Questions: Determine the specific solutions (if any) on the interval ([0,2 pi) ). [ tan theta-1/sqrt3=0 ]

Solution Steps

The given equation is:

\[ \tan \theta - \frac{1}{\sqrt{3}} = 0 \]

To find the solutions, we set the equation to zero:

\[ \tan \theta = \frac{1}{\sqrt{3}} \]

The tangent function is positive in the first and third quadrants. We know that:

\[ \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \]

Thus, the general solutions for \(\theta\) are:

\[ \theta = \frac{\pi}{6} + n\pi \]

where \(n\) is an integer.

We need to find the values of \(\theta\) within the interval \([0, 2\pi)\). Using the general solution:

1. For \(n = 0\): \[ \theta = \frac{\pi}{6} \]

2. For \(n = 1\): \[ \theta = \frac{\pi}{6} + \pi = \frac{7\pi}{6} \]

3. For \(n = 2\): \[ \theta = \frac{\pi}{6} + 2\pi = \frac{13\pi}{6} \quad (\text{not in the interval}) \]

Thus, the specific solutions in the interval \([0, 2\pi)\) are \(\frac{\pi}{6}\) and \(\frac{7\pi}{6}\).

Final Answer