Question

If \( \sin 3A = \cos (A - 26^\circ) \), where \( 3A \) is an acute angle, find the value of \( A \).

Hint:

Use trigonometric identities like \( \sin \theta = \cos (90^\circ - \theta) \) to simplify equations involving both sine and cosine.

Answer 1

We are given: \[ \sin 3A = \cos (A - 26^\circ). \] Using the identity \( \sin \theta = \cos (90^\circ - \theta) \), we can rewrite \( \sin 3A \) as: \[ \sin 3A = \cos (90^\circ - 3A). \] Thus, the equation becomes: \[ \cos (90^\circ - 3A) = \cos (A - 26^\circ). \] Since \( \cos \theta_1 = \cos \theta_2 \), we know that: \[ 90^\circ - 3A = A - 26^\circ. \] Step 1: Solve the equation. Rearranging the terms: \[ 90^\circ + 26^\circ = 3A + A \quad \implies \quad 116^\circ = 4A \quad \implies \quad A = \frac{116^\circ}{4} = 29^\circ. \]
Conclusion:
The value of \( A \) is \( 29^\circ \).