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 find the value of \((f + g)\left(\frac{\pi}{3}\right)\), we first need to evaluate \(f\left(\frac{\pi}{3}\right)\) and \(g\left(\frac{\pi}{3}\right)\).

### Step 1: Calculate \(f\left(\frac{\pi}{3}\right)\)

Given:
\[ f(x) = 2 \sin x \]
Substituting \(x = \frac{\pi}{3}\):
\[ f\left(\frac{\pi}{3}\right) = 2 \sin\left(\frac{\pi}{3}\right) \]
We know that \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\):
\[ f\left(\frac{\pi}{3}\right) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \]

### Step 2: Calculate \(g\left(\frac{\pi}{3}\right)\)

Given:
\[ g(x) = \cos^2 x \]
Substituting \(x = \frac{\pi}{3}\):
\[ g\left(\frac{\pi}{3}\right) = \cos^2\left(\frac{\pi}{3}\right) \]
We know that \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\):
\[ g\left(\frac{\pi}{3}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 3: Calculate \((f + g)\left(\frac{\pi}{3}\right)\)

Now we can find \((f + g)\left(\frac{\pi}{3}\right)\):
\[ (f + g)\left(\frac{\pi}{3}\right) = f\left(\frac{\pi}{3}\right) + g\left(\frac{\pi}{3}\right) \]
Substituting the values we found:
\[ (f + g)\left(\frac{\pi}{3}\right) = \sqrt{3} + \frac{1}{4} \]

### Step 4: Final Calculation

To combine these values, we can express \(\sqrt{3}\) in terms of a common denominator:
\[ \sqrt{3} = \frac{4\sqrt{3}}{4} \]
Thus,
\[ (f + g)\left(\frac{\pi}{3}\right) = \frac{4\sqrt{3}}{4} + \frac{1}{4} = \frac{4\sqrt{3} + 1}{4} \]

### Final Answer
\[ (f + g)\left(\frac{\pi}{3}\right) = \frac{4\sqrt{3} + 1}{4} \]