Deriving the Period of a Tan Function - Trig Graphs

[datafiend]

I know the general equation for trig functions and how to manipulate them:
y=A sin [B (x-c)] + D
howver , the tan function has a period of ∏/b. how is this derived? I know it has to do with tan = y/x right? but I just don't understand how to derive the period when you're graphing a tan function.

Thanks,
Randy

 

[micromass]

You're asking about the function $f(x) = \tan(x)$? Then what is $b$?

Also, how did you define the tangent function?

 

[Mark44]

I know the general equation for trig functions and how to manipulate them:
y=A sin [B (x-c)] + D
howver , the tan function has a period of ∏/b. how is this derived? I know it has to do with tan = y/x right? but I just don't understand how to derive the period when you're graphing a tan function.

Thanks,
Randy

In addition to what micromass said, "tan = y/x" is not correct. There's an angle involved that you don't show. Tangent of what? It's a little like saying "√ = 4". Square root of what?

 

[datafiend]

hmmm...maybe I'm not being clear. If using the "unit circle" the sin function is y/r, cos is x/r, tan is y/x, where r=1. Is this not how you plot a sin/cos/tan function on the x/y plane?

 

[micromass]

hmmm...maybe I'm not being clear. If using the "unit circle" the sin function is y/r, cos is x/r, tan is y/x, where r=1. Is this not how you plot a sin/cos/tan function on the x/y plane?

Well first of all, if you take a unit circle definition, then I don't see why you bother to write the $r$.

Second, you ignored the post by Mark. There is no such thing as a $\tan$. You need to take the $\tan$ of some angle.

Anyway, let's move on to you question. You know that

\tan(x) = \frac{\sin(x)}{\cos(x)}

holds for all $x$ for which the fraction on the right is defined.
Can you try to show that

\tan(x+\pi) = \tan(x)

To do this, do you know some formulas for $\sin(x+\pi)$ and $\cos(x+\pi)$?

 

[datafiend]

tan(x) formula

I'm sorry, but I don't see how this is germane to a tan function. In a standard problem that asks to A:graph a tangent function B:show the period of the function C: show the asymptotes D: give the domain and range. I don't see how the formulas \sin(x+\pi) and \cos(x+\pi) help me get there.

Thanks

 

[micromass]

Seeing as this is a standard problem, I moved it to the homework forums. Now, please provide an attempt at solving the problem before we can continue.

 

[datafiend]

graph tan(4t)

ok. here is one I missed. graph y= tan(4t)
A: find the period.
B: find the phase shift
c: give the domain/range
d: find the asymptotes
the general formula for the tan/cot funcit is y= A tan [B (x-C)] + D. A is amp, ∏/B is the period, C will give the phase shift, and D is the vertical translation.
I know that at the points on the unit circle (0,1) and (0,-1) the function is UNDEFINED, so this is the asymptote. Now WHAT IS THE 4t? This is what I missed.
Thanks,

 

[vanceEE]

$4t$ is your independent variable, instead of an $x$ or $\theta$ that you might usually see. You can use this 4t to find the period of this particular function. $tan\theta$ contains a period of $\pi$ but since you're dealing w/ $tan(n\theta$), your period will be $\frac{\pi}{n}$ which will give you a ratio of $\pi$ relative to your function, in other words n($\frac{\pi}{n}$) = $\pi$