Transcript • 5.1 Fundamental Identities Statements like “ tan   sin  / cos ” and “ csc  1/ sin  ”

•
5.1 Fundamental Identities
Statements like “ tan   sin  / cos ” and
“ csc  1/ sin  ” are trigonometric identities
because they are true for all values of the variable for
which both sides of the equation are defined.
• The set of all such values is called the domain of validity of
the identity.
• Basic Trigonometric Identities
– Some trigonometric identities follow directly from the
definitions of the six basic trigonometric functions.
– These basic identities consists of the reciprocal identities
and the quotient identities.
Basic Trigonometric Identities
• Reciprocal Identities
1
1
1
csc  
sec  
cot  
sin 
cos 
tan 
1
sin  
csc 
1
cos  
sec 
1
tan  
cot 
Basic Trigonometric Identities
• Quotient Identities
sin 
tan  
cos 
cos 
cot  
sin 
Basic Trigonometric Identities
• Pythagorean Identities
cos   sin   1
2
2
1  tan   sec 
2
2
cot   1  csc 
2
2
Using Identities
• Find sin  and cos  if tan   5 and
cos  0.
sec   1  tan   1  5  1  25  26
2
2
sec   26
cos   1
26
tan   5
sin 
5
cos 
2
sin   5cos
 1 
 5

 26 
5 26
sin  
26
Cofunction Identities


sin      cos 
2



cos      sin 
2



tan      cot 
2



cot      tan 
2



sec      csc 
2



csc      sec 
2

Odd-Even Identities
sin( x)   sin x
cos( x)   cos x
tan( x)   tan x
csc( x)   csc x
sec( x)   sec x
cot( x)   cot x
Using More Identities
• If cos   0.34 , find sin    2  .
sin   


2    sin    
2

  cos 
 0.34
Simplifying by Factoring and Using Identities
sin x  sin x cos x.
3
• Simplify the expression
2
sin x  sin x cos x
3
2
 sin x  sin x  cos x 
2
 sin x(1)
 sin x
2
Simplifying by Expanding and Using Identities
 sec x  1 sec x  1 / sin 2 x.
• Simplify the expression
 sec x  1 sec x  1 / sin x
2
1
sec x  1 tan x sin x

 2


2
2
2
sin x
sin x cos x sin x
2
2
1
2

 sec x
2
cos x
2
Simplifying by Combining Fractions and Using Identities
• See Example 5 on p. 448 – 449.
Solving a Trigonometric Equation
• See Example 6 and Example 7 on p. 449 – 450.
More Practice!!!!!
• Homework – Textbook p. 451 – 452 #1 – 31
ODD.