Trigonometric Identities 20 December 2010
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)