First slide

Compound angles

Question

$I n a Δ P Q R, i f 3 \sin P + 4 \cos Q = 6 a n d 4 \sin Q + 3 \cos P = 1$ then the acute angle $R$ is equal to

Moderate

A

$\frac{π}{6}$
B

$\frac{5 π}{6}$
C

$\frac{3 π}{4}$
D

$\frac{π}{4}$

Solution

$W e h a v e 3 \sin P + 4 \cos Q = 6, 4 \sin Q + 3 \cos P = 1 . S q u a r i n g o n b o t h s i d e s a n d a d d i n g, w e g e t 9 (\sin^{2} P + \cos^{2} P) + 16 (\sin^{2} Q + \cos^{2} Q) + 24 (\sin P \cos Q + \cos P \sin Q) = 37 \Rightarrow 24 \sin (P + Q) = 37 - 25 = 12 \Rightarrow \sin (π - R) = \frac{1}{2} (∵ P + Q + R = π) \Rightarrow \sin R = \frac{1}{2} \Rightarrow R = \frac{π}{6} (∵ R i s a c u t e)$

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