find the exact value. do not use a calculator.\n\ncos\\frac { 23\\pi } { 2 }\n\ncos\\frac { 23\\pi } { 2 } =…

Explanation:

Step1: Express ( \frac{23\pi}{2} ) as ( 12\pi-\frac{\pi}{2} )

( \frac{23\pi}{2}=12\pi-\frac{\pi}{2} )

Step2: Use the cosine periodicity ( \cos(x + 2k\pi)=\cos x,k\in\mathbb{Z} )

Since ( \cos(x + 2k\pi)=\cos x ) and here ( x =-\frac{\pi}{2},k = 6) (because (12\pi=2\times6\pi)), then ( \cos\frac{23\pi}{2}=\cos\left(12\pi-\frac{\pi}{2}\right)=\cos\left(-\frac{\pi}{2}\right) )

Step3: Use the property ( \cos(-x)=\cos x )

( \cos\left(-\frac{\pi}{2}\right)=\cos\frac{\pi}{2} )

Step4: Recall the value of ( \cos\frac{\pi}{2} )

We know that ( \cos\frac{\pi}{2}=0 )

Answer:

(0)