Trigonometry/Using Fundamental Identities

Current revision (unreviewed)

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

By the Pythagorean Theorem,

$a^2+b^2=c^2\,$

Dividing through by c² gives

$\bigg(\frac{a}{c}\bigg)^2+\bigg(\frac{b}{c}\bigg)^2=\bigg(\frac{c}{c}\bigg)^2=1$

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 the second equation to get

$\sin^2 A + \cos^2 A =1\,$

Related identities include:

$\sin^2 A =1 - \cos^2 A\mbox{ or }\cos^2 A =1-\sin^2 A\,$

$\tan^2 A + 1 =\sec^2 A\mbox{ or }\tan^2 A =\sec^2 A -1\,$

$1 + \cot^2 A =\csc^2 A\mbox{ or }\cot^2 A =\csc^2 A -1\,$

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

Reciprocal identities

$\csc A =\frac{1}{\sin A}\quad\sec A =\frac{1}{\cos A}\quad\cot A =\frac{1}{\tan A}\,$

Ratio identities

$\tan A =\frac{\sin A}{\cos A}\quad\cot A =\frac{\cos A}{\sin A}\,$

Co-function identities (in radians)

$\cos A =\sin\left(\frac{\pi}{2}-A\right)\quad \csc A =\sec\left(\frac{\pi}{2}-A\right)\quad \cot A =\tan\left(\frac{\pi}{2}-A\right)\,$