Download 5-1: TRIGONOMETRIC IDENTITIES TRIGONOMETRIC IDENTITIES What is an identity? Identities already known:

5-1: TRIGONOMETRIC IDENTITIES
CP Precalculus
Mr. Gallo
TRIGONOMETRIC IDENTITIES
What is an identity?
 An equation in which the left side is equal to the right side:
x2  9
 x3
x 3
Identities already known:
Examples:
If cos
, find sec
4
3
If sec
and tan
, find sin
3
5
1
PYTHAGOREAN IDENTITIES
Based on definition of trigonometric functions on the unit circle and the
Pythagorean Theorem
 Three Pythagorean Identities:
a  sin  b  cos 
c 1
1
sin
a2  b2  c2
sin 2   cos 2   1
cos 
sin 2   cos 2   1
sin 2   cos 2   1
sin 2  cos 2 
1


2
2
cos  cos  cos 2 
sin 2  cos 2 
1


2
2
sin  sin  sin 2 
tan 2   1  sec 2 
cot 2   1  csc 2 
EXAMPLE 1:
If
, then cos
and sin
.
Show the Pythagorean identity holds:
EXAMPLE 2:
If sin
cos .
sin 2   cos 2   1
2
2
1  3
 1
   
 2   2 
1 3
 1
4 4
, use the Pythagorean Identity to find
sin 2   cos 2   1
2
1
2
   cos   1
3
8
cos 2  
9
cos 2  
8
9
cos  
2 2
3
2
COFUNCTION IDENTITIES
A trigonometric function is a cofunction of another trigonometric function
if
when and are complementary angles.
y
r
y
tan   cot   cot  90    
x
r
sec   csc   csc  90    
y
sin   cos   cos  90    

r
y

x
Cofunction Identities:
ODD-EVEN IDENTITIES
Each of the trigonometric are either odd or even
 Can see on the unit circle
sin   y
sin      y
cos   x
cos     x
tan  
y
x
tan     
y
x
(x,y)

-
(x,-y)
Odd-Even Identities
3
EXAMPLE 3:
If sin
0.37, find cos
.


 cos   x 
2

 sin x
 0.37
EXAMPLE 4:
If cos
.75, find sin

 


cos  x    cos     x  
2


 2
.

 


sin  x    sin     x  
2


 2


  sin   x 
2

  cos x
 0.75
HOMEWORK: P. 317 #1-21 ODD
4
USING IDENTITIES TO SIMPLIFY AND REWRITE TRIGONOMETRIC EQUATIONS
Various methods to try:
 Simplify by rewriting using only sine and cosine
 Simplify by factoring
 Simplify by combining fractions
 Rewrite so fractions are eliminated
To be successful:
 Know the identities
 Persevere and don’t give up after only a brief try
REWRITE USING ONLY SINE AND COSINE
Simplify
1
1
1  sin 2 x  
cos 2 x 


cos x
cos x

Simplify sec x  tan x sin x 
cos 2 x
 cos x
cos x
1
sin x

sin x
cos x cos x

1
sin 2 x

cos x cos x

1  sin 2 x
cos x

cos 2 x
 cos x
cos x
5
SIMPLIFY BY FACTORING
Simplify cos x tan x  sin x cos 2 x  cos x 
sin x 
2
  sin x cos x
 cos x 
 sin x  sin x cos 2 x
 sin x 1  cos 2 x 
 sin x  sin 2 x   sin 3 x
2


 x   tan 2 x sec x   sec x  sin 2 x  1 
cos x  cos x 
2

Simplify  csc 


1
sin 2 x

cos x cos3 x
 cos 2 x  sin 2 x
cos3 x
 1 
3
 
   sec x
3
x
cos


SIMPLIFY BY COMBINING FRACTIONS
Simplify

sec x
sec x

1  sec x 1  sec x
sec x  sec 2 x  sec x  sec 2 x
1  sec 2 x
 2sec 2 x 
 2sec 2 x 
  2
  

2
 sec x  1 
 tan x 
2
2
2
 cos2 x   2
sin x
sin x
2
cos x

 2 csc 2 x
Simplify
cos x 1  sin x

1  sin x
cos x

cos 2 x  1  2sin x  sin 2 x
cos x 1  sin x 
cos 2 x  sin 2 x  1  2sin x

cos x 1  sin x 

1  1  2sin x
2  2sin x

cos x 1  sin x  cos x 1  sin x 

2 1  sin x 
cos x 1  sin x 

2
 2sec x
cos x
6
REWRITE TO ELIMINATE
FRACTIONS
2
Simplify
1  tan x
csc 2 x

Simplify
sec 2 x
1
sin 2 x
1
2
 cos x
1
sin 2 x

4
sec x  tan x

sec x  tan x sec x  tan x

4sec x  4 tan x
sec 2 x  tan 2 x

4sec x  4 tan x
1
 4sec x  4 tan x
sin 2 x
 tan 2 x
cos 2 x
HOMEWORK: P. 317 #23-35 ODD, 39-47 ODD
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