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:

Simplifying the expression analytically before substituting numerical values is highly recommended.
By changing $\sin^2 A + 2\cos^2 A$ to $1 + \cos^2 A$, you avoid computing $\sin A$ entirely, saving valuable time and reducing the chances of computational errors.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The topic is Trigonometric Identities and Evaluation.
We are given the value of $\cos A$ and need to find the value of the algebraic expression $\sin^2 A + 2\cos^2 A$.

Step 2: Key Formula or Approach:
We can simplify the given expression first using the basic Pythagorean identity:
\[ \sin^2 A + \cos^2 A = 1 \implies \sin^2 A = 1 - \cos^2 A \]
This reduces the entire expression to depend only on $\cos A$, eliminating the need to calculate $\sin A$ separately.
Alternatively, we can find the value of $\sin^2 A$ directly using $\sin^2 A = 1 - \cos^2 A = 1 - \left(\frac{1}{2}\right)^2$ and substitute both values.

Step 3: Detailed Explanation:

• Let us rewrite the expression to make calculation simpler:
Let $E = \sin^2 A + 2\cos^2 A$
Split $2\cos^2 A$ into $\cos^2 A + \cos^2 A$:
\[ E = \sin^2 A + \cos^2 A + \cos^2 A \]

• Use the identity $\sin^2 A + \cos^2 A = 1$:
\[ E = 1 + \cos^2 A \]

• Now substitute the given value $\cos A = \frac{1}{2}$ into this simplified expression:
\[ E = 1 + \left(\frac{1}{2}\right)^2 \]
\[ E = 1 + \frac{1}{4} \]

• Solve the addition:
\[ E = \frac{4 + 1}{4} = \frac{5}{4} \]


Step 4: Final Answer:
The value of the expression is $\frac{5}{4}$, which corresponds to option (B).