Question

Compute \(\arcsin \left(\sin 66^{\circ}-\sin 54^{\circ}\right)\) in degrees.

Answer 1

From the sum-to-product formula,
\[
\sin x-\sin z=2 \sin \frac{x-z}{2} \cos \frac{x+z}{2} .
\]
Applying this with \(x=66^{\circ}\) and \(z=54^{\circ}\), we have
\[
\begin{aligned}
\arcsin \left(\sin 66^{\circ}-\sin 54^{\circ}\right) &=\arcsin \left(2 \sin \frac{66^{\circ}-54^{\circ}}{2} \cos \frac{66^{\circ}+54^{\circ}}{2}\right) \\
&=\arcsin \left(2 \sin 6^{\circ} \cos 60^{\circ}\right) \\
&=\arcsin \left(\sin 6^{\circ}\right) \\
&=6^{\circ} .
\end{aligned}
\]
Final Answer: The final answer is \(6^{\circ}\)