Function FrequenciesDate: 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/ |
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