Trigonometric+Identities

toc =Pythagorean Identities =
 * Trigonometric Identities **

This is a ** Pythagorean Identity **:



Using this first Pythagorean Identity, two additional Pythagorean Identities can be created. The second ** Pythagorean Identity ** is:



Divide each term by.
 * [[image:http://www.regentsprep.org/Regents/math/algtrig/ATT9/pythag7.gif width="229" height="166"]] || Start with the first Pythagorean Identity.

Remember: Reduce and Substitute. ||  ||   ||   || The third ** Pythagorean Identity ** is:


 * [[image:http://www.regentsprep.org/Regents/math/algtrig/ATT9/pythag12.gif align="left"]] || Start with the first Pythagorean Identity.

Divide each term by.

Remember: Reduce and Substitute. ||  ||   ||   ||

If you have the opposite trigonometric functions, they cancel each other out to equal one. tanx cotx= 1 ;because cotx=1/tanx then tanx times 1/tanx= tanx/tanx=1 cosx secx= 1 ;because secx=1/cosx then cosx times 1/cosx= cosx/cosx=1 sinx cscx= 1 ;because cscx=1/sinx then sinx times 1/sinx= sinx/sinx=1 

These three formulas can be algebraically altered in various ways to make other equations, as listed below:
 * **sin²x + cos²x = 1  ---► 1 - sin²x = cos²x **
 * **sin²x + cos²x = 1  ---► 1 - cos²x = sin²x **
 * **tan²x + 1 = sec²x  ---► sec²x - tan²x = 1 **
 * **tan²x + 1 = sec²x ---► sec²x - 1 = tan²x **
 * **1 + cot²x = csc²x  ---► csc²x - cot²x = 1 **
 * **1 + cot²x = csc²x <span style="font-family: Tahoma,Geneva,sans-serif; font-size: 120%;"> ---► csc²x - 1 = cot²x **

=**<span style="color: #000080; font-family: Tahoma,Geneva,sans-serif;">Quotient Identities **=

cos θ ||  || cot θ = || __cos θ __ sin θ || <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">This identity is used for changing tangent/cotangent into terms of sine and cosine, or vice-versa.
 * tan θ = || __sin θ __

=**<span style="color: #000080; font-family: Tahoma,Geneva,sans-serif;">Reciprocal Identities **=

<span style="font-family: Tahoma,Geneva,sans-serif;">When multiplying a trigonometric function by its inverse, the result is 1. For instance...
 * sin θ || **=** || __1__ csc θ ||  || csc θ || **=** || __1__ sin θ ||
 * cos θ || **=** || __ 1 __sec θ ||  || sec θ || **=** || __1__ cos θ ||
 * tan θ || **=** || __1__ cot θ ||  || cot θ || **=** || __1__ tan θ ||
 * tan θ || **=** || __1__ cot θ ||  || cot θ || **=** || __1__ tan θ ||
 * tan θ || **=** || __1__ cot θ ||  || cot θ || **=** || __1__ tan θ ||
 * sin( θ ) x csc( θ ) = 1

<span style="font-family: Tahoma,Geneva,sans-serif;">As a numerical example, 1/2 x 2/1 = 1. In this case, the same principle applies.

This can also be used to move a denominator to the numerator (or vise-versa). Ex: tan(x)/sec(x) can be changed to cos(x) tan(x).

=<span style="color: #000080; font-family: Tahoma,Geneva,sans-serif;">Odd-Even Identities =

= = <span style="color: #000000; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">Odd functions have opposite end behavior. Sine, cosecant, tangent, and co tangent are odd functions.
 * sin(-x)= -sinx || cos(-x) = cosx || tan(-x) = -tanx ||
 * csc(-x)= -cscx || sec(-x)= secx || cot(-x)= -cotx ||

Even functions have the same end behavior. <span style="font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif;">Cosine and secant are even functions

= = =<span style="color: #000080; font-family: Tahoma,Geneva,sans-serif;">Videos = = =

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An Easy to Understand Lesson on Identities