If \(\tan 2x=\cot(2x-30^\circ)\), where \(2x\) and \(2x-30^\circ\) are acute angles, then what is the value of \(\sin 3x\)?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 1
The tangent of an angle equals the cotangent of its complement: \(\tan A=\cot(90^\circ-A)\). So for two acute angles, \(\tan A=\cot B\) means \(A+B=90^\circ\).
\(2x+(2x-30^\circ)=90^\circ\)
\(4x=120^\circ\), so \(x=30^\circ\)
Both angles are acute: \(2x=60^\circ\) and \(2x-30^\circ=30^\circ\)
\(\sin 3x=\sin 90^\circ=1\)