L2 Trigonometric Identities - Part 8
Completion requirements
Unit 4A
Trigonometry Part 1
Lesson 2: Trigonometric Identities
Proofs differ from a lot of other types of math that have been previously studied because there is not a set of rules that can be followed to guarantee a solution. Here are some suggestions to help prove an identity.
Watch the video Proving Identities for an additional example on proving an equation is an identity using two different strategies.
Proofs often require several attempts, different approaches, and even a bit of luck! Pick a starting point that makes sense and begin manipulating both sides of a given equation. If something does not work, try a different approach.
Step 1:
Replace «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»tan«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»csc«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»sec«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»,«/mi»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»cot«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math» with equivalent expressions that contain
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mi»x«/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»«mi»x«/mi»«/mrow»«/mstyle»«/math».
Step 2:
If a second degree trigonometric ratio is used «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«msup»«mi»sec«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mi
mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»etc«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», try one of the Pythagorean identities.
Step 3:
Manipulate algebraic expressions algebraically. Factoring, creating common denominators, and multiplying by an expression equivalent to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math»
are some of the most common strategies.
Step 4:
Be inventive. There is more than one correct way to show two expressions are equivalent. If one strategy fails, try another. With practice, you will begin to see what can be done to simplify an expression!
Watch the video Proving Identities for an additional example on proving an equation is an identity using two different strategies.
Proofs often require several attempts, different approaches, and even a bit of luck! Pick a starting point that makes sense and begin manipulating both sides of a given equation. If something does not work, try a different approach.
Prove the equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mn»1«/mn»«mrow»«mn»1«/mn»«mo»+«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi»sec«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mi»sin«/mi»«mi»x«/mi»«mi»sec«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
is an identity for all permissible values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math».
The left side equals the right side, so the identity has been proven.
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The left side equals the right side, so the identity has been proven.