Question

find the exact value. do not use a calculator.
\\( \sin (9 \pi) \\)
\\( \sin (9 \pi)= \\) (simplify your answer, including any radicals. use integers or fractions for any numbe

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:

\(0\)