What is the period of the trigonometric function f(x) = 3 \cos\left(\frac{\pi}{5}x + \frac… — ACT

What is the period of the trigonometric function $f(x) = 3 \cos\left(\frac{\pi}{5}x + \frac{\pi}{2}\right) - 4$ ?

The period of a function of the form

f(x) = A \cos(Bx + C) + D

is given by

T = \frac{2\pi}{|B|}

. Substituting

B = \frac{\pi}{5}

yields

T = \frac{2\pi}{\frac{\pi}{5}} = 2\pi \cdot \frac{5}{\pi} = 10

Identify the coefficient of

x

(

B

) in the given trigonometric function.

B = \frac{\pi}{5}

The standard form for a transformed cosine function is

f(x) = A \cos(Bx + C) + D

, where

B

controls the horizontal stretch or compression.

Apply the period formula for cosine,

T = \frac{2\pi}{|B|}

T = \frac{2\pi}{\frac{\pi}{5}}

The fundamental period of

\cos(x)

2\pi

, which is scaled inversely by

|B|

Simplify the fraction by multiplying by the reciprocal.

T = 2\pi \cdot \frac{5}{\pi} = 10

Dividing by a fraction is equivalent to multiplying by its reciprocal, and the factor of

\pi

cancels out.

Period of Transformed Cosine Functions

Question