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:

Using trigonometric identities to simplify the expression before substituting values is highly recommended.
It reduces fractional arithmetic and prevents potential radical calculation mistakes.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given that $\cos A = \frac{1}{2}$. We need to calculate the numerical value of the trigonometric expression $\sin^2 A + 2\cos^2 A$.

Step 2: Key Formula or Approach:
We can approach this problem in two different ways:
Method 1: Express the entire expression in terms of $\cos A$ using the identity $\sin^2 A = 1 - \cos^2 A$. This is faster as we only need to plug in the given value of $\cos A$.
Method 2: Identify the angle $A$. Since $\cos A = \frac{1}{2}$, for an acute angle $A$, $A = 60^\circ$. We can then compute $\sin 60^\circ$ and evaluate the expression.

Step 3: Detailed Explanation:
Let us use Method 1 (Algebraic substitution):

• Write down the expression to be evaluated:
\[ E = \sin^2 A + 2\cos^2 A \]

• Substitute the trigonometric identity $\sin^2 A = 1 - \cos^2 A$ into the expression:
\[ E = (1 - \cos^2 A) + 2\cos^2 A \]

• Simplify the terms:
\[ E = 1 + \cos^2 A \]

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

• Calculate the square of the fraction:
\[ E = 1 + \frac{1}{4} \]

• Add the numbers using a common denominator:
\[ E = \frac{4}{4} + \frac{1}{4} = \frac{5}{4} \]

Let us verify using Method 2 (Standard angle values):

• Since $\cos A = \frac{1}{2}$, the angle is $A = 60^\circ$.

• The value of $\sin A$ is:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]

• Substitute these values into the original expression:
\[ E = \left(\frac{\sqrt{3}}{2}\right)^2 + 2\left(\frac{1}{2}\right)^2 \]
\[ E = \frac{3}{4} + 2\left(\frac{1}{4}\right) \]
\[ E = \frac{3}{4} + \frac{2}{4} = \frac{5}{4} \]

Both methods consistently yield the same result.

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