find the period of this function.\n\ny = cos \\(\\frac{\\pi}{3}\\)x\n\n?

Answer

Explanation:

Step1: Recall the period formula for cosine function

The general form of a cosine function is ( y = \cos(Bx) ), and its period ( T ) is given by the formula ( T=\frac{2\pi}{|B|} ).

Step2: Identify the value of B in the given function

In the function ( y = \cos\frac{\pi}{3}x ), we can see that ( B = \frac{\pi}{3} ).

Step3: Substitute B into the period formula

Substitute ( B=\frac{\pi}{3} ) into the formula ( T = \frac{2\pi}{|B|} ). We get ( T=\frac{2\pi}{\left|\frac{\pi}{3}\right|} ). Since ( \frac{\pi}{3}>0 ), the absolute value doesn't change it, so ( T=\frac{2\pi}{\frac{\pi}{3}} ).

Step4: Simplify the expression

When dividing by a fraction, we multiply by its reciprocal. So ( \frac{2\pi}{\frac{\pi}{3}}=2\pi\times\frac{3}{\pi} ). The ( \pi ) terms cancel out, and we are left with ( 2\times3 = 6 ).