find the exact value. do not use a calculator. \n\\( \\sin (9 \\pi) \\) \n\\( \\sin (9 \\pi)= \\) (simplify…

Explanation:

Step1: Use the periodicity of the sine function

The sine function (y = \sin(x)) has a period of (2\pi), i.e., (\sin(x + 2k\pi)=\sin(x)) for any integer (k). We can rewrite (9\pi) as (9\pi= 4\times2\pi+\pi). So, (\sin(9\pi)=\sin(4\times2\pi + \pi)). Since (\sin(x + 2k\pi)=\sin(x)) with (k = 4) and (x=\pi), we have (\sin(9\pi)=\sin(\pi)).

Step2: Recall the value of (\sin(\pi))

We know from the unit - circle definition (or the graph of (y = \sin(x))) that when (x=\pi) (the angle in radians), the (y) - coordinate of the point on the unit circle ((\cos x,\sin x)) is (0). So, (\sin(\pi)=0).

Answer:

(0)