If `tan A=-sqrt(3)`, find all the possible values of A between `0^(@)` and `360^(@)`.

Answer

To solve the problem where \( \tan A = -\sqrt{3} \) and find all possible values of \( A \) between \( 0^\circ \) and \( 360^\circ \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Quadrants**:
The tangent function is negative in the second and fourth quadrants. Therefore, we need to find angles in these quadrants where \( \tan A = -\sqrt{3} \).

2. **Reference Angle**:
The reference angle for \( \tan A = \sqrt{3} \) is \( 60^\circ \) because \( \tan 60^\circ = \sqrt{3} \).

3. **Find the Angle in the Second Quadrant**:
To find the angle in the second quadrant where \( \tan A = -\sqrt{3} \), we use the formula:
\[ A = 180^\circ - \text{reference angle} \]
Substituting the reference angle:
\[ A = 180^\circ - 60^\circ = 120^\circ \]

4. **Find the Angle in the Fourth Quadrant**:
To find the angle in the fourth quadrant where \( \tan A = -\sqrt{3} \), we use the formula:
\[ A = 360^\circ - \text{reference angle} \]
Substituting the reference angle:
\[ A = 360^\circ - 60^\circ = 300^\circ \]

5. **Conclusion**:
The two possible values of \( A \) between \( 0^\circ \) and \( 360^\circ \) are:
\[ A = 120^\circ \quad \text{and} \quad A = 300^\circ \]

### Final Answer:
The possible values of \( A \) are \( 120^\circ \) and \( 300^\circ \).

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