Question

Prove the following: If $\tan A=\frac{3}{4}$, then $\sin A·\cos A=\frac{12}{25}$

Answer 1

Proof:

A right angle triangle $∆ABC$ is shown,

By using the Pythagoras theorem,

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

The trigonometric ratios $\sin A\mathrm{and}\cos A$ are computed as,

$$\begin{gathered} \sin A=\frac{\mathrm{Perpendicular}}{\mathrm{Hypotenuse}} \\ =\frac{BC}{AC} \\ =\frac{3}{5} \\ \cos A=\frac{\mathrm{Base}}{\mathrm{Hypotenuse}} \\ =\frac{AB}{AC} \\ =\frac{4}{5} \end{gathered}$$

$$\begin{gathered} \therefore \sin A·\cos A=\frac{3}{5}·\frac{4}{5} \\ =\frac{12}{25} \end{gathered}$$

Hence proved.