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Precalculus : Find the Period of a Sine or Cosine Function

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Example Questions

Example Question #1 : Find The Period Of A Sine Or Cosine Function

Given $F(t)=sin\left(\frac{x}{2}-4\right)$ , what is the period for the function?

Possible Answers:

$2\pi$

$4\pi$

$\pi$

$\frac{\pi}{2}$

Correct answer:

$4\pi$

Explanation:

The formula for the period of a sine/cosine function is $\frac{2\pi}{|B|}$ .

With the standard form being:

Asin(Bx-C)+D

Since $B=\frac{1}{2}$ , the formula becomes $\frac{2\pi}{\frac{1}{2}}$ .

Simplified, the period is $4\pi$ .

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Example Question #1 : Find The Period Of A Sine Or Cosine Function

What is the period of this graph?

3cos etc with pi and two pi

Possible Answers:

$\frac{2}{3}\pi$

$\pi$

$\frac{3}{2}\pi$

$2\pi$

Correct answer:

$2\pi$

Explanation:

One wave of the graph goes exactly from 0 to $2\pi$ before repeating itself. This means that the period is $2\pi$ .

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Example Question #1 : Find The Period Of A Sine Or Cosine Function

Please choose the best answer from the following choices.

Find the period of the following function in radians:

y=cos(x)

Possible Answers:

$\pi/2$

$\pi$

$2\pi$

180

Correct answer:

$2\pi$

Explanation:

If you look at a graph, you can see that the period (length of one wave) is $2\pi$ . Without the graph, you can divide $2\pi$ with the frequency, which in this case, is 1.

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Example Question #1 : Find The Period Of A Sine Or Cosine Function

Please choose the best answer from the following choices.

Find the period of the following function.

y=5sin(2x)

Possible Answers:

$\pi/2$

$2\pi$

$\pi$

Correct answer:

$\pi$

Explanation:

The period is defined as the length of one wave of the function. In this case, one full wave is 180 degrees or $\pi$ radians. You can figure this out without looking at a graph by dividing $2\pi$ with the frequency, which in this case, is 2.

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Example Question #1 : Find The Period Of A Sine Or Cosine Function

What is the period of this sine graph?

Trig period 1

Possible Answers:

$\frac{3}{2}\pi$

$3\pi$

$\frac{2}{3}\pi$

$\pi^2$

Correct answer:

$\frac{2}{3}\pi$

Explanation:

The graph has 3 waves between 0 and $2\pi$ , meaning that the length of each of the waves is $2\pi$ divided by 3, or $\frac{2}{3}\pi$ .

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Example Question #1 : Find The Period Of A Sine Or Cosine Function

Write the equation for a cosine graph with a minimum at $\left(\frac{ 3 \pi }{ 8 } , -5\right)$ and a maximum at $\left(\frac{ 11 \pi }{ 8 } , -1 \right)$ .

Possible Answers:

$y = 3 \cos (x - \frac{ 11 \pi }{8 }) - 2$

$y = -5 \cos (x + \frac{ 3 \pi }{8 })$

$y = 2 \cos (x + \frac{ 5 \pi }{ 8 }) - 3$

$y = 2 \cos (x - \frac {3 \pi }{ 8 }) - 3$

$y = 3 \cos (x - \frac{ 5 \pi }{ 8 }) - 2$

Correct answer:

$y = 2 \cos (x + \frac{ 5 \pi }{ 8 }) - 3$

Explanation:

The equation for this graph will be in the form $y = A \cos (f (x - h)) + k$ where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

To write this equation, it is helpful to sketch a graph:

Trig graph 4

From sketching the maximum and the minimum, we can see that the graph is centered at y = -3 and has an amplitude of 2.

The distance between the maximum and the minimum is half the wavelength. Here, it is $\frac{ 11 \pi }{ 8 } - \frac{ 3 \pi }{ 8 } = \frac{ 8 \pi }{ 8 } = \pi$ . That means that the full wavelength is $2 \pi$ , so the frequency is 1.

