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Find an Equation for the Sine or Cosine Wave - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

When finding the equation for a trig function, try to identify if it is a sine or cosine graph. To find the equation of sine waves given the graph, find the amplitude which is half the distance between the maximum and minimum. Next, find the period of the function which is the horizontal distance for the function to repeat. If the period is more than 2pi, B is a fraction; use the formula period=2pi/B to find the exact value. Last, find any phase shift, h.

I want to find an equation of a sine or a cosine wave. I've got an example. Here I'm given a graph and I have to come up with the equation. So he's given an equation of the graph that I have here.
I first have to decide whether this is going to be a sine or cosine curve. From the look of it, I would say cosine because it starts at a maximum point. You know, the sine graph looks something like this. It starts on a 0, on an intercept and then goes up whereas the cosine curve starts like this. So I'm going to guess that this is some kind of transformed cosine curve with an equation of the form y=a cosine bx. So the question is what's a and what's b.
Well, this is really our only clue. We have 2 pi as the x value of this point and the -3 as the y value and I'm going to assume that this this length is the same as this length and in that case that means the amplitude is 3. Right? Because amplitude gives me half the distance between the maximum and minimum. So if the amplitude is 3, that means a is either plus or minus 3. And I'm going to say that it's +3 because cosine normally starts from its maximum. And for this it's starting from its minimum. I would say that a was -3.
So we have that y=3 cosine bx. We just have to find b. So it's going to have something to do with the x coordinate here. We have to figure out what the period of this function is and it looks like it's going to be 4 pi. If this is 2 pi then this is going to be 4 pi. Now if the period is 4 pi, I can use this formula for period to find out what b is. Plug in 4 pi for the period then I can multiply both sides by b and then divide by 4 pi. And it looks like I'm going to get one half. b is one half and so our final answer is y=3 times the cosine of one half x.
When you're finding the equation for the graph of a sine or cosine curve, you have to first try recognize whether it's a sine or cosine curve. A sine curve would start at the origin like this. It starts on an intercept whereas a cosine curve starts at its max or its mean if it's reflected.

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