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Transforming the Cotangent Graph - Concept
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Cotangent is the reciprocal trig function of tangent function and can be defined as cot(theta)=cos(theta)/sin(theta). It is an odd function, meaning cot(-theta)=-cot(theta), and it has the property that cot(theta+pi)=cot(theta). Because sine is the denominator, and the function is undefined when sin(theta)=0, the cotangent graph has vertical asymptotes at all integer multiples of pi, when sin(theta)=0.
I want to talk about transformations of the cotangent graph but first let's review some things about cotangent. y equals the cotangent theta remember that cotangent is the same as cosine theta over sine theta that's an important identity and we'll need to develop a few facts about cotangent, for example is cotangent even or is it odd? Well we can use the fact that cosine is even and sine is odd. Cotangent of negative theta would equal cosine of negative theta over sine of negative theta. Cosine of negative theta equals cosine theta and sine of negative theta is negative sine theta and so this equals negative cotangent theta now we have the cotangents nod function. So opposite inputs give opposite outputs that's important to know. Then I want to find out what happens when I add pi, now I know that this equals cosine of theta plus pi over sine of theta plus pi.
Now both cosine and sine of the property that if you add pi to the input you get the opposite output. So this could be minus cosine of theta over minus sine of theta. Now the minuses will cancel and you'll just get cotangent theta. So if you add pi you get the exact same output, this means cotangent is periodic with period pi, so we'll need to know that. Okay let's start making a table of values for cotangent; let's start with 0, now cotangent is cosine over sine, sine of 0 is 0. So cotangent is going to be undefined at 0, what about let's say pi over 4, cosine and sine both have the value root 2 over 2. So this is going to be 1, root 2 over 2 over root 2 over 2, 1 pi root 2. Cosine of pi over 2 is 0, sine of pi root 2 is 1 so we get 0 over 1 which is 0.
Now let me actually work backwards, let me give us a little room here cotangent theta I'll work back to negative pi over 4 because cotangents node function, the cotangent of negative pi over 4 will be the opposite of the cotangent of pi over 4. So we'll get negative 1 here also because it has period pi if I add pi to this which will give me 3 pi over 4 I'll get the same output. And I can actually use periodicity to get the rest of the output, if I add pi to this, to 0 I get pi. I get the same output undefined, this is actually one complete period of the cotangent function. And so I can start graphing, let me start by plotting the vertical asymptotes, there's going to be a vertical asymptote to x=0 and to x=pi. So one of them is here and one of them is here, now in between what happens?
At pi over 2 we get 0, this is pi over 2, at pi over 4 we get 1 and at 3 pi over 4 we get negative 1. Now recall the shape of tangent, cotangent has a very similar shape, it's got this kind of curvy shape and of course it has the asymptotic behavior. So this is the rough shape of the cotangent graph. Now if you want to graph more of it remember this is one period all you have to do is duplicate this period. I can take this whole thing and shift it to the right, and shifting to the right everything shifts right including the asymptote. So for example when I shift this to the right, this asymptote shifts to the right as well. So I get another one at x=2 pi and I can also shift to the left, this asymptote shifted to the left gives me x equals negative pi. So let me shift each of these points to the right, this point shifted to the right gives me a point at 3 pi over 2, this one here, this one I'll put here and we're just using periodicity at this point.
I'll shift this to the left I get a point here, here, here and this is 3 periods of the cotangent function. So you'll notice it was actually really easy to develop the key points for this function all I needed to do was to remember that cotangent was cosine over sine and that allowed me to find zeros of the function and also where the function was undefined. Very easy to know that cotangent of pi over 4 is 1 because cosine and sine are both root 2 over 2. I also used the fact that cotangent was odd to get this value and then I used periodicity and turned this value into the value at 3 pi over 4. And that's all you need values at 0, pi over 4, pi over 2, 3 pi over 4 and pi and once you get a full period use periodicity to extend the graph in both directions.
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