How to find the period of the cosine - ACT Math

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Question

A function with period will repeat its solutions in intervals of length .

What is the period of the function ?

Answer

For a trigonometric function $f(x) = \alpha cos(bx -c) +d$ , the period is equal to $\frac{2\pi}{b}$ . So, for , $P = \frac{2\pi}{1} = 2\pi$ .

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Question

A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Find the period of the function

$cos, (\pi x)$ .

Answer

For the function

the period is equal to

$\frac{2\pi}{A}$

or in this case

$\frac{2\pi}{\pi}$

which reduces to .

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Question

A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Find the period of the function

.

Answer

For the function

the period is equal to

$\frac{2\pi}{A}$

or in this case

$\frac{2\pi}{2}$

which reduces to $\pi$ .

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Question

A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Find the period of the function

$\cos \bigg(\frac{1}{4}\pi x\bigg)$ .

Answer

For the function

the period is equal to

$\frac{2\pi}{A}$

or in this case

$\frac{2\pi}{\frac{1}{4}\pi}=2\pi \cdot \frac{4}{1\pi}$

which reduces to .

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Question

A function with period will repeat its solutions in intervals of length .

What is the period of the function $f(x) = 3cos(4\pi x-4) -2$ ?

Answer

For a trigonometric function $f(x) = \alpha cos(bx -c) +d$ , the period is equal to $\frac{2\pi}{b}$ . So, for $f(x) = 3cos(4\pi x-4) -2$ , $P = \frac{2\pi}{4\pi} = \frac{1}{2}$ .

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Question

A function with period will repeat its solutions in intervals of length .

What is the period of the function $f(x) = 2cos(\frac{1}{2}\pi x)$ ?

Answer

For a trigonometric function $f(x) = \alpha cos(bx -c) +d$ , the period is equal to $\frac{2\pi}{b}$ . So, for $f(x) = 2cos(\frac{1}{2}\pi x)$ , $P = \frac{2\pi}{\frac{1}{2}\pi} = 4$ .

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