Trigonometric Identities

20 December 2010

Remember:

y = sin αx = cos α

α = alpha (just another variable, like x or θ)

Quotient Identities

cos

sintan

sin

coscot

Reciprocal Identities

csc

1sin

sec

1cos

sin

1csc

cos

1sec

cot

1tan

tan

1cot

Reciprocal Identities

sin

1csc

1csc

y

Reciprocal Identities

cos

1sec

1sec

x

Reciprocal Identities

sin

coscot

cot

y

x

Simplify v. Prove

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.

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

Strategies:

1. Use your trig identities to get the expression/equation in terms of one trig function (ideally sine or cosine).

2. Consider expanding tan, csc, sec, and cot in order to find common terms.

3. Cancel terms in order to simplify.

4. 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

a2 + b2 = c2

Special Case: y2 + x2 = 12

y2 + x2 = 1

1

x

y

Pythagoras + Unit Circle = Pythagorean Identities!!!

If: y = sin α and x = cos αThen: y2 + x2 = 1

sin2 α + cos2 α = 1

sin2 α + cos2 α = 1

sin2 α + cos2 α = 1 sin2 α + cos2 α = 1

Other Pythagorean Identities

tan2 α + 1 = sec2 α

1 + cot2 α = csc2 α

tan2 α + 1 = sec2 α

tan2 α + 1 = sec2 α

1 + cot2 α = csc2 α

1 + cot2 α = csc2 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

8. Consider expanding terms raised to powers in order to find common terms and/or cancel out terms.

(1 + cos α)2 = 1 + 2cos α + cos2 α

9. For exponents larger than 2, consider factoring. (Remember the difference of squares!!!)

sin4 α – cos4 α = (sin2 α + cos2 α)(sin2 α cos2 α)