Unit 4A

Trigonometry Part 1

Lesson 2: Trigonometric Identities

Pythagorean Identities

Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»P«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#952;«/mo»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»y«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» be the point of intersection of the terminal arm of angle «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#952;«/mo»«/mstyle»«/math», in standard position, and on the unit circle. Recall this point can be used to draw a right triangle with sides «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi»y«/mi»«mo»,«/mo»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math».


It is also true that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mi»y«/mi»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»cos«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math». From this information, an identity using the Pythagorean theorem can be derived.


Dividing this identify by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math» gives a second identity.


Dividing the original identity by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mo»§#952;«/mo»«/mrow»«/mstyle»«/math» gives a third identity.


The Pythagorean identities are as follows.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable»«mtr»«mtd»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mo»§#952;«/mo»«mo»+«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mo»§#952;«/mo»«mo»=«/mo»«mn»1«/mn»«/mtd»«mtd/»«mtd/»«mtd»«mn»1«/mn»«mo»+«/mo»«msup»«mi»cot«/mi»«mn»2«/mn»«/msup»«mo»§#952;«/mo»«mo»=«/mo»«msup»«mi»csc«/mi»«mn»2«/mn»«/msup»«mo»§#952;«/mo»«/mtd»«mtd/»«mtd/»«mtd»«mn»1«/mn»«mo»+«/mo»«msup»«mi»tan«/mi»«mn»2«/mn»«/msup»«mo»§#952;«/mo»«mo»=«/mo»«msup»«mi»sec«/mi»«mn»2«/mn»«/msup»«mo»§#952;«/mo»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»

Trigonometric expressions can often be simplified in more than one way, as is shown in Example 3. As a general guideline, simplifying should involve rewriting an expression with as few terms as possible, often, but not always, using only the primary trigonometric ratios.