If f(x) = 2 sinx, `g(x) = cos^(2) x`, then the value of `(f+g)((pi)/(3))`

Options

A. 1
B. `(2sqrt(3)+1)/(4)`
C. `sqrt(3)+(1)/(4)`
D. 0

Answer

To solve the problem, we need to find the value of \((f + g)\left(\frac{\pi}{3}\right)\) given that \(f(x) = 2 \sin x\) and \(g(x) = \cos^2 x\).

### Step-by-Step Solution:

1. **Calculate \(f\left(\frac{\pi}{3}\right)\)**:
\[ f\left(\frac{\pi}{3}\right) = 2 \sin\left(\frac{\pi}{3}\right) \]
We know that \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\).
Therefore,
\[ f\left(\frac{\pi}{3}\right) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \]

2. **Calculate \(g\left(\frac{\pi}{3}\right)\)**:
\[ g\left(\frac{\pi}{3}\right) = \cos^2\left(\frac{\pi}{3}\right) \]
We know that \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\).
Therefore,
\[ g\left(\frac{\pi}{3}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

3. **Add \(f\left(\frac{\pi}{3}\right)\) and \(g\left(\frac{\pi}{3}\right)\)**:
\[ (f + g)\left(\frac{\pi}{3}\right) = f\left(\frac{\pi}{3}\right) + g\left(\frac{\pi}{3}\right) \]
Substituting the values we calculated:
\[ (f + g)\left(\frac{\pi}{3}\right) = \sqrt{3} + \frac{1}{4} \]

4. **Final Result**:
The value of \((f + g)\left(\frac{\pi}{3}\right)\) is:
\[ \sqrt{3} + \frac{1}{4} \]