Question

sin(180)
find reference angle

Explanation:

Step1: Determine the quadrant

The angle \(180^{\circ}\) lies on the negative \(x -\)axis. It is not in any of the four standard quadrants (\(0^{\circ}<\theta <90^{\circ}\) - first, \(90^{\circ}<\theta <180^{\circ}\) - second, \(180^{\circ}<\theta <270^{\circ}\) - third, \(270^{\circ}<\theta <360^{\circ}\) - fourth).

Step2: Apply the reference - angle formula

For an angle \(\theta\) on the \(x -\)axis (\(\theta = n\cdot180^{\circ},n\in\mathbb{Z}\)), the reference angle \(\theta_{r}\) is calculated as follows. If \(\theta = 180^{\circ}\), and we use the formula \(\theta_{r}=\vert\theta - 180^{\circ}\vert\) (since for angles on the \(x -\)axis \(\theta = k\cdot180^{\circ}\), and the reference angle is the smallest non - negative angle between the terminal side of \(\theta\) and the \(x -\)axis).

$$ \theta_{r}=\vert180^{\circ}- 180^{\circ}\vert=0^{\circ} $$

Answer:

\(0^{\circ}\)