In a right triangle, one of the acute angles is `cos ((pi)/(3)), anc cos ((pi)/(3))= sin x.` What is the measure of x ?

Options

A. `(pi)/(12)`
B. `(pi)/(6)`
C. `pi/3`
D. `(2pi)/(3)`

Answer

To solve the problem step by step, we need to find the measure of \( x \) given that \( \cos\left(\frac{\pi}{3}\right) = \sin x \).

### Step 1: Find the value of \( \cos\left(\frac{\pi}{3}\right) \)
We know that:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Step 2: Set up the equation
From the problem, we have:
\[ \cos\left(\frac{\pi}{3}\right) = \sin x \]
Substituting the value we found:
\[ \frac{1}{2} = \sin x \]

### Step 3: Find the angle \( x \)
We need to find \( x \) such that:
\[ \sin x = \frac{1}{2} \]
The angles for which \( \sin x = \frac{1}{2} \) in the range of \( 0 \) to \( \frac{\pi}{2} \) (since \( x \) is an acute angle) is:
\[ x = \frac{\pi}{6} \]

### Conclusion
Thus, the measure of \( x \) is:
\[ x = \frac{\pi}{6} \]