Question : PQR is a triangle right-angled at Q and PQ : QR = 3 : 4. What is the value of $\sin P + \sin Q + \sin R?$
Option 1: $\frac{4}{5}$
Option 2: $\frac{3}{5}$
Option 3: $\frac{12}{5}$
Option 4: $\frac{2}{5}$
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Correct Answer: $\frac{12}{5}$
Solution :
Given: $\angle PQR = 90°$ and $PQ:QR = 3:4$
Let, $PQ = 3k$, $QR = 4k$.
As $PR$ is the hypotenuse, $PR = \sqrt{(3k^{2} + 4k^{2})}=5k$
$\sin P = \frac{QR}{PR} = \frac{4k}{5k}=\frac{4}{5}$
$\sin Q = \sin 90° = 1$
$\sin R = \frac{PQ}{PR} = \frac{3k}{5k}=\frac{3}{5}$
So, $\sin P + \sin Q + \sin R =\frac{4}{5} + 1 + \frac{3}{5} = \frac{12}{5}$
Hence, the correct answer is $\frac{12}{5}$.
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