Section 5.1
Simplifying More Trig Expressions
Fundamental Trigonometric Identities:Reciprocal Identities:
1sin
csc
1cos
sec
1tan
cot
1csc
sin
1
seccos
1
cottan
Quotient Identities:
sintan
cos
coscot
sin
Pythagorean Identities:2 2sin cos 1 2 21 tan sec 2 21 cot csc
Cofunction Identities:
sin cos2
cos sin2
tan cot2
cot tan2
sec csc2
csc sec2
Even / Odd Identities:
sin( ) sin
csc( ) csc
sec( ) sec
cos( ) cos
tan( ) tan
cot( ) cot
Trig Identities
•Any relationship that is true for all values of the variable for which each side is defined is called an identity.
•We can use trig identities to simplify trig expressions, prove other identities and solve more complex trig equations.
Strategies for Simplifying Trig Expressions:
1.
2.
3.
4.
5.
We can factor trigonometric expressions, too!
Ex 1: Factor each trig expression
A. B. C. 2sec 1 24 tan tan 3 2csc 2csc 3
sin cotx x
Ex 2: Use trig identities to simplify
2 2csc 1 cosx x
Ex 3: Use trig identities to simplify
Ex 4: Simplify by factoring:
2cos 4
cos 2
x
x
Ex 5: Use trig identities to simplify
2sin cos sinx x x
Ex 6: Simplify by adding the fractions first:
1 1
sec 1 sec 1x x
Ex 7: Simplify by factoring
4 2csc 2csc 1
Ex 8: Simplify by adding the fractions first:
sec tan
cos cot
x x
x x
HomeworkPage 379
(2, 27-36 multiples of 3, 45-53 odd, 61, 64, 65)
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