Attempting to intuitively understand why the graph of cos(x^2) looks the way it does.

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cosx and cosx^2 are dependent on x. X represents the arc length (radian measure) or the angle measure of a right triangle. The cosine of the input represents the resultant ratio between the adjacent side of the right triangle with this angle measure and the hypotenuse.

We observe a change in wavelength (or frequency) as x grows. Let’s first focus on other graphs which differ in frequency. Let us say that cosx has a period of 2pi. If so, cos2x has a period of pi. This means that these two equations cycle through their inputs at different rates. To achieve the same output of -1, cosx must go through twice as many values as cos2x.

Rather than thinking of cosx^2 as so, we should think of it as cosx*x. This means that the coefficient of x varies as x does (obviously). This means that your period changes as x does (as opposed to being constant as in the case of cos2x). So you can imagine the output of cosx^2 at each point as being cos(that point *x). In this case, that point is always equal to x. The important thing to notice is that this point is varying, which explains the aforementioned variation in wavelength and frequency. This means that the period would be given by 2pi/x. Rate of change of period is -2pi*1/x^2. I think that the derivative of cos(x^2) is -sin(x^2).

I’m thinking about solving multivariable equations relates to three dimensional graphing.