explain how to find the exact value of sin(570°).

Explanation:

Step1: Reduce the angle

Since the sine - function has a period of (360^{\circ}), we can subtract (360^{\circ}) from (570^{\circ}). So, (570^{\circ}-360^{\circ}=210^{\circ}).

Step2: Rewrite the angle in terms of a reference angle

We know that (210^{\circ}=180^{\circ} + 30^{\circ}). In the third - quadrant, (\sin(180^{\circ}+\alpha)=-\sin\alpha). Here (\alpha = 30^{\circ}).

Step3: Use the known value of sine

We know that (\sin(30^{\circ})=\frac{1}{2}). So, (\sin(210^{\circ})=\sin(180^{\circ}+30^{\circ})=-\sin(30^{\circ})).

Answer:

(-\frac{1}{2})