Questions: Use the unit circle to find all values of θ between 0 and 2π for which the given statement is true. (Enter your answers as a comma-separated sin θ=1 / 2 θ= rad

Solution Steps

To find all values of \(\theta\) between 0 and \(2\pi\) for which \(\sin \theta = \frac{1}{2}\), we can use the unit circle. The sine function equals \(\frac{1}{2}\) at specific angles in the first and second quadrants. These angles are \(\frac{\pi}{6}\) and \(\frac{5\pi}{6}\).

To solve the equation \(\sin \theta = \frac{1}{2}\), we need to find the angles \(\theta\) in the interval \([0, 2\pi]\) where this condition holds true.

From the unit circle, we know that \(\sin \theta = \frac{1}{2}\) at the angles:

• \(\theta_1 = \frac{\pi}{6}\)
• \(\theta_2 = \frac{5\pi}{6}\)

The values of \(\theta\) that satisfy the equation \(\sin \theta = \frac{1}{2}\) are:

• \(\theta_1 \approx 0.524\)
• \(\theta_2 \approx 2.618\)

Final Answer