Find all the angles between `0^(@)` and `720^(@)` whose tangent is `-(1)/(sqrt(3))`.

Answer

To find all angles between \(0^\circ\) and \(720^\circ\) whose tangent is \(-\frac{1}{\sqrt{3}}\), we can follow these steps:

### Step 1: Understand the Tangent Function
The tangent function is negative in the second and fourth quadrants. We need to find the reference angle whose tangent is \(\frac{1}{\sqrt{3}}\).

### Step 2: Find the Reference Angle
The reference angle for \(\tan \theta = \frac{1}{\sqrt{3}}\) is:
\[ \theta = 30^\circ \quad \text{(or } \frac{\pi}{6} \text{ radians)} \]
Thus, for \(\tan \theta = -\frac{1}{\sqrt{3}}\), the angles in the second and fourth quadrants are:
- Second quadrant: \(180^\circ - 30^\circ = 150^\circ\)
- Fourth quadrant: \(360^\circ - 30^\circ = 330^\circ\)

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

### Step 4: Find Specific Solutions in the Range \(0^\circ\) to \(720^\circ\)
Now, we will find the specific angles by substituting integer values for \(k\).

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

2. For \(k = 1\):
- From \(150^\circ\): \(150^\circ + 180^\circ = 330^\circ\) (already counted)
- From \(330^\circ\): \(330^\circ + 180^\circ = 510^\circ\)

3. For \(k = 2\):
- From \(150^\circ\): \(150^\circ + 2 \cdot 180^\circ = 630^\circ\)
- From \(330^\circ\): \(330^\circ + 2 \cdot 180^\circ = 810^\circ\) (out of range)

### Step 5: List All Valid Angles
The valid angles between \(0^\circ\) and \(720^\circ\) are:
- \(150^\circ\)
- \(330^\circ\)
- \(510^\circ\)
- \(630^\circ\)

### Final Answer
Thus, the angles between \(0^\circ\) and \(720^\circ\) whose tangent is \(-\frac{1}{\sqrt{3}}\) are:
\[ 150^\circ, \quad 330^\circ, \quad 510^\circ, \quad 630^\circ \]