Sections 2.1 & 2.2 Fundamental Trigonometric Identities
- Use fundamental trigonometric identities to simplify and evaluate trigonometric expressions.
- Verify (prove) trigonometric identities.
Using Basic Identities to Simplify
Basic identities are extremely useful for simplifying expressions. Here are some examples....
- Suppose and .
Find the exact values of all six trigonometric functions at .
(We could use a right triangle, but let's use a Pythagorean identity.)
- Use the substitution , , to write in terms of .
Verifying trig identities can be very difficult---it's hard to figure out where to start, and it's easy to get stuck in cycles. Here is some advice:
- Always remember that there is no single correct approach.
- Work on one side at a time.
- Look for opportunities to factor, get common denominators and add fractions, expand algebraic expressions, multiply by conjugates, etc.
- Be on the lookout for opportunities to use fundamental identities.
- If all else fails, consider converting to sines and cosines.
- Do something! Quite often, the action of just trying something will lead to useful results.