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I want to know why 3 types of trig graphs have the shape that they do:

  1. Secant graphs. Why would $\dfrac {1}{\cos \theta}$ result in a graph like this? I get the cosine graph - it makes sense when you compare it to a unit circle. But I don't understand it at the same level for secant graphs. Can someone please explain why the secant graph is shaped like that given it's equation?

  2. Cosecant graphs. Same as above.

  3. Cotangent graphs. Why would $\dfrac {1}{\tan \theta}$ cause the tangent graph to flip? Because when you look at a cotangent graph, it's basically a reflected tangent graph. Why would $\dfrac {1}{\cos \theta}$ create a graph that looks like that?

Can you please give the explanation for why the graphs are shaped like that not too rigorously, and at the level of a Precalculus student who hasn't learnt Calculus yet? Thank you.

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  • $\begingroup$ When $\cos(\theta) = \pm 1$, then $\sec(\theta) = \pm 1$. When $\cos(\theta) = 0$, then $\sec(\theta)$ is undefined---this implies the possible existence of an asymptote (you need limits to confirm this, and that is usually a topic for the end of pre-calc, or for calculus). Between the ones and the asymptotes, the cosine function either increases or decreases monotonically, so the secant function will have the same behaviour, only in the opposite direction. This can be made rigorous without calculus (though calculus makes it easier). $\endgroup$ – Xander Henderson Jul 26 '18 at 14:31
  • $\begingroup$ That being said, I think that this is a question which might be more appropriate for the Mathematics Educators SE. $\endgroup$ – Xander Henderson Jul 26 '18 at 14:32
  • $\begingroup$ @XanderHenderson I'm not an educator though; I'm a student. But yeah, thanks for the response. Last thing though, what about for cotangent graphs? Why would it be a perfect reversal of tangent graph? $\endgroup$ – Ethan Chan Jul 26 '18 at 14:35
  • $\begingroup$ Because $\tan(\dfrac{\pi}{2}-x) = \cot(x)$ $\endgroup$ – steven gregory Jul 26 '18 at 15:16
  • $\begingroup$ " Can someone please explain why the secant graph is shaped like that given it's equation? " Can you explain why you don't understand it? It's $\frac 1{\cos x}$ and as $\cos x$ is periodic from 0 to 1 to 0 to -1 back to 0, It's reciprical will be periodic from $+\infty$ to $1$ back to $\infty$then an axis jump to $-\infty$ up to $-1$ and back down. .... And that is exactly what it does do. So from my point of view $\frac 1{\cos x}$ exactly as I expect others would expect it to do. So ... I'm confused why you are confused. $\endgroup$ – fleablood Jul 26 '18 at 16:03
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consider how behavior in a graph of $f(x)$ will affect behavior in a graph $\frac 1 {f(x)}$. For values where $0 < f(x) < 1$ then we will have $\infty > \frac 1{f(x)}$, with little changes from $teensy$ to $\frac 12 teensy$ resulting in huge stretches from $huge$ to $2 \times huge$. And if $f(x)$ goes through from $teensy$ to $0$ to $-teensy$ then $\frac 1{f(x)}$ will get asymptotic to infinity, be infinite(undefined) at a point, and jump to "negative infinity" and then imediate start reducing to more moderate negative values as $f(x)$ because more siginificant negative values.

If $1 < f(x)$ we will have $1 > \frac 1{f(x)} > 0$ and if $f(x)$ grows blithely huge toward $huge$, $double-huge$ and $huge^2$ then $\frac 1{f(x)}$ will tend to flatline toward $0$.

So bearing that in mind:

$\cos x$ is periodic from $0$ to $1$ back down $0$ and and to $-1$ to $0$ to $1$ etc. and $\frac 1{\cos x}$ behaves exactly as we'd expect, for $+\infty$ down to a minimum of $1$ (as $\cos x$ goes from $0$ to $1$ and peaks) and then back to $\infty$. Then as $\cos x$ goes from tiny positive through $0$, to tiny negative. $\frac 1{\cos x}$ goess to $+\infty$ and jumps to $-\infty$ and starts surfaces up to $-1$ and then (as $\cos x$ reaches a nadir of $-1$ and heads back to $0$) $\frac 1{\cos x}$ reaches max at $-1$ and starts plumeting back to $-\infty.

As for $\tan x$ and $\frac 1{\tan x}$. $\tan x$ will go from $\infty$ where $\cos x = 1; \sin x= 0$ to $1$ where $\cos x = \sin x = \frac 1{\sqrt 2}$. In this stage $\sin x$ is increasing faster than $\cos x$ is decreasing. So there will be a bulge to the right in the graph. Then as $\cos x\to 0$ and $\sin x \to 1$ we have $\frac {\sin x}{\cos x} \to \infty$. and as $\cos x$ is not decreasing faster than $\sin x$ is increasing, the graph stretches to be long and skinny. Then as $\cos x$ goes through $0$ to negative, we jump to negative infinity.

In the same interval $\cot x = \frac {\cos x}{\sin x}$ starts and $\frac 10 = \infty$ and decreases down to $1$ and then to $0$. The exact same but in reverse. Then as $\cos x$ goes threough $0$ to negative $\frac {\cos x}{\sin x}$ goes ther $0$ to negative.

Useful to realize $\cot x =\frac {\cos x}{\sin x } = \frac {\sin(\frac \pi 2 - x)}{\cos \frac \pi 2 - x)} = \tan (\frac \pi 2 - x)$ so that explains more rigororously why their graphs are symmetriic.

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Pictures often help.

Here's a graphical depiction of $\cos \theta$ and $\sec \theta = \frac{1}{\cos \theta}$ for $\theta = \frac{\pi}{3} = 60$ degrees.

enter image description here

The situation for $\cos \theta$ is represented by the right triangle $\triangle ABC$. The radius is $CA = 1$, and the cosine of $\theta$ is represented by the length of $CB = \frac12$.

Now, extend $CA$ rightward to $A'$, which lies on the vertical line tangent to the unit circle. I submit that the secant of $\theta$ is represented by the length $CA'$, on this basis:

  • $\triangle ABC$ and $\triangle A'B'C$ are similar.
  • Therefore $CA'$ is to $CB'$ as $CA$ is to $CB$.
  • Since $CB' = CA = 1$, we have that $CA'$ is to $1$ as $1$ is to $CB$.
  • Symbolically, $CA' = \frac{CA'}{1} = \frac{1}{CB}$.
  • And since $CB$ is $\cos \theta$, $CA' = \frac{1}{\cos \theta} = \sec \theta$.

Finally, imagine increasing $\theta$ toward $\frac{\pi}{2} = 90$ degrees. It is evident that $A'$ must move vertically upward—without bound, in fact, and that is indeed what the plot of $\sec \theta$ does.

If you increase $\theta$ beyond $\frac{\pi}{2}$, but continue to extend $AC$ rightward toward the same vertical tangent, you find that $\sec \theta$ is negative and initially large, but decreasing toward $0$ as you approach $\theta = \pi = 180$ degrees.

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