Question

Two circles of radii $4$ cm and $1$ cm touches each other externally and $\theta$ is the angle contained by their direct common tangents. If $\sin \theta =\frac{4b}{25}$
Find $b$.

Answer 1

From Fig., we have
$\sin \frac{\theta }{2}=\frac{3}{5}$
$\cos \frac{\theta }{2}=\frac{4}{5}$
$\therefore$ $\sin \theta =2\times \frac{3}{5}\times \frac{4}{5}=\frac{24}{25}$
Ans: 6