If `sin((2)/(9))pi=cosx`, then x=

Options

A. `(7)/(9)pi`
B. `(5)/(18)pi`
C. `(pi)/(3)`
D. `(13)/(18)pi`

Answer

To solve the equation \( \sin\left(\frac{2}{9}\pi\right) = \cos x \), we can follow these steps:

### Step 1: Use the co-function identity
We know that:
\[ \sin \theta = \cos\left(\frac{\pi}{2} - \theta\right) \]
Applying this identity to our equation, we can rewrite \( \sin\left(\frac{2}{9}\pi\right) \) as:
\[ \sin\left(\frac{2}{9}\pi\right) = \cos\left(\frac{\pi}{2} - \frac{2}{9}\pi\right) \]

### Step 2: Simplify the right-hand side
Now, we need to simplify \( \frac{\pi}{2} - \frac{2}{9}\pi \):
\[ \frac{\pi}{2} = \frac{9\pi}{18} \]
So,
\[ \frac{\pi}{2} - \frac{2}{9}\pi = \frac{9\pi}{18} - \frac{4\pi}{18} = \frac{5\pi}{18} \]
Thus, we have:
\[ \sin\left(\frac{2}{9}\pi\right) = \cos\left(\frac{5\pi}{18}\right) \]

### Step 3: Set the equations equal
Since we have:
\[ \cos x = \cos\left(\frac{5\pi}{18}\right) \]
This implies that:
\[ x = \frac{5\pi}{18} \quad \text{or} \quad x = 2\pi - \frac{5\pi}{18} \]

### Step 4: Calculate the second possible value of x
Calculating the second possible value:
\[ 2\pi - \frac{5\pi}{18} = \frac{36\pi}{18} - \frac{5\pi}{18} = \frac{31\pi}{18} \]

### Step 5: Conclusion
Thus, the possible values for \( x \) are:
\[ x = \frac{5\pi}{18} \quad \text{or} \quad x = \frac{31\pi}{18} \]
However, since \( x \) is typically considered in the range \( [0, 2\pi) \), we take:
\[ x = \frac{5\pi}{18} \]

### Final Answer
Therefore, the value of \( x \) is:
\[ \boxed{\frac{5\pi}{18}} \]