Question

If in a $∆\mathrm{ABC}$, $a=6\mathrm{cm},b=8\mathrm{cm},c=10\mathrm{cm}$, then $\sin 2A$ is

• A. $\frac{6}{25}$
• B. $\frac{8}{25}$
• C. $\frac{10}{25}$
• D. $\frac{24}{25}$

Answer 1

The correct option is D

$\frac{24}{25}$

Explanation for the correct option.

Find the value of $\sin 2A$:

Given, $a=6\mathrm{cm},b=8\mathrm{cm},c=10\mathrm{cm}$.

From cosine rule,

$\begin{array}{rcl}\cos A& =& \frac{\left(b^2+c^2-a^2\right)}{2bc}\\ & =& \frac{\left(8^2+{10}^2-6^2\right)}{2\times 8\times 10}\\ & =& \frac{\left(64+100-36\right)}{160}\\ & =& \frac{128}{160}\\ & =& \frac{4}{5}\end{array}$

We know that,

$\begin{array}{rcl}{\sin }^{2}A& =& 1-{\cos }^{2}A\\ \sin A& =& \sqrt{\left(1-{\left(\frac{4}{5}\right)}^2\right)}\\ & =& \sqrt{1-\frac{16}{25}}\\ & =& \frac{3}{5}\end{array}$

Now,

$\begin{array}{rcl}\sin 2A& =& 2\sin A\cos A\\ & =& 2\times \frac{3}{5}\times \frac{4}{5}\\ & =& \frac{24}{25}\end{array}$

Hence, the correct option is D.