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Pythagorean trigonometric identity

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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: \sin^2(\theta)+\cos^2(\theta)=1 , for all \theta\in\R
Theorem. '
Pythagorean theorem: a^2+b^2=c^2

Proof. Assume a point C in the unit circle centered at the origin forming an angle \theta . Let B be a point on the x-axis corresponding to the x-coordinate of C . Let A be the origin. A right triangle would be formed: \triangle ABC . The length of \overline{AB} would be equal to \cos(\theta) and the length of \overline{BC} would be equal to \sin(\theta) . The length of \overline{AC} is 1 (since we're using the unit circle). Using the Pythagorean theorem, we have: \sin^2(\theta)+\cos^2(\theta)=1


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:

Identity: \tan^2(\theta)+1=\sec^2(\theta)
Theorem. '

Identity: 1+\cot^2(\theta)=\csc^2(\theta)
Theorem. '


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