Function Frequencies

Date: 02/03/2001 at 23:00:54
From: danah worrell
Subject: Difference in frequency

Why does the tangent function have twice the frequency of the sine 
and cosine functions?

Date: 02/04/2001 at 02:15:46
From: Doctor Greenie
Subject: Re: Difference in frequency

Hi, Danah -

The fact that the frequency of the tangent function is twice that of 
the sine and cosine functions can be explained by two facts:

     (1) tan(x)     = sin(x)/cos(x)

     (2) sin(x+180) = -sin(x)  
         cos(x+180) = -cos(x)

Here are some words to help show why these two facts explain the 
frequency of the tangent function. It will help if you have good 
mental (or physical) pictures of the graphs of the sine and cosine 
functions and/or of the unit circle, which is often used to define the 
sine and cosine functions.

Choose any arbitrary angle x, and determine the values of the sine 
and cosine functions; then determine the value of the tangent 
function:

     tan(x) = sin(x)/cos(x)

Now choose another angle y exactly 180 degrees larger than x: 
y = x+180. Using either your graphs of the sine and cosine functions 
or your unit circle, you should see that

     sin(y) = sin(x+180) = -sin(x)
     cos(y) = cos(x+180) = -cos(x)

You can also show the truth of these identities - the identities in 
(2) above - by using the formulas for sin(a+b) and cos(a+b), if you 
have encountered those formulas in your studies.

Now let's find the value for tan(y):

     tan(y) = sin(y)/cos(y)
            = [-sin(x)]/[-cos(x)]
            = sin(x)/cos(x)
            = tan(x)

We have shown that, given any angle x, the tangent of the angle x and 
the tangent of an angle 180 degrees larger than x are the same. This 
means that the tangent function completes one cycle every 180 
degrees, while the sine and cosine functions complete a cycle every 
360 degrees.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/04/2001 at 03:21:56
From: Doctor Mitteldorf
Subject: Re: Difference in frequency between tan function

One way to look at it is that the tangent is sine divided by cosine.  
After half their cycle, sine and cosine both repeat the same pattern 
in negative territory:

     sin(180 + x) = -sin(x)
     cos(180 + x) = -cos(x)

So the quotient of these two, (-sin(x))/(-cos(x)), is the same as 
sin(x)/cos(x).

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/