Question

Evaluate : \(\frac{3 \cos^2 30^{\circ} - 6 \csc^2 30^{\circ}}{\tan^2 60^{\circ}}\).

Hint:

Double check the square terms! It's common to forget to square the numbers in the denominator of the fraction (like squaring \(2\) to get \(4\) in \(\cos^2 30^{\circ}\)).

Answer 1

Step 1: Understanding the Concept:
Substitute the standard values for trigonometric ratios of \(30^{\circ}\) and \(60^{\circ}\) into the expression.
Step 2: Key Formula or Approach:
\(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\).
\(\csc 30^{\circ} = \frac{1}{\sin 30^{\circ}} = \frac{1}{1/2} = 2\).
\(\tan 60^{\circ} = \sqrt{3}\).
Step 3: Detailed Explanation:
Substitute the values into the expression:
\[ \text{Expression} = \frac{3 \left( \frac{\sqrt{3}}{2} \right)^2 - 6(2)^2}{(\sqrt{3})^2} \]
Simplify the numerator and denominator:
\[ = \frac{3 \left( \frac{3}{4} \right) - 6(4)}{3} \]
\[ = \frac{\frac{9}{4} - 24}{3} \]
Find a common denominator for the numerator:
\[ = \frac{\frac{9 - 96}{4}}{3} \]
\[ = \frac{-\frac{87}{4}}{3} \]
Divide by 3:
\[ = -\frac{87}{4 \times 3} = -\frac{29}{4} \]
Step 4: Final Answer:
The evaluated value is \(-\frac{29}{4}\).