An acute angle of a right triangle measures x radians. Which of the following is equal to cosx when `x=(pi)/(9)?`

Options

A. `-cos((11pi)/(18))`
B. `-sin((pi)/(9))`
C. `sin((8pi)/(9))`
D. `sin((7pi)/(18))`

Answer

To solve the problem, we need to find the value of \( \cos x \) when \( x = \frac{\pi}{9} \). We will also explore the options provided to determine which one is equal to \( \cos x \).

### Step-by-Step Solution:

1. **Identify the Value of x**:
We are given that \( x = \frac{\pi}{9} \).

2. **Write the Cosine Function**:
We need to find \( \cos x \):
\[ \cos x = \cos\left(\frac{\pi}{9}\right) \]

3. **Use the Co-function Identity**:
We can use the identity that relates cosine and sine:
\[ \cos x = \sin\left(\frac{\pi}{2} - x\right) \]
Substituting \( x = \frac{\pi}{9} \):
\[ \cos\left(\frac{\pi}{9}\right) = \sin\left(\frac{\pi}{2} - \frac{\pi}{9}\right) \]

4. **Simplify the Argument of Sine**:
To simplify \( \frac{\pi}{2} - \frac{\pi}{9} \), we need a common denominator:
\[ \frac{\pi}{2} = \frac{9\pi}{18} \]
\[ \frac{\pi}{2} - \frac{\pi}{9} = \frac{9\pi}{18} - \frac{2\pi}{18} = \frac{7\pi}{18} \]
Thus, we have:
\[ \cos\left(\frac{\pi}{9}\right) = \sin\left(\frac{7\pi}{18}\right) \]

5. **Conclusion**:
Therefore, we find that:
\[ \cos\left(\frac{\pi}{9}\right) = \sin\left(\frac{7\pi}{18}\right) \]
This means that the correct option is:
**Option 4: \( \sin\left(\frac{7\pi}{18}\right) \)**.