L2 Trigonometric Identities - Part 6
Completion requirements
Unit 4A
Trigonometry Part 1
Lesson 2: Trigonometric Identities
Verifying an Identity
An identity can be verified numerically by substituting a value for the variable(s). If the resulting values on each side of the equation are the same, the equation may be an identity. If the two sides are not equal, the equation cannot be an identity.Watch the video Verifying Identities. Note the second example uses «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mn»30«/mn»«mo»§#8728;«/mo»«/msup»«/mstyle»«/math» instead of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mstyle»«/math». Try this verification on your own using «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mstyle»«/math».
Verify the equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»cot«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math»
numerically using «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mrow»«/mstyle»«/math».
Set up a chart to see if the left side of the equation is equal to the right side of the equation when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mrow»«/mstyle»«/math»
and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mrow»«/mstyle»«/math».
The left side equals the right side, so the equation has been verified for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mrow»«/mstyle»«/math». This result means the given equation is a potential identity.
The left side equals the right side, so the equation has been verified for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mrow»«/mstyle»«/math». This result means the given equation is a potential identity.
| Left Side
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Right Side
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| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mrow»«mi»cos«/mi»«mfenced»«mstyle displaystyle=¨true¨»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»6«/mn»«/mfrac»«/mstyle»«/mfenced»«/mrow»«mrow»«mi»cot«/mi»«mfenced»«mstyle displaystyle=¨true¨»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»6«/mn»«/mfrac»«/mstyle»«/mfenced»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mfenced»«mstyle displaystyle=¨true¨»«mfrac»«msqrt»«mn»3«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«/mstyle»«/mfenced»«mfenced»«msqrt»«mn»3«/mn»«/msqrt»«/mfenced»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mfenced»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mfenced»«mo»=«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mrow»«/mstyle»«/math» |
The left side equals the right side, so the equation has been verified for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mrow»«/mstyle»«/math». This result means the given equation is a potential identity.
| Left Side
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Right Side
|
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mrow»«mi»cos«/mi»«mfenced»«mstyle displaystyle=¨true¨»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mstyle»«/mfenced»«/mrow»«mrow»«mi»cot«/mi»«mfenced»«mstyle displaystyle=¨true¨»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mstyle»«/mfenced»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mfenced»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mstyle»«/mfenced»«mfenced»«mstyle displaystyle=¨true¨»«mfrac»«mn»1«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«/mstyle»«/mfenced»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«msqrt»«mn»3«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mfenced»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mfenced»«mo»=«/mo»«mfrac»«msqrt»«mn»3«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«/mrow»«/mstyle»«/math» |
The left side equals the right side, so the equation has been verified for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mrow»«/mstyle»«/math». This result means the given equation is a potential identity.