Question

If \(\cot 2A = \tan (A - 22^\circ)\), where \(2A\) is an acute angle, find the value of \(A\).

Hint:

Use complementary angle relationships: \(\cot \theta = \tan (90^\circ - \theta)\).

Answer 1

Step 1: Recall the identity.
\[ \cot \theta = \tan (90^\circ - \theta) \] So, \[ \tan (A - 22^\circ) = \cot 2A = \tan (90^\circ - 2A) \]
Step 2: Equate the angles.
\[ A - 22^\circ = 90^\circ - 2A \]
Step 3: Simplify.
\[ 3A = 112^\circ \implies A = \frac{112^\circ}{3} = 37^\circ 20' \]
Step 4: Conclusion.
Hence, the value of \(A\) is \(\boxed{37^\circ 20'}\).