Question

If \(\cos A = \frac{1}{2}\), then the value of \(\sin^{2} A + 2 \cos^{2} A\) is :

• A. \(\frac{3}{2}\)
• B. \(\frac{5}{4}\)
• C. \(-1\)
• D. \(\frac{1}{2}\)

Hint:

You can also identify that if \(\cos A = 1/2\), then \(A = 60^{\circ}\).
Then \(\sin^{2} 60^{\circ} + 2 \cos^{2} 60^{\circ} = (\sqrt{3}/2)^{2} + 2(1/2)^{2} = 3/4 + 2/4 = 5/4\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We can solve this by either finding the angle \(A\) or using trigonometric identities.
Step 2: Key Formula or Approach:
Identity: \(\sin^{2} A = 1 - \cos^{2} A\).
Step 3: Detailed Explanation:
Given \(\cos A = \frac{1}{2}\).
Then \(\cos^{2} A = \left(\frac{1}{2}\right)^{2} = \frac{1}{4}\).
Using the identity, \(\sin^{2} A = 1 - \frac{1}{4} = \frac{3}{4}\).
Now, calculate the required expression:
\[ \text{Value} = \sin^{2} A + 2 \cos^{2} A \]
\[ \text{Value} = \frac{3}{4} + 2 \left( \frac{1}{4} \right) \]
\[ \text{Value} = \frac{3}{4} + \frac{2}{4} = \frac{5}{4} \]
Step 4: Final Answer:
The value of the expression is \(\frac{5}{4}\).