Question

If \( \tan 27^\circ \cdot \tan 63^\circ = \sin A \), then the value of \(A\) is:

• A. \(27^\circ\)
• B. \(63^\circ\)
• C. \(90^\circ\)
• D. \(36^\circ\)

Hint:

\(\tan(90^\circ-\theta)=\cot\theta\). Hence \(\tan\theta\cdot\tan(90^\circ-\theta)=1\), a handy shortcut for many angle products.

Answer 1

The Correct Option is C

Step 1: Use co-function identity for tangent.
Since \(63^\circ = 90^\circ - 27^\circ\), we have \(\tan 63^\circ = \cot 27^\circ\).
Step 2: Evaluate the product.
\[ \tan 27^\circ \cdot \tan 63^\circ = \tan 27^\circ \cdot \cot 27^\circ = 1. \]
Step 3: Equate to \(\sin A\) and solve.
Given \( \tan 27^\circ \cdot \tan 63^\circ = \sin A \Rightarrow \sin A = 1 \).
Therefore \( A = 90^\circ \) (in the principal range).