Find the values of `theta` lying between `0^(@)` and `360^(@)` when
`sec theta=-2`

Answer

To solve the equation \( \sec \theta = -2 \) for values of \( \theta \) in the interval \( [0^\circ, 360^\circ] \), we can follow these steps:

### Step 1: Understand the relationship of secant
The secant function is defined as the reciprocal of the cosine function:
\[ \sec \theta = \frac{1}{\cos \theta} \]
Thus, if \( \sec \theta = -2 \), we can rewrite it as:
\[ \cos \theta = -\frac{1}{2} \]

### Step 2: Determine the quadrants
The cosine function is negative in the second and third quadrants. Therefore, we need to find the angles \( \theta \) in these quadrants where \( \cos \theta = -\frac{1}{2} \).

### Step 3: Find reference angle
The reference angle for \( \cos \theta = \frac{1}{2} \) is \( 60^\circ \) (or \( \frac{\pi}{3} \) radians). Since we need the angles where cosine is negative, we can find the angles in the second and third quadrants.

### Step 4: Calculate angles in the second and third quadrants
1. **Second Quadrant**:
\[ \theta = 180^\circ - 60^\circ = 120^\circ \]

2. **Third Quadrant**:
\[ \theta = 180^\circ + 60^\circ = 240^\circ \]

### Step 5: Final values
Thus, the values of \( \theta \) that satisfy \( \sec \theta = -2 \) in the interval \( [0^\circ, 360^\circ] \) are:
\[ \theta = 120^\circ \quad \text{and} \quad \theta = 240^\circ \]

### Summary of the solution:
The values of \( \theta \) are:
- \( \theta = 120^\circ \)
- \( \theta = 240^\circ \)