Which of the following statements is/are correct about the angle`(10pi)/(3)`radians?
I. It is equivalent to `600^(@)`.
II. The angle falls in the third quadrant.
III.`(Cos)(10pi)/(3)` is positive.

Options

A. Only I
B. Only II
C. Both I and III
D. Both I and II

Answer

To solve the question regarding the angle \(\frac{10\pi}{3}\) radians, we will analyze each of the statements provided.

### Step 1: Convert \(\frac{10\pi}{3}\) radians to degrees
To convert radians to degrees, we use the conversion factor \(180^\circ = \pi\) radians. Thus, we can set up the conversion as follows:

\[ \text{Degrees} = \frac{10\pi}{3} \times \frac{180^\circ}{\pi} \]

### Step 2: Simplify the conversion
Now, we simplify the expression:

\[ \text{Degrees} = \frac{10 \times 180}{3} = \frac{1800}{3} = 600^\circ \]

### Step 3: Determine the quadrant of the angle
Next, we need to determine the quadrant in which the angle \(\frac{10\pi}{3}\) lies. To do this, we can find the equivalent angle within the range of \(0\) to \(2\pi\) by subtracting \(2\pi\) until we reach a value in this range.

First, we calculate \(2\pi\):

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

Now, we subtract \(2\pi\) from \(\frac{10\pi}{3}\):

\[ \frac{10\pi}{3} - \frac{6\pi}{3} = \frac{4\pi}{3} \]

The angle \(\frac{4\pi}{3}\) is in the third quadrant since it is between \(\pi\) and \(\frac{3\pi}{2}\).

### Step 4: Evaluate \(\cos\left(\frac{10\pi}{3}\right)\)
Now, we need to evaluate \(\cos\left(\frac{10\pi}{3}\right)\). Since \(\frac{10\pi}{3}\) is equivalent to \(\frac{4\pi}{3}\), we can find:

\[ \cos\left(\frac{4\pi}{3}\right) \]

In the third quadrant, the cosine function is negative. Therefore, \(\cos\left(\frac{4\pi}{3}\right)\) is negative.

### Conclusion
Now we can summarize the correctness of the statements:

- **Statement I**: It is equivalent to \(600^\circ\) - **Correct**
- **Statement II**: The angle falls in the third quadrant - **Correct**
- **Statement III**: \(\cos\left(\frac{10\pi}{3}\right)\) is positive - **Incorrect**

Thus, the correct statements are I and II.