Question

find the exact value. cos 3π

Explanation:

Step1: Use the cosine function property

The cosine function has a period of \(2\pi\), so \(\cos(x + 2k\pi)=\cos x\) for any integer \(k\). Also, \(\cos(\pi + \theta)=-\cos\theta\).
We can write \(3\pi = 2\pi+\pi\). Then \(\cos(3\pi)=\cos(2\pi + \pi)\).
Since \(\cos(x + 2\pi)=\cos x\) (periodicity of cosine function, \(k = 1\) here), we have \(\cos(2\pi+\pi)=\cos\pi\).

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

We know that on the unit - circle, for the angle \(\theta=\pi\), the coordinates of the point on the unit - circle are \((- 1,0)\). And by the definition of the cosine function \(\cos\theta=x\) (where \((x,y)\) is the point on the unit - circle corresponding to the angle \(\theta\)). So \(\cos\pi=-1\).

Answer:

\(-1\)