Question : If $\tan\theta+\sec\theta=3$, $\theta$ being acute, the value of $5\sin\theta$ is:
Option 1: $\frac{5}{2}$
Option 2: $\frac{\sqrt{3}}{5}$
Option 3: $\frac{5}{\sqrt{3}}$
Option 4: $4$
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Correct Answer: $4$
Solution :
Given: $\theta$ being acute.
$⇒\tan\theta+\sec\theta=3$ ... (1)
We know the identity,
$⇒\sec^2\theta-\tan^2\theta=1$
$⇒(\tan\theta+\sec\theta)( \sec\theta-\tan\theta)=1$
$⇒\sec\theta-\tan\theta=\frac{1}{3}$ ... (2)
Solving equation (1) and (2),
$⇒2\sec\theta=3+\frac{1}{3}$
$⇒2\sec\theta=\frac{10}{3}$
$\therefore\sec\theta=\frac{5}{3}$
$⇒\cos \theta=\frac{3}{5}$
$⇒ {\sin \theta=\sqrt{1-\cos^2\theta}=\sqrt{1-\frac{9}{25}}=\sqrt{\frac{16}{25}}=\frac{4}{5}}$
$\therefore 5\sin \theta=5×\frac{4}{5}=4$
Hence, the correct answer is $4$.
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