The minimum occurs in the middle of the graph, so to figure out where it starts, subtract $\pi$ from the minimum's x-coordinate:

$\frac{ 3 \pi }{ 8 } - \pi = \frac{ 3 \pi }{8 } - \frac{ 8 \pi }{ 8 } = \frac{ -5 \pi }{8 }$

This graph's equation is

$y = 2 \cos (x + \frac{ 5 \pi }{8 }) -3$ .

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Example Question #1 : Find The Period Of A Sine Or Cosine Function

Give the period and frequency for the equation $y = 4 \sin (6x - \pi ) + 2$ .

Possible Answers:

Period: $6 \pi$ , Frequency:

Period: $\frac{ \pi }{ 2 }$ , Frequency:

Period: $12 \pi$ , Frequency:

Period: $\pi$ , Frequency:

Period: $\frac{ \pi }{3}$ , Frequency:

Correct answer:

Period: $\frac{ \pi }{3}$ , Frequency:

Explanation:

Our equation is in the form $y = A \cdot \sin ( fx - h ) + k$

where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

We can look at the equation and see that the frequency, , is .

The period is $\frac{ 2 \pi }{f }$ , so in this case $\frac{ 2 \pi }{6} = \frac{ \pi }{3}$ .

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Example Question #1 : Find The Period Of A Sine Or Cosine Function

What is the period of the graph $y = -2 \cos (\frac { \pi }{8 } x + 1 )$ ?

Possible Answers:

$\frac{ \pi }{ 8 }$

$\frac{ 8}{\pi }$

$\frac{ \pi }{4}$

Correct answer:

Explanation:

The equation for this function is in the form $y = A \cos (fx - h) + k$

where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

By looking at the equation, we can see that the frequency, , is $\frac{ \pi }{ 8 }$ .

The period is $\frac{ 2 \pi }{f }$ , so in this case $\frac{ 2 \pi }{ \frac{ \pi }{ 8 }} = 2 \pi \div \frac{ \pi }{ 8 } = \frac{ 2 \pi }{1} \cdot \frac{ 8}{\pi } = 16$ .

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Example Question #9 : Find The Period Of A Sine Or Cosine Function

What could be the function for the following graph?

4sin

Possible Answers:

$y=-2\sin(x)+4$

$y=4\cos(x)+2$

$y=\sin(x)-2$

$y=4\sin(x)+2$

Correct answer:

$y=4\sin(x)+2$

Explanation:

What could be the function for the following graph?

4sin

Begin by realizing we are dealing with a periodic function, so sine and cosine are your best bet.

Next, note that the range of the function is $-2\leq y \leq 6$ and that the function goes through the point (0,2) .

From this information, we can find the amplitude:

$A=\frac{6--2}{2}=\frac{8}{2}=4$

So our function must have a out in front.

Also, from the point (0,2) , we can deduce that the function has a vertical translation of positive two.

The only remaining obstacle, is whether the function is sine or cosine. Recall that sine passes through (0,0) , while cosine passes through (0,1) . this means that our function must be a sine function, because in order to be a cosien graph, we would need a horizontal translation as well.

Thus, our answer is:

$y=4\sin(x)+2$

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Precalculus : Find the Period of a Sine or Cosine Function

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find The Period Of A Sine Or Cosine Function

Example Question #1 : Find The Period Of A Sine Or Cosine Function

Example Question #1 : Find The Period Of A Sine Or Cosine Function

Example Question #1 : Find The Period Of A Sine Or Cosine Function

Example Question #1 : Find The Period Of A Sine Or Cosine Function

Example Question #1 : Find The Period Of A Sine Or Cosine Function

Example Question #1 : Find The Period Of A Sine Or Cosine Function

Example Question #1 : Find The Period Of A Sine Or Cosine Function

Example Question #9 : Find The Period Of A Sine Or Cosine Function

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