Trigonometry/Using Fundamental Identities

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Some of the fundamental trigometric identities are those derived from the Pythagorean Theorem. These are defined using a right triangle:

By the Pythagorean Theorem,

A 2 + B 2 = C 2

{1}

Dividing through by C² gives

$\left(\frac{A}{C}\right)^2+\left(\frac{B}{C}\right)^2=\left(\frac{C}{C}\right)^2=1$ {2}

We have already defined the sine of a in this case as A/C and the cosine of a as B/C. Thus we can substitute these into {2} to get

$\sin^2 a + \cos^2 a \equiv 1$

Related identities include:

$\sin^2 a \equiv 1 - \cos^2 a\mbox{ or }\cos^2 a \equiv 1-\sin^2 a$

$\tan^2 a + 1 \equiv \sec^2 a\mbox{ or }\tan^2 a \equiv \sec^2 a -1$

$1 + \cot^2 a \equiv \csc^2 a\mbox{ or }\cot^2 a \equiv \csc^2 a -1$

Other Fundamental Identities include the Reciprocal, Ratio, and Co-function identities

Reciprocal identities

$\csc a \equiv \frac{1}{\sin a}\quad\sec a \equiv \frac{1}{\cos a}\quad\cot a \equiv \frac{1}{\tan a}$

Ratio identities

$\tan a \equiv \frac{\sin a}{\cos a}\quad\cot a \equiv \frac{\cos a}{\sin a}$

Co-function identities (in radians)

$\cos a \equiv \sin\left(\frac{\pi}{2}-a\right)\quad \csc a \equiv \sec\left(\frac{\pi}{2}-a\right)\quad \cot a \equiv \tan\left(\frac{\pi}{2}-a\right)$

Next Page: Verifying Trigonometric Identities | Previous Page: Applications and Models

Home: Trigonometry

Trigonometry/Using Fundamental Identities

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