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# Amplitude & period of sinusoidal functions from equation

Sal finds the amplitude and the period of y=-0.5cos(3x). Created by Sal Khan and Monterey Institute for Technology and Education.

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• what does speeding up the rotation by 3 times mean? if one period is 2pi how can it speed up the rotation without completing 3 circles? I'm like stuck in this thought, can anyone help me understand what is a period? Every time when the period is 2 or more, I think of it completing 2pi twice, but apparently it does not. How can you complete 2pi 3 times faster without going around 2pi 3 times? I don't know if I'm explaining my struggle right, but that is pretty much it, can anyone help me? Does going 3 times faster literally means condensing the graph into three 2pis? Which means like doing the same length of 2pi 3 times in one circle? • The x-axis shows the measure of an angle. We know y=cos(x) completes a full cycle or period for every change of 2π radians along the x-axis, and as a consequence cos(2π) = cos(0). y=cos(2x) completes a full cycle for every change of π radians along the x-axis, and when x = π, cos(2x) = cos(2 * π) = cos(0). So, for a given change in x, cos(2x) completes more cycles than cos(x).

I would say you are right to think of this as meaning that cos(2x) completes 2π twice over the interval that cos(x) completes only one cycle, but wrong to say that "one period is 2π". One period of cos(x) is 2π, but one period of cos(2x) is π. In other words, "period" is descriptive of a specific function, not of whatever function you perceive to be the "underlying function". In some sense, cos(2x) does behave very like cos(x), but its period is different, judging by our definition of period, which is the change in x over one complete cycle of a function.
• Can someone please illustrate what is happening with the unit circle for the equation in this video?

I think Sal and the people below did an excellent job explaining the "periodic" concept verbally and mathematically, but what is happening on the unite circle/cartesian graph? It makes sense visually on a sinusoid graph (x axis in units of Pi), but not the unit circle/Cartesian graph as theta goes in circles.

The cos(x) graph repeats because it completes a cycle(2pi). How can a value repeat it'self 1/3 of the rotation, without completing a cycle?

I've looked online at other sources, and everyone is more than happy describing what's happening on the unit circle when y=sin(x), or cos(x) with a cycle of 2pi. But when looking at an equation of y=sin (3x) they completely avoid describing it in terms of the unit circle graph, and only allude to the sinusoid graph. Can someone please illustrate what is happening with the unit circle for the equation in this video? • I've done a lot of work and understand the math and sinusoid graph throughly, but there's something fundamental/visual that's off. I spoke to a friend, and he says the issue is, I'm thinking of the unit circle as a function, and I should not interpret something like y=cos(1/3x), which has a period of 6pi back into the unit circle. I have to wrap my head around this a little more, sometimes the x axis sounds like a function of time, and some times it's a representation of angles. That's the best way I can explain it right now,but feel free to give me more input if you can understand this issue better..
• • What would the amplitude of a tangent function be? • Why is the amplitude always the number the trig function is multiplied with? • Both the normal sine and cosine functions sway between 1 and -1. When you add a coefficient, you are multiplying that positive one or negative one by the coefficient, giving you a new amplitude equal to the absolute value of your coefficient.

Example: y= 2 sin (x)
The normal sine function is bound between 1 and -1, so the 2 coefficient multiplies those values by two, giving us a function with an amplitude of 2.
• Could someone please tell me how does the number before x make the rotation faster?
E.g. in
y = -3sine(4x)
How does the 4 make x rotate around the unit circle four times faster? What I thought was that surely the 4 only increases x by four times, then it will be a different angle so the period is still going to be 2pi.

Please could someone help me. I'm really confused.

Thank you very much! • Normally the period of sin(x) would be 2pi long. Lets compare sin(4x) and sin(x):
At x=1
sin(x) = sin(1) , sin(4x) = sin(4)
At x=2
sin(x) = sin(2) , sin(4x) = sin(8)

As you can see in sin(x) x goes up by plus 1 every time (0, 1, 2, 3, 4,...).
On the other hand sin(4x) x goes up by plus 4 every time( 0, 4, 8, 12, 16,...).

In sin(4x) the gap between the inputs are 4 times as large (4/1=4, 8/2=4,...) so it technically jumps over 1/4 of sin(x) (1/4 = 2/8 = 3/12,...).

I'm sorry if my answer is confusing, but I hope it helps.
• So amplitude refers to the highest point the graph of the sine/cosine function reaches on the y axis while period is the length on the x axis in one cycle, am I right? Thanks. • You are partially correct: the period is the length on the x axis in one cycle. However, the amplitude does not refer to the highest point on the graph, or the distance from the highest point to the x axis. The amplitude is 1/2 the distance from the lowest point to the highest point, or the distance from the midline to either the highest or lowest point. This is an important distinction when the trig function is shifted up or down.
• How do I find amplitude and period for a tangent function? • Is there a "proof of sinusoidal functions" video? I see the one where we graph from a unit circle, but is there a video explaining how each of these parts of the sinusoidal function cause the manipulations the way they do? • Everything Vader said is correct. Just want to add that I had an insight as to what's really happening when I realized what b is doing. If you write a function for temperature over a 24 hr period, then 2π/24 really just breaks the unit circle into 24 equal parts, and t chooses one of those parts. Once I figured that out, the rest of it just began to fall into place. Hope that helps.
• At , I don't understand why we start at y=-1/2 when x=0. When I read Y=-1/2cos3X, I get Y= 1/2(times)cos3(times)X, wich should be Y=0 when X=0. Where do I get it wrong ?
(1 vote) • A point on the Y axis is given by running a value for the variable x through the equation Y=1/2 * cos(3*x)

A careful point to note here. Lets look at a cosine function all by itself. it is written as cos(x). whatever is in the parenthesis is called the argument of the function. Now recall the order of operations. Parenthesis, then exponents, then multiplication and division, then addition and subtraction. You must clear parenthesis first. So for a basic cosine function of cos(x) you would do whatever operations exist inside of the parentheses.

Returning to our original equation and just looking at the cosine function for now, we see that we have cos(3*x). The variable x here can represent any point on the x axis, lets choose 0.
We begin by substituting 0 for x in the cosine function.
We now have cos(3*0). This evaluates to cos(0) because 3*0 =0.
Now we have the equation Y=-1/2 * cos(0).
Lets continue evaluating our cosine function, we aren't done yet!What is the cosine of 0? it is 1.
So now we have Y= -1/2 * 1.
Continuing to solve and simplify and you see that -1/2 * 1 = -1/2.
So we now have Y=-1/2.
Thus for the coordinate 0 on the x-axis, the y coordinate is -1/2.

Its important to do the order of operations, especially with trig functions!

Good luck!