Trigonometric Identities 20 December 2010 Remember y sin

Trigonometric Identities 20 December 2010

Remember: y = sin α l x = cos α l l α = alpha (just another variable, like x or θ)

Quotient Identities

Reciprocal Identities

Reciprocal Identities

Reciprocal Identities

Reciprocal Identities

Simplify v. Prove l l Simplify – get into simplest possible terms (no equal sign) Prove – demonstrate that both sides of the equation equal the same thing (equal sign)

Q. E. D. l l l quod erat demonstrandum Means “what was to be demonstrated” Write at the end of a proof.

Strategies: 1. 2. 3. 4. Use your trig identities to get the expression/equation in terms of one trig function (ideally sine or cosine). Consider expanding tan, csc, sec, and cot in order to find common terms. Cancel terms in order to simplify. When proving identities, deal with 1 side of the equation until it matches the other side of the equation. (Very useful when one side of the equation is much simpler than the other side. )

Return of Pythagoras l l a 2 + b 2 = c 2 Special Case: l l y 2 + x 2 = 12 y 2 + x 2 = 1 1 y x

Pythagoras + Unit Circle = Pythagorean Identities!!! If: y = sin α and x = cos α l Then: y 2 + x 2 = 1 l sin 2 α + cos 2 α = 1

sin 2 α + cos 2 α = 1

Other Pythagorean Identities l tan 2 l 1 α + 1 = sec 2 α + cot 2 α = csc 2 α

tan 2 α + 1 = sec 2 α

1 + cot 2 α = csc 2 α

Strategies 5. Try using Pythagorean Identities when you have squared terms.

Strategies 6. Factor out common terms, especially if the result is a trig identity.

Strategies Consider expanding terms raised to powers in order to find common terms and/or cancel out terms. (1 + cos α)2 = 1 + 2 cos α + cos 2 α 9. For exponents larger than 2, consider factoring. (Remember the difference of squares!!!) sin 4 α – cos 4 α = (sin 2 α + cos 2 α)(sin 2 α cos 2 α) 8.