Question

find the exact value.

\\( \sin \frac { 7 \pi } { 2 } \\)

Explanation:

Step1: Use the periodicity of sine function

The sine function has a period of \(2\pi\), so \(\sin(x + 2k\pi)=\sin x\) for any integer \(k\).
We can rewrite \(\frac{7\pi}{2}\) as \(\frac{7\pi}{2}= 2\pi+\frac{3\pi}{2}\).
So, \(\sin\frac{7\pi}{2}=\sin(2\pi+\frac{3\pi}{2})\).
Since \(\sin(x + 2\pi)=\sin x\) (periodicity of sine function, \(k = 1\), \(x=\frac{3\pi}{2}\)), then \(\sin(2\pi+\frac{3\pi}{2})=\sin\frac{3\pi}{2}\).

Step2: Recall the value of sine at special angle

We know that on the unit - circle, for the angle \(\theta=\frac{3\pi}{2}\), the coordinates of the point on the unit - circle are \((0, - 1)\).
Since \(\sin\theta=y\) (where \((x,y)\) is the point on the unit - circle corresponding to the angle \(\theta\)), when \(\theta=\frac{3\pi}{2}\), \(y=-1\). So, \(\sin\frac{3\pi}{2}=-1\).

Answer:

\(-1\)