The value of `("sin"(pi)/(9))(4+"sec"(pi)/(9))` is :

Options

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

Answer

To solve the expression \( \sin\left(\frac{\pi}{9}\right) \left(4 + \sec\left(\frac{\pi}{9}\right)\right) \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sin\left(\frac{\pi}{9}\right) \left(4 + \sec\left(\frac{\pi}{9}\right)\right) \]
Recall that \( \sec\theta = \frac{1}{\cos\theta} \), so we can rewrite the expression as:
\[ \sin\left(\frac{\pi}{9}\right) \left(4 + \frac{1}{\cos\left(\frac{\pi}{9}\right)}\right) \]

### Step 2: Simplify the expression
Now, we can combine the terms inside the parentheses:
\[ \sin\left(\frac{\pi}{9}\right) \left(4 + \frac{1}{\cos\left(\frac{\pi}{9}\right)}\right) = \sin\left(\frac{\pi}{9}\right) \left(\frac{4\cos\left(\frac{\pi}{9}\right) + 1}{\cos\left(\frac{\pi}{9}\right)}\right) \]
This gives us:
\[ \frac{\sin\left(\frac{\pi}{9}\right) \left(4\cos\left(\frac{\pi}{9}\right) + 1\right)}{\cos\left(\frac{\pi}{9}\right)} \]

### Step 3: Use trigonometric identities
We can use the double angle identity for sine:
\[ 2\sin\theta\cos\theta = \sin(2\theta) \]
Thus, we can express \( 4\sin\left(\frac{\pi}{9}\right)\cos\left(\frac{\pi}{9}\right) \) as:
\[ 2 \cdot 2\sin\left(\frac{\pi}{9}\right)\cos\left(\frac{\pi}{9}\right) = 2\sin\left(\frac{2\pi}{9}\right) \]
Now, our expression becomes:
\[ \frac{2\sin\left(\frac{2\pi}{9}\right) + \sin\left(\frac{\pi}{9}\right)}{\cos\left(\frac{\pi}{9}\right)} \]

### Step 4: Combine sine terms
Next, we can express \( \sin\left(\frac{2\pi}{9}\right) \) using the sine addition formula:
\[ \sin\left(\frac{2\pi}{9}\right) = \sin\left(\frac{\pi}{3} - \frac{\pi}{9}\right) = \sin\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{9}\right) - \cos\left(\frac{\pi}{3}\right)\sin\left(\frac{\pi}{9}\right) \]
Substituting \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \) and \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \):
\[ \sin\left(\frac{2\pi}{9}\right) = \frac{\sqrt{3}}{2}\cos\left(\frac{\pi}{9}\right) - \frac{1}{2}\sin\left(\frac{\pi}{9}\right) \]

### Step 5: Substitute back into the expression
Now we substitute back into our expression:
\[ \frac{2\left(\frac{\sqrt{3}}{2}\cos\left(\frac{\pi}{9}\right) - \frac{1}{2}\sin\left(\frac{\pi}{9}\right)\right) + \sin\left(\frac{\pi}{9}\right)}{\cos\left(\frac{\pi}{9}\right)} \]
This simplifies to:
\[ \frac{\sqrt{3}\cos\left(\frac{\pi}{9}\right) - \sin\left(\frac{\pi}{9}\right) + \sin\left(\frac{\pi}{9}\right)}{\cos\left(\frac{\pi}{9}\right)} = \frac{\sqrt{3}\cos\left(\frac{\pi}{9}\right)}{\cos\left(\frac{\pi}{9}\right)} = \sqrt{3} \]

### Final Answer
Thus, the value of \( \sin\left(\frac{\pi}{9}\right) \left(4 + \sec\left(\frac{\pi}{9}\right)\right) \) is \( \sqrt{3} \).