FInd the sign of `sec2000^@`

Answer

To find the sign of \( \sec(2000^\circ) \), we can follow these steps:

### Step 1: Reduce the angle
First, we need to reduce \( 2000^\circ \) to an equivalent angle within the standard range of \( 0^\circ \) to \( 360^\circ \). We can do this by subtracting multiples of \( 360^\circ \).

\[ 2000^\circ - 360^\circ \times 5 = 2000^\circ - 1800^\circ = 200^\circ \]

### Step 2: Identify the quadrant
Next, we determine in which quadrant \( 200^\circ \) lies. The angle \( 200^\circ \) is between \( 180^\circ \) and \( 270^\circ \), which means it is in the third quadrant.

### Step 3: Determine the sign of secant
In the third quadrant, the cosine function is negative. Since secant is the reciprocal of cosine, we have:

\[ \sec(200^\circ) = \frac{1}{\cos(200^\circ)} \]

Since \( \cos(200^\circ) < 0 \), it follows that:

\[ \sec(200^\circ) < 0 \]

### Conclusion
Thus, the sign of \( \sec(2000^\circ) \) is negative.

### Summary of Steps:
1. Reduce \( 2000^\circ \) to \( 200^\circ \).
2. Identify that \( 200^\circ \) is in the third quadrant.
3. Conclude that \( \sec(200^\circ) \) is negative.