Transcript Sullivan Algebra and Trigonometry: Section 8.3

Sullivan Algebra and
Trigonometry: Section 7.3
Trigonometric Identities
Objectives of this Section
• Use Algebra to Simplify Trigonometric Expressions
•Establish Identities
Two functions f and g are said to be
identically equal if
f x   g x 
for every value of x for which both
functions are defined. Such an equation
is referred to as an identity. An
equation that is not an identity is called
a conditional equation.
Quotient Identities
sin 
tan  
cos
cos
cot  
sin 
Reciprocal Identities
1
csc 
sin 
1
sec =
cos
1
cot  
tan 
Pythagorean Identities
sin   cos   1
2
tan   1  sec 
2
2
2
1  cot   csc 
2
2
Even-Odd Identities
sin     sin 
csc     csc 
cos    cos
sec    sec 
tan     tan 
cot     cot 
The directions Establish the identity means to
show, through the use of basic identities and
algebraic manipulation, that one side of an
equation is the same as the other side of the
equation.
Establish the identity:
sin  csc   cos   sin 
2
2
1
2
 cos 
sin  csc   cos   sin  
sin 
2
 1  cos 
2
 sin 2 
Establish the identity:
cos  1 1  sec 

cos  1 1  sec 
1
1

1
1
1  sec
cos
cos 



1  sec 1  1
1

1

cos
cos 

cos  1

cos  1


 cos 
 cos 


Establish the identity:
1  sin  1  sin 

 4 tan  sec
1  sin  1  sin 
1  sin  1  sin 

1  sin  1  sin 
 1  sin   1  sin    1  sin   1  sin  





 1  sin   1  sin    1  sin   1  sin  
1  2 sin   sin  1  2 sin   sin 


2
2
1  sin 
1  sin 
2
2
1  2 sin   sin  1  2 sin   sin 


2
1  sin 
1  sin 2 
2

1  2 sin   sin   1  2 sin   sin 
2

2
2
cos2 
1  2 sin   sin   1  2 sin   sin 

2
cos 
2
4 sin 
1
 4 tan  sec 


cos  cos 
2

Guidelines for Establishing Identities
1. It is almost always preferable to start with
the side containing the more complicated
expression.
2. Rewrite sums or differences of quotients as a
single quotient.
3. Sometimes rewriting one side in terms of
sines and cosines only will help.
4. Always keep your goal in mind.