Sin t =1/2

To solve the equation \( \sin t = \frac{1}{2} \), we need to find all angles \( t \) for which the sine function equals \( \frac{1}{2} \).

The sine of an angle is \( \frac{1}{2} \) at the following angles in the interval \( [0, 2\pi) \):

1. \( t = \frac{\pi}{6} \) (or 30 degrees)
2. \( t = \frac{5\pi}{6} \) (or 150 degrees)

Since the sine function is periodic, we can express the general solutions using the periodic nature of sine. The general solutions will include all angles where the sine is equal to \( \frac{1}{2} \):

\[ t = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad t = \frac{5\pi}{6} + 2k\pi \]

where \( k \) is any integer (to account for the periodicity).

In summary, the solutions to \( \sin t = \frac{1}{2} \) are:

\[ t = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad t = \frac{5\pi}{6} + 2k\pi, \quad k \in \mathbb{Z} \]