determine the period of the following trigonometric function. \n\n\\( y = -8 cos left( dfrac{8}{5}x +…

Explanation:

Step1: Recall the period formula for cosine function

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

Step2: Identify the value of ( B ) from the given function

In the function ( y = -8\cos\left(\frac{8}{5}x+\frac{72}{5}\right)+10 ), we can see that ( B = \frac{8}{5} ).

Step3: Calculate the period

Using the period formula ( T=\frac{2\pi}{|B|} ), substitute ( B = \frac{8}{5} ) into the formula. We get ( T=\frac{2\pi}{\left|\frac{8}{5}\right|} ). Since the absolute value of ( \frac{8}{5} ) is ( \frac{8}{5} ), we have ( T = \frac{2\pi}{\frac{8}{5}} ). When dividing by a fraction, we multiply by its reciprocal, so ( T=2\pi\times\frac{5}{8}=\frac{5\pi}{4} ).

Answer:

The period of the function ( y = -8\cos\left(\frac{8}{5}x+\frac{72}{5}\right)+10 ) is ( \frac{5\pi}{4} ).