Basic Trig Identities

Next let's see how these functions are related. These are identities you will need for calculus (there are many others you learned in your trig class that we generally do not use in calculus).

  1. $\sin(-\theta)=-\sin(\theta)$

  2. $\cos(-\theta)=+\cos(\theta)$

  3. $\sin(2\theta)=2\sin(\theta)\cos(\theta)$

  4. $\sin^2(\theta)+\cos^2(\theta)=1$

  5. $\tan^2(\theta)+1=\sec^2(\theta)$

  6. $1+\cot^2(\theta)=\csc^2(\theta)$ 

  7. $\cos^2(\theta)=\frac{1+\cos(2\theta)}{2}$

  8. $\sin^2(\theta)=\frac{1-\cos(2\theta)}{2}$

The first two say that sine is an odd function, and that cosine is an even function. 

DO:  Play with positive and negative angles on the unit circle to see why the first two identities make sense.

The next three are called Pythagorean identities, since they're all based on the Pythagorean theorem.  Notice that if you take the primary Pythagorean identity, $\sin^2(\theta)+\cos^2(\theta)=1$, and divide all terms by $\cos^2(\theta)$ you get the tangent/secant Pythagorean identity.  Similarly, if you divide all terms of the primary identity by $\cos^2(\theta)$ you get the cotangent/cosecant identity.  You need not memorize the last two since they are so easily computed.

The last two identities will be used to help compute integrals, during the second semester of calculus.