Which of the following is equal to `cos((pi)/(7))`?

Options

A. `-cos(-(pi)/(7))`
B. `-sin((pi)/(7))`
C. `sin((5pi)/(14))`
D. `-cos((5pi)/(14))`

Answer

To find which of the following is equal to \( \cos\left(\frac{\pi}{7}\right) \), we can use the co-function identity of sine and cosine. Here’s a step-by-step solution:

### Step 1: Identify the angle
We start with the angle \( \frac{\pi}{7} \). We need to determine its position on the unit circle.

### Step 2: Determine the quadrant
The angle \( \frac{\pi}{7} \) is approximately \( 25.7^\circ \), which lies in the first quadrant where both sine and cosine values are positive.

### Step 3: Use the co-function identity
In the first quadrant, we can use the identity:
\[ \sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta) \]
Here, we can let \( \theta = \frac{\pi}{7} \). Thus, we have:
\[ \sin\left(\frac{\pi}{2} - \frac{\pi}{7}\right) = \cos\left(\frac{\pi}{7}\right) \]

### Step 4: Simplify the angle
Now we simplify \( \frac{\pi}{2} - \frac{\pi}{7} \):
\[ \frac{\pi}{2} = \frac{7\pi}{14} \]
So,
\[ \frac{\pi}{2} - \frac{\pi}{7} = \frac{7\pi}{14} - \frac{2\pi}{14} = \frac{5\pi}{14} \]
This means:
\[ \sin\left(\frac{5\pi}{14}\right) = \cos\left(\frac{\pi}{7}\right) \]

### Step 5: Conclusion
Thus, we conclude that:
\[ \cos\left(\frac{\pi}{7}\right) = \sin\left(\frac{5\pi}{14}\right) \]

### Final Answer
The correct expression that is equal to \( \cos\left(\frac{\pi}{7}\right) \) is \( \sin\left(\frac{5\pi}{14}\right) \).

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