What is the greatest value of `theta` lying between `0^@` and `720^@` whose tangent is `-1/sqrt3` ?

Answer

To find the greatest value of \( \theta \) lying between \( 0^\circ \) and \( 720^\circ \) for which \( \tan \theta = -\frac{1}{\sqrt{3}} \), we can follow these steps:

### Step 1: Identify the reference angle
The value \( -\frac{1}{\sqrt{3}} \) corresponds to angles where the tangent function is negative. The reference angle for \( \tan \theta = \frac{1}{\sqrt{3}} \) is \( 30^\circ \) or \( \frac{\pi}{6} \).

### Step 2: Determine the quadrants
Since tangent is negative in the second and fourth quadrants, we can find the angles:
- In the second quadrant:
\[ \theta = 180^\circ - 30^\circ = 150^\circ \]
- In the fourth quadrant:
\[ \theta = 360^\circ - 30^\circ = 330^\circ \]

### Step 3: Find the general solutions
The general solutions for \( \tan \theta = -\frac{1}{\sqrt{3}} \) can be expressed as:
\[ \theta = 150^\circ + n \cdot 180^\circ \quad \text{and} \quad \theta = 330^\circ + n \cdot 180^\circ \]
where \( n \) is any integer.

### Step 4: Calculate specific values of \( \theta \)
Now, we will calculate the values of \( \theta \) for \( n = 0, 1, 2, \ldots \) until we exceed \( 720^\circ \).

1. For \( n = 0 \):
- From \( 150^\circ \): \( \theta = 150^\circ \)
- From \( 330^\circ \): \( \theta = 330^\circ \)

2. For \( n = 1 \):
- From \( 150^\circ \): \( \theta = 150^\circ + 180^\circ = 330^\circ \)
- From \( 330^\circ \): \( \theta = 330^\circ + 180^\circ = 510^\circ \)

3. For \( n = 2 \):
- From \( 150^\circ \): \( \theta = 150^\circ + 2 \cdot 180^\circ = 510^\circ \)
- From \( 330^\circ \): \( \theta = 330^\circ + 2 \cdot 180^\circ = 690^\circ \)

4. For \( n = 3 \):
- From \( 150^\circ \): \( \theta = 150^\circ + 3 \cdot 180^\circ = 690^\circ \)
- From \( 330^\circ \): \( \theta = 330^\circ + 3 \cdot 180^\circ = 870^\circ \) (exceeds \( 720^\circ \))

### Step 5: Identify the greatest value
The values we have calculated that are within the range \( 0^\circ \) to \( 720^\circ \) are:
- \( 150^\circ \)
- \( 330^\circ \)
- \( 510^\circ \)
- \( 690^\circ \)

The greatest value of \( \theta \) is:
\[ \theta = 690^\circ \]

### Final Answer
Thus, the greatest value of \( \theta \) lying between \( 0^\circ \) and \( 720^\circ \) for which \( \tan \theta = -\frac{1}{\sqrt{3}} \) is:
\[ \boxed{690^\circ} \]