Question : If $0 \leq \theta \leq 90^{\circ}$, and $\sin \left(2 \theta+50^{\circ}\right)=\cos \left(4 \theta+16^{\circ}\right)$, then what is the value of $\theta$ (in degrees)?
Option 1: $10^{\circ}$
Option 2: $8^{\circ}$
Option 3: $4^{\circ}$
Option 4: $12^{\circ}$
Correct Answer: $4^{\circ}$
Solution :
Given,
$\sin \left(2 \theta+50°\right)=\cos \left(4 \theta+16°\right)$
We know, $\sin(90°-\theta)=\cos\theta$
So, $\sin \left(2 \theta+50°\right)=\sin(90°-(4 \theta+16°)$
⇒ $2\theta+50°=90°-4\theta-16°$
⇒ $2\theta+4\theta=90°-16°-50°$
⇒ $6\theta=24°$
$\therefore\theta=4°$
Hence, the correct answer is $4°$.
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