Question

find the exact value. do not use a calculator.

cos\frac{9\pi}{2}

cos\frac{9\pi}{2}=\square
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Rewrite the angle

We know that \(\frac{9\pi}{2}=4\pi+\frac{\pi}{2}\).
Since the cosine function has a period of \(2\pi\), \(\cos(x + 2k\pi)=\cos x\) for any integer \(k\). Here \(k = 2\) and \(x=\frac{\pi}{2}\), so \(\cos\frac{9\pi}{2}=\cos(4\pi+\frac{\pi}{2})\).
By the periodicity of the cosine function \(\cos(4\pi+\frac{\pi}{2})=\cos\frac{\pi}{2}\).

Step2: Evaluate \(\cos\frac{\pi}{2}\)

We know from the unit - circle definition of the cosine function. For an angle \(\theta\) in standard position, \(\cos\theta=x\) where \((x,y)\) is the point on the unit - circle corresponding to \(\theta\).
When \(\theta=\frac{\pi}{2}\), the point on the unit - circle is \((0,1)\), so \(\cos\frac{\pi}{2}=0\).

Answer:

\(0\)