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Let's solve each trigonometric equation within the interval \([0, 2\pi]\) as required.

### Problem 1:
\[
\tan(t) = -\frac{1}{\sqrt{3}}
\]

The tangent of an angle is \(-\frac{1}{\sqrt{3}}\) at specific standard angles where \(t\) is in either the second or fourth quadrant (where tangent is negative).

The reference angle for \(\tan(t) = \frac{1}{\sqrt{3}}\) is \(\frac{\pi}{6}\), so:
- In the second quadrant: \(t = \pi - \frac{\pi}{6} = \frac{5\pi}{6}\)
- In the fourth quadrant: \(t = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}\)

Thus, the solution for \(t\) is:
\[
t = \frac{5}{6}\pi, \frac{11}{6}\pi
\]

### Problem 2:
\[
\tan(t) = \sqrt{3}
\]

The tangent of an angle is \(\sqrt{3}\) at specific angles where \(t\) is in either the first or third quadrant (where tangent is positive).

The reference angle for \(\tan(t) = \sqrt{3}\) is \(\frac{\pi}{3}\), so:
- In the first quadrant: \(t = \frac{\pi}{3}\)
- In the third quadrant: \(t = \pi + \frac{\pi}{3} = \frac{4\pi}{3}\)

Thus, the solution for \(t\) is:
\[
t = \frac{1}{3}\pi, \frac{4}{3}\pi
\]

### Final Answers:
- For \(\tan(t) = -\frac{1}{\sqrt{3}}\): \( t = \frac{5}{6}, \frac{11}{6} \)
- For \(\tan(t) = \sqrt{3}\): \( t = \frac{1}{3}, \frac{4}{3} \)

Do you need further details or have questions?

#### Related Questions:
1. How do we determine the reference angle in tangent problems?
2. Why is tangent positive in the first and third quadrants?
3. How do we convert between radians and degrees for reference angles?
4. How can we solve for angles outside the \([0, 2\pi]\) range?
5. What changes when solving similar problems with \(\sin\) or \(\cos\) instead of \(\tan\)?

#### Tip:
For tangent equations, remember that \(\tan(\theta + \pi) = \tan(\theta)\) due to the periodicity of \(\tan\) with a period of \(\pi\).

Solve the following equations in the interval [0, 2π]. tan(t) = -1/√3 and tan(t) = √3

Learn to solve trigonometric equations involving the tangent function within the interval [0, 2π]. This solution covers key trigonometric concepts, including finding reference angles, using the unit circle, and understanding the periodicity of the tangent function. Suitable for high school students in Grades 10-12.

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