The angles \(4A-10^\circ\) and \(A+25^\circ\) are both acute, and \(\sin(4A-10^\circ)=\cos(A+25^\circ)\). Find \(A\).
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(15^\circ\)
Since \(\sin(4A-10^\circ)=\cos(100^\circ-4A)\), the condition says that two acute angles have the same cosine. Two acute angles with equal cosines are equal, so \(100^\circ-4A=A+25^\circ\).
\(\cos(100^\circ-4A)=\cos(A+25^\circ)\), so \(100^\circ-4A=A+25^\circ\)
\(5A=75^\circ\), so \(A=15^\circ\)
Check: \(4A-10^\circ=50^\circ\) and \(A+25^\circ=40^\circ\) are both acute and add up to \(90^\circ\)