PC 3.8 Fundamental Identitiesx
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Transcript PC 3.8 Fundamental Identitiesx
3.8 Fundamental Identities
–A trig identitiy is a trig equation that is always true
–We can prove an identity using the definitions of trig
functions
(they use x, y, and r)
Ex 1) Use definitions to prove: sin cot cos
y x
yx
x
cos
r y
ry
r
We also have the Pythagorean Identities
cos sin 1
2
2
1 tan sec
2
2
“I tan in a second”
(get by ÷ by cos2θ)
1 cot csc
2
2
(get by ÷ by sin2θ)
“I cotan in a cosecond”
We can prove identities (using θ, ϕ, β, etc) or verify the
identity using specific values.
Ex 2) Use exact values to verify the identity for the given θ
a) tan( ) tan ;
3
LHS: tan 3
3
RHS: tan 3 3
3
1
2
60°
1
3
2
Ex 2) Use exact values to verify the identity for the given θ
b) 1 tan 2 sec2 ; 150
1
2
2
1 3 1 4
1
LHS: 1
1 3 3 3 3
3
2
4 4
1 1
3 1
RHS:
3
3 3
4
2
1
30°
3
2
150°
Other Identities to use:
Reciprocal:
csc
1
sin
1
sec
cos
sin
tan
cos
cos
cot
sin
Ratio:
1
cot
tan
Pythagorean Identities: (already mentioned)
Odd/ Even:
sin( ) sin
csc( ) csc
tan( ) tan
cot( ) cot
cos( ) cos
sec( ) sec
Ex 3) Simplify by writing in terms of sine & cosine
a) cot sec
(try ratio & reciprocal)
cos
1
1
sin cos
sin
b) sec2 1 Pythag (1 + tan2θ = sec2θ)
tan( ) odd/even
21
tan
sin
tan
tan
cos
Homework
#308 Pg 169 #1–45 odd
Hints for HW Make sure calculator is in correct MODE
Draw those reference triangle pictures!