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!