FANDOM


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. '
Prerequisites:
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
Tombstone

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:

Identity: \tan^2(\theta)+1=\sec^2(\theta)
Theorem. '
Proof.
\sin^2(\theta)+\cos^2(\theta)=1
\frac{\sin^2(\theta)}{\cos^2(\theta)}+\frac{\cos^2(\theta)}{\cos^2(\theta)}=\frac{1}{\cos^2(\theta)}
\tan^2(\theta)+1=\sec^2(\theta)

Tombstone
Identity: 1+\cot^2(\theta)=\csc^2(\theta)
Theorem. '
Proof.
\sin^2(\theta)+\cos^2(\theta)=1
\frac{\sin^2(\theta)}{\sin^2(\theta)}+\frac{\cos^2(\theta)}{\sin^2(\theta)}=\frac{1}{\sin^2(\theta)}
1+\cot^2(\theta)=\csc^2(\theta)

Tombstone

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.