TEF - Amplitude and Period Lesson

Amplitude and Period

One important feature of a sine or cosine curve is the midline. The midline is a horizontal line that is halfway between the maximum and minimum. In both equation image indicator $LaTeX: f\left(t\right)=\sin\left(t\right)$ $f\left(t\right)=\sin\left(t\right)$ and $LaTeX: f\left(t\right)=\cos\left(t\right)$ $f\left(t\right)=\cos\left(t\right)$ the midline is y = 0.

image of sine curve on graph

image of cosine curve graph

The amplitude of a sine or cosine curve is the distance from the midline to either the maximum or minimum value. In both, equation image indicator $LaTeX: f\left(t\right)=\sin\left(t\right)$ $f\left(t\right)=\sin\left(t\right)$ and $LaTeX: f\left(t\right)=\cos t$ $f\left(t\right)=\cos t$ the amplitude is 1.

image of sine curve with amplitude indicated on graph

image of cosine curve with amplitude indicated on graph.png

So, what do you think will happen if we take equation image indicator $LaTeX: f\left(t\right)=\sin\left(t\right)$ $f\left(t\right)=\sin\left(t\right)$ and multiply it by 2, creating $LaTeX: f\left(t\right)=2\sin\left(t\right)$ $f\left(t\right)=2\sin\left(t\right)$ ?

We can see that the amplitude grows, so instead of being 1, the distance from the midline to the maximum or minimum becomes 2.

A function of the form, equation image indicator $LaTeX: f\left(t\right)=A\sin\left(t\right)\:or\:f\left(t\right)=A\cos\left(t\right)$ $f\left(t\right)=A\sin\left(t\right)\:or\:f\left(t\right)=A\cos\left(t\right)$ , then LaTeX: |A|= $|A|=$ amplitude. Notice that the amplitude is always positive. The amplitude is the distance from a maximum point to the midline or a minimum point to the midline.

You may have noticed that sine and cosine are cyclical - this is because coterminal angles have the same trig ratios.

For instance:

equation image indicator $LaTeX: \sin\left(\frac{\pi}{3}\right)=\frac{1}{2}\:and\:\sin\left(\frac{7\pi}{3}\right)=\frac{1}{2}\:because\:\frac{\pi}{3}\:and\:\frac{7\pi}{3}\:are\:coterminal\:angles.$ $\sin\left(\frac{\pi}{3}\right)=\frac{1}{2}\:and\:\sin\left(\frac{7\pi}{3}\right)=\frac{1}{2}\:because\:\frac{\pi}{3}\:and\:\frac{7\pi}{3}\:are\:coterminal\:angles.$

Recall: coterminal angles are $LaTeX: 2\pi$ $2\pi$ radians apart!

So, what we notice in the sine and cosine graphs is that they repeat every equation image indicator $LaTeX: 2\pi$ $2\pi$ radians. In the images below, we can see sine and cosine are repeating every $LaTeX: 2\pi$ $2\pi$ radians.

image of sine curve repeated with consistent radians indicated on graph

image of cosine curve repeated with consistent radians indicated on graph

We can change the period of a sine or cosine function by dividing the argument by a constant. Watch the video below as we change the graph from equation image indicator $LaTeX: f\left(t\right)=\cos\left(t\right)\:to\:f\left(t\right)=\cos\left(2t\right)$ $f\left(t\right)=\cos\left(t\right)\:to\:f\left(t\right)=\cos\left(2t\right)$

Remember for f(t)=A sin(Bt) or f(tO=A cos(Bt)
amplitude= |A|
Period = (2pi)/|B|
Distance between critical points = Period/4

A function of the form, equation image indicator $LaTeX: f\left(t\right)=\sin\left(Bt\right)\:or\:f\left(t\right)=\cos\left(Bt\right)$ $f\left(t\right)=\sin\left(Bt\right)\:or\:f\left(t\right)=\cos\left(Bt\right)$ , then $LaTeX: \frac{2\pi}{|B|}=period$ $\frac{2\pi}{|B|}=period$ .

Watch this video to practice graphing:

IMAGES CREATED BY GAVS