Question : If $\cot \theta=\frac{4}{3}, 0<\theta<\frac{\pi}{2}$, and $5 p \cos ^2 \theta \sin \theta=\cot ^2 \theta$, then the value of $p$ is:
Option 1: $\frac{7}{27}$
Option 2: $\frac{125}{27}$
Option 3: $\frac{5}{27}$
Option 4: $\frac{25}{27}$
Correct Answer: $\frac{25}{27}$
Solution :
Here, $\cot \theta = \frac{\text {Base}} {\text{Perpendicular}}$
Let Base = 4 units and Perpendicular = 3 units
Now, Using the Pythagoras theorem,
$H^2 = P^2 + B^2$
⇒ $H = \sqrt{(P2 + B2)}$
⇒ $H = \sqrt{(32 + 42)}$
⇒ $H = \sqrt{(9 + 16)}$
⇒ $H = \sqrt{(25)}$
⇒ H = 5
So, $\cos \theta = \frac{4}{5}$ and $\sin \theta = \frac{3}{5}$
Now, $5 p \cos ^2 \theta \sin \theta=\cot ^2 \theta$
⇒ $5p (\frac{4}{5})^2\frac{3}{5} = (\frac{4}{3})^2$
⇒ $5p (\frac{16}{25})(\frac{3}{5}) = (\frac{16}{9})$
⇒ $p (\frac{3}{25})= (\frac{1}{9})$
⇒ $p = (\frac{1}{9})(\frac{25}{3})$
⇒ $p = (\frac{25}{27})$
Hence, the correct answer is $\frac{25}{27}$.
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