# Amplitude, Period and Frequency

Amplitude and Period of Sine and Cosine

Tip: This isn't the place to ask a question because the teacher can't reply.

## Key Questions

• If $f \left(x\right) = a \sin \left(b x\right)$ or $g \left(x\right) = a \cos \left(b x\right)$, then their amplitudes are $| a |$, and the periods are $\frac{2 \pi}{|} b |$.

I hope that this was helpful.

• Frequency and period are related inversely. A period $P$ is related to the frequency $f$
$P = \frac{1}{f}$

Something that repeats once per second has a period of 1 s. It also have a frequency of $\frac{1}{s}$. One cycle per second is given a special name Hertz (Hz). You may also say that it has a frequency of 1 Hz.

A sin function repeats regularly. Its frequency (and period) can be determined when written in this form:

$y \left(t\right) = \sin \left(2 \pi f t\right)$

• The coefficient of the angle is key to both; it is the number of cycles in $2 \pi$ radians (or $360$ degrees) and hence represents the frequency on $2 \pi$, which in graphing is commonly referred to as just the "frequency," while simultaneously being the divisor of $2 \pi$ radians (or $360$ degrees) that results in the length to complete one cycle, which is called the period (my one behemoth sentence answer).

In short, if you have a trigonometric function in the form:
$A \sin \left(B \left(x - C\right)\right) + D$ or
$A \cos \left(B \left(x - C\right)\right) + D$

$P e r i o d = \frac{2 \pi}{B}$

$F r e q u e n c y = \frac{1}{P e r i o d}$

For tangent: $A \tan \left(B \left(x - C\right)\right) + D$, the period would be $\frac{\pi}{B}$

Let's try an example:

graph{3sin(2x) [-16.02, 16.02, -8.35, 8.34]}

$y = 3 \sin 2 x$

In short terms, the period is $\frac{2 \pi}{2} = \pi$
The frequency is $\frac{1}{\pi}$

In longer in-depth terms, it has a frequency (on $2 \pi$ or $360$ degrees) of $2$, meaning that there will be $2$ complete cycles from $0$ to $2 \pi$ (or $360$ degrees), or any domain with that width (like $- \pi$ to $\pi$).

The period, or length of each cycle can be found by dividing $2 \pi$ or $360$ by that coefficient revealing that in the above example, each complete cycle is $\pi$ or $180$ degrees in width.

As a side note: in real world applications (word problems) frequency is actually the reciprocal of period and so is the angles coefficient divided by $2 \pi$ (or possibly $360$), or$\frac{2}{2} \pi$ ($\frac{2}{360}$) in the above example. This is because period is time or distance per cycle, while frequency should be cycles per unit time (not per $2 \pi$ units of time because the world doesn't count like that--we want to know how many revolutions per second our tires are making, not how many revolutions in $6.28$ seconds, usually.)

Thanks for the question.

• "(Amplitude)"=1/2["(Highest Value)"-"(Lowest Value)"]

graph{4sinx [-11.25, 11.25, -5.62, 5.625]}

In this sine wave the highest value is $4$ and the lowest is $- 4$

So the maximum deflection from the middle is $4$k.

This is called the amplitude

If the middle value is different from $0$ then the story still holds
graph{2+4sinx [-16.02, 16.01, -8, 8.01]}

You see the highest value is 6 and the lowest is -2,
The amplitude is still $\frac{1}{2} \left(6 - - 2\right) = \frac{1}{2} \cdot 8 = 4$

## Questions

• · Yesterday
• · 4 days ago
• · 5 days ago
• · 1 week ago
• · 1 week ago
• · 2 weeks ago
• · 2 weeks ago
• · 2 weeks ago
• · 2 weeks ago
• · 2 weeks ago
• · 3 weeks ago
• · 3 weeks ago
• · 3 weeks ago
• · 3 weeks ago
• · 4 weeks ago
• · 1 month ago
• · 1 month ago
• · 1 month ago
• · 1 month ago
• · 1 month ago
• · 1 month ago

## Videos on topic View all (1)

• No other videos available at this time.