Question

If \( \dfrac{A}{5} = 12^\circ \), then the value of \( 3\csc^{2} A \) will be:

• A. \(2\sqrt{3}\)
• B. \(3\)
• C. \(4\)
• D. \(4\sqrt{3}\)

Hint:

Convert the given angle first; with special angles like \(30^\circ, 45^\circ, 60^\circ\), use exact trig values to simplify quickly.

Answer 1

The Correct Option is C

Step 1: Find angle \(A\).
\(\dfrac{A}{5}=12^\circ \Rightarrow A=60^\circ.\)
Step 2: Evaluate \( \csc^2 A \).
\(\sin 60^\circ=\dfrac{\sqrt{3}}{2} \Rightarrow \csc 60^\circ=\dfrac{1}{\sin 60^\circ}=\dfrac{2}{\sqrt{3}}.\)
Hence, \(\csc^2 60^\circ=\left(\dfrac{2}{\sqrt{3}}\right)^2=\dfrac{4}{3}.\)
Step 3: Multiply by 3.
\(3\csc^2 60^\circ=3\cdot\dfrac{4}{3}=4.\)