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− | The Pythagorean |
+ | The '''Pythagorean trigonometric identity''' is a [[trigonometry|trigonometric]] identity based on the application of the [[Pythagorean Theorem]] to the [[unit circle|unit triangle]]. |
The fundamental identity is: |
The fundamental identity is: |
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+ | {{Identity |
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+ | |{{Proof |
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+ | |reqs = [[Pythagorean theorem]]: <math>a^2+b^2=c^2</math> |
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+ | |Assume a [[point]] <math>C</math> in the [[unit circle]] centered at the origin forming an angle <math>\theta</math> . Let <math>B</math> be a point on the [[x-axis]] corresponding to the [[x-coordinate]] of <math>C</math> . Let <math>A</math> be the origin. A [[right triangle]] would be formed: <math>\triangle ABC</math> . The length of <math>\overline{AB}</math> would be equal to <math>\cos(\theta)</math> and the length of <math>\overline{BC}</math> would be equal to <math>\sin(\theta)</math> . The length of <math>\overline{AC}</math> is 1 (since we're using the unit circle). Using the [[Pythagorean theorem]], we have: <math>\sin^2(\theta)+\cos^2(\theta)=1</math> |
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+ | }} |
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+ | }} |
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Due to this fundamental relationship, other Pythagorean Identities emerge through the use of: |
Due to this fundamental relationship, other Pythagorean Identities emerge through the use of: |
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− | * |
+ | *the complimentary and cofunction properties |
− | * |
+ | *the reciprocal functions |
− | * |
+ | *the quotient identities |
The other identities include: |
The other identities include: |
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+ | {{Identity |
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− | <math>\tan^2 |
+ | |subtitle = <math>\tan^2(\theta)+1=\sec^2(\theta)</math> |
+ | |{{Proof| |
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− | <math>\ |
+ | :<math>\sin^2(\theta)+\cos^2(\theta)=1</math> |
+ | :<math>\frac{\sin^2(\theta)}{\cos^2(\theta)}+\frac{\cos^2(\theta)}{\cos^2(\theta)}=\frac{1}{\cos^2(\theta)}</math> |
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+ | :<math>\tan^2(\theta)+1=\sec^2(\theta)</math> |
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+ | }} |
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+ | }} |
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+ | {{Identity |
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+ | |subtitle = <math>1+\cot^2(\theta)=\csc^2(\theta)</math> |
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+ | |{{Proof| |
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+ | :<math>\sin^2(\theta)+\cos^2(\theta)=1</math> |
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+ | :<math>\frac{\sin^2(\theta)}{\sin^2(\theta)}+\frac{\cos^2(\theta)}{\sin^2(\theta)}=\frac{1}{\sin^2(\theta)}</math> |
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+ | :<math>1+\cot^2(\theta)=\csc^2(\theta)</math> |
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+ | }} |
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+ | }} |
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+ | [[Category:Trigonometric identities]] |
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+ | [[Category:Trigonometry]] |
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+ | [[Category:Triangles]] |
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+ | [[Category:Geometry]] |
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+ | [[Category:Mathematics]] |
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+ | [[Category:Articles containing proofs]] |
Latest revision as of 23:42, 28 June 2018
The Pythagorean trigonometric identity is a trigonometric identity based on the application of the Pythagorean Theorem to the unit triangle.
The fundamental identity is:
Identity: , for all | ||
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Implications
Due to this fundamental relationship, other Pythagorean Identities emerge through the use of:
- the complimentary and cofunction properties
- the reciprocal functions
- the quotient identities
The other identities include: