Trigonometry/Using Fundamental Identities

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

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:

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

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

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

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

Reciprocal identities

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

Ratio identities

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

Co-function identities (in radians)

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