Find all solutions to the equation \( \tan (t)=\frac{3}{\tan (t)} \) in the interval \( 0 \leq t \leq 2 \pi \). Find exact answers (given as fractions, not decimals). Et answers as a comma separated list. \( t=1 \) help (fractions)

To solve the equation \( \tan(t) = \frac{3}{\tan(t)} \), let's start by multiplying both sides by \( \tan(t) \) (assuming \( \tan(t) \neq 0 \)) to eliminate the fraction: \[ \tan^2(t) = 3. \] Next, we can solve for \( \tan(t) \): \[ \tan(t) = \pm \sqrt{3}. \] Now we need to find the values of \( t \) in the interval \( 0 \leq t \leq 2\pi \) that correspond to these values of the tangent function. 1. **For \( \tan(t) = \sqrt{3} \)**: - The reference angle where \( \tan(t) = \sqrt{3} \) is \( \frac{\pi}{3} \). - In the interval \( [0, 2\pi] \), the angles satisfying this are: \[ t = \frac{\pi}{3} \quad \text{and} \quad t = \frac{\pi}{3} + \pi = \frac{4\pi}{3}. \] 2. **For \( \tan(t) = -\sqrt{3} \)**: - The reference angle for \( \tan(t) = -\sqrt{3} \) is also \( \frac{\pi}{3} \). - The angles satisfying this in the interval \( [0, 2\pi] \) are: \[ t = \frac{\pi}{3} + \frac{3\pi}{2} = \frac{2\pi}{3} \quad \text{and} \quad t = \frac{4\pi}{3} + \frac{3\pi}{2} = \frac{5\pi}{3}. \] So, the solutions to the equation \( \tan(t) = \frac{3}{\tan(t)} \) in the interval \( [0, 2\pi] \) are: \[ t = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}. \]