Download 6.1A Reciprocal, Quotient, and Pythagorean Identities

6.1A Reciprocal, Quotient, and Pythagorean Identities
Recall the basic
definitions for a
unit circle:
y
sin  
r
x
cos  
r
y
tan  
x
sin 
.
Prove that tan  
cos 
y
x
y x

r r
y r

r x
y
x
L.S. = R.S.
A trigonometric identity is a trigonometric equation that is true
for all permissible values of the variable in the expressions on
both sides of the equation.
Both sides of the equation have the same value for all permissible values of the
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variable.
Trigonometric Identities
Reciprocal Identities
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csc  
sin 
sec  
1
sin  
csc 
1
cos 
cot  
1
cos  
sec 
1
tan 
1
tan  
cot 
Quotient Identities
sin 
tan  
cos 
Pythagorean Identities
cos 
cot  
sin 
sin2 + cos2 = 1
tan2 + 1 = sec2
cot2 + 1 = csc2
sin2 = 1 - cos2
cos2 = 1 - sin2
tan2 = sec2 - 1
cot2 = csc2 - 1
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Simplifying Trigonometric Expressions
Identities can be used to simplify trigonometric expressions.
Simplify to a single trigonometric expression.
a)
b)
sin  cot 
 cos  
 sin  

sin



 cos 
cot 2 
1  sin2 
cos 2 
2
sin


cos 2 
cos 2 
1


2
sin  cos2 

1
2
sin 
 csc 2 
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Simplify to a single trigonometric expression.
c)
d)
sin 2   cos 2   tan 2 
 1  tan 2 
1  cos 2 
sin  cos 
sin 2 

sin  cos 
 sec 2 

sin 
cos 
 tan 
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Simplifing Trigonometric Expressions
e)
(1 + tan x)2 - 2 sin x sec x
1
cos x
sin x
2
 1  2 tan x  tan x  2
cos x
2
 (1  tan x)  2 sin x
 1  tan2 x  2tanx  2 tanx
 sec2 x
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csc x
f)
tan x  cot x
1

sin x
sin x cos x

cos x sin x
1

sin x
sin 2 x  cos 2 x
sin xcos x
1

sin x
1
sin x cos x
1
sin x cos x


sin x
1
 cos x
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Proving an Identity using a Two Column Proof
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1
2

 2 cs c x
1  cos x 1  cos x
(1  cos x)  (1  cos x) 2 csc2 x
(1  cos x)(1  cos x)
2
(1  cos 2 x)
2
sin 2 x
2 csc 2 x
L.S. = R.S.
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Possible Strategies for Simplifying using Identities
Trigonometric Simplifications:
Look for familiar trig relationships and substitute
Rewrite in terms of sine or cosine.
If the expression contains squared terms, try using
the Pythagorean Identities.
Algebraic Simplifications:
Multiply, expand, factor, reduce or square.
Common denominator to add or subtract.
Multiply by the conjugate of a binomial.
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