High School Trigonometry/Fundamental Identities

Reciprocal Identities

$\sin u=\left({\frac {1}{\csc u}}\right)$$\sin u=\left({\frac {1}{\csc u}}\right)$

$\cos u=\left({\frac {1}{\sec u}}\right)$$\cos u=\left({\frac {1}{\sec u}}\right)$

$\tan u=\left({\frac {1}{\ cotu}}\right)$$\tan u=\left({\frac {1}{\ cotu}}\right)$

$\csc u=\left({\frac {1}{\sin u}}\right)$$\csc u=\left({\frac {1}{\sin u}}\right)$

$\sec u=\left({\frac {1}{\cos u}}\right)$$\sec u=\left({\frac {1}{\cos u}}\right)$

$\cot u=\left({\frac {1}{\tan u}}\right)$$\cot u=\left({\frac {1}{\tan u}}\right)$

Pythagorean Identities

$\sin ^{2}u+\cos ^{2}u=1$$\sin ^{2}u+\cos ^{2}u=1$

$1+\tan ^{2}u=\sec ^{2}$$1+\tan ^{2}u=\sec ^{2}$

$1+\cot ^{2}u=\csc ^{2}u$$1+\cot ^{2}u=\csc ^{2}u$

Quotient Identities

$\tan u=\left({\frac {\sin u}{\cos u}}\right)$$\tan u=\left({\frac {\sin u}{\cos u}}\right)$

$\cot u=\left({\frac {\cos u}{\sin u}}\right)$$\cot u=\left({\frac {\cos u}{\sin u}}\right)$

Co-Function Identities

$\sin(\left({\frac {\pi }{2}}\right)-u)=\cos u$$\sin(\left({\frac {\pi }{2}}\right)-u)=\cos u$

$\cos(\left({\frac {\pi }{2}}\right)-u)=\sin u$$\cos(\left({\frac {\pi }{2}}\right)-u)=\sin u$

$\tan(\left({\frac {\pi }{2}}\right)-u)=\cot u$$\tan(\left({\frac {\pi }{2}}\right)-u)=\cot u$

$\csc(\left({\frac {\pi }{2}}\right)-u)=\sec u$$\csc(\left({\frac {\pi }{2}}\right)-u)=\sec u$

$\sec(\left({\frac {\pi }{2}}\right)-u)=\csc u$$\sec(\left({\frac {\pi }{2}}\right)-u)=\csc u$

$\cot(\left({\frac {\pi }{2}}\right)-u)=\tan u$$\cot(\left({\frac {\pi }{2}}\right)-u)=\tan u$

Even-Odd Identities

$\sin(-u)=-\sin(u)$$\sin(-u)=-\sin(u)$

$\cos(-u)=cos(u)$$\cos(-u)=cos(u)$

$\tan(-u)=-\tan(u)$$\tan(-u)=-\tan(u)$

$\csc(-u)=-\csc(u)$$\csc(-u)=-\csc(u)$

$\sec(-u)=\sec(u)$$\sec(-u)=\sec(u)$

$\cot(-u)=-\cot(u)$$\cot(-u)=-\cot(u)$

This material was adapted from the original CK-12 book that can be found here. This work is licensed under the Creative Commons Attribution-Share Alike 3.0 United States License