Question

What is the period of the function f left parenthesis t right parenthesis equals cosine left parenthesis 5 t plus Start Fraction pi over 2 End Fraction right parenthesis minus 3?
(1 point)
Responses

two fifths
Image with alt text: two fifths

three fifths
Image with alt text: three fifths

Start Fraction 2 pi over 5 End Fraction
Image with alt text: Start Fraction 2 pi over 5 End Fraction

π
π

2π

Answer 1

To find the period of the function \( f(t) = \cos(5t + \frac{\pi}{2}) - 3 \), we focus on the cosine component, since the \(-3\) does not affect the period.

The general form of the cosine function is \( \cos(kx) \), where the period is given by:

\[
\text{Period} = \frac{2\pi}{|k|}
\]

In your case, the argument of the cosine function is \( 5t + \frac{\pi}{2} \). Here, \( k = 5 \).

Now, applying the formula for the period:

\[
\text{Period} = \frac{2\pi}{|5|} = \frac{2\pi}{5}
\]

Therefore, the period of the function \( f(t) \) is:

\[
\frac{2\pi}{5}
\]

So the correct response is:

\(\frac{2\pi}{5}\)