If `sin x = cos ((13pi)/(6)),` which os the following could be the value of x ?

Options

A. `pi/6`
B. `pi/4`
C. `pi/3`
D. `pi/2`

Answer

To solve the equation \( \sin x = \cos \left( \frac{13\pi}{6} \right) \), we will follow these steps:

### Step 1: Simplify \( \cos \left( \frac{13\pi}{6} \right) \)

To simplify \( \cos \left( \frac{13\pi}{6} \right) \), we can use the periodic property of cosine. The cosine function has a period of \( 2\pi \).

\[ \frac{13\pi}{6} = 2\pi + \frac{\pi}{6} \]

Thus, we can write:

\[ \cos \left( \frac{13\pi}{6} \right) = \cos \left( 2\pi + \frac{\pi}{6} \right) = \cos \left( \frac{\pi}{6} \right) \]

### Step 2: Set up the equation

Now we have:

\[ \sin x = \cos \left( \frac{\pi}{6} \right) \]

### Step 3: Use the identity \( \sin x = \cos \left( \frac{\pi}{2} - x \right) \)

We know that:

\[ \sin x = \cos \left( \frac{\pi}{2} - x \right) \]

Thus, we can equate:

\[ \cos \left( \frac{\pi}{2} - x \right) = \cos \left( \frac{\pi}{6} \right) \]

### Step 4: Equate the angles

Since the cosines are equal, we can set the angles equal to each other:

\[ \frac{\pi}{2} - x = \frac{\pi}{6} \]

### Step 5: Solve for \( x \)

Now, we will solve for \( x \):

1. Rearranging gives:

\[ x = \frac{\pi}{2} - \frac{\pi}{6} \]

2. To subtract these fractions, we need a common denominator. The least common multiple of 2 and 6 is 6.

\[ \frac{\pi}{2} = \frac{3\pi}{6} \]

Thus,

\[ x = \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3} \]

### Conclusion

The value of \( x \) is:

\[ x = \frac{\pi}{3} \]