question

If $\sin \left(A\right)=\frac{3}{4}$ , calculate $\sec \left(A\right)$?

Answer 1

Computing the required values:

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 $△ABC$.

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

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

By definition,

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

Hence, the required value is $\frac{4}{\sqrt{7}}$.