The value of `sin(pi/18)+sin(pi/9)+sin((2pi)/9)+sin((5pi)/18)` is

Options

A. `sin""(7pi)/(18)+sin""(4pi)/(9)`
B. 1
C. `cos""(pi)/(6)+cos""(3pi)/(7)`
D. `cos""(pi)/(9)+sin""(pi)/(9)`

Explanation

To find the value of \( \sin\left(\frac{\pi}{18}\right) + \sin\left(\frac{\pi}{9}\right) + \sin\left(\frac{2\pi}{9}\right) + \sin\left(\frac{5\pi}{18}\right) \), we can follow these steps:

### Step 1: Convert Radians to Degrees
First, we convert the angles from radians to degrees for easier manipulation:
- \( \frac{\pi}{18} = 10^\circ \)
- \( \frac{\pi}{9} = 20^\circ \)
- \( \frac{2\pi}{9} = 40^\circ \)
- \( \frac{5\pi}{18} = 50^\circ \)

Now, we can rewrite the expression as:
\[ \sin(10^\circ) + \sin(20^\circ) + \sin(40^\circ) + \sin(50^\circ) \]

### Step 2: Group the Sine Terms
Next, we can group the terms:
\[ (\sin(10^\circ) + \sin(50^\circ)) + (\sin(20^\circ) + \sin(40^\circ)) \]

### Step 3: Apply the Sine Addition Formula
We can use the sine addition formula:
\[ \sin x + \sin y = 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]

#### For \( \sin(10^\circ) + \sin(50^\circ) \):
Let \( x = 10^\circ \) and \( y = 50^\circ \):
\[ \sin(10^\circ) + \sin(50^\circ) = 2 \sin\left(\frac{10 + 50}{2}\right) \cos\left(\frac{50 - 10}{2}\right) = 2 \sin(30^\circ) \cos(20^\circ) \]
Since \( \sin(30^\circ) = \frac{1}{2} \):
\[ = 2 \cdot \frac{1}{2} \cos(20^\circ) = \cos(20^\circ) \]

#### For \( \sin(20^\circ) + \sin(40^\circ) \):
Let \( x = 20^\circ \) and \( y = 40^\circ \):
\[ \sin(20^\circ) + \sin(40^\circ) = 2 \sin\left(\frac{20 + 40}{2}\right) \cos\left(\frac{40 - 20}{2}\right) = 2 \sin(30^\circ) \cos(10^\circ) \]
Again, since \( \sin(30^\circ) = \frac{1}{2} \):
\[ = 2 \cdot \frac{1}{2} \cos(10^\circ) = \cos(10^\circ) \]

### Step 4: Combine the Results
Now, we combine the results:
\[ \sin(10^\circ) + \sin(20^\circ) + \sin(40^\circ) + \sin(50^\circ) = \cos(20^\circ) + \cos(10^\circ) \]

### Step 5: Final Expression
Thus, the final expression is:
\[ \cos(20^\circ) + \cos(10^\circ) \]

### Conclusion
The value of \( \sin\left(\frac{\pi}{18}\right) + \sin\left(\frac{\pi}{9}\right) + \sin\left(\frac{2\pi}{9}\right) + \sin\left(\frac{5\pi}{18}\right) \) simplifies to \( \cos(20^\circ) + \cos(10^\circ) \).