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Chapter 5
Trigonometric Identities
Objective:
SWBAT use the fundamental identities of Trigonometry functions to find
trigonometric functions values given one value and the quadrant.
Trigonometric Identities
(write these down on an index card)
Quotient Identities:
sin 
tan  
cos 
cos 
cot  
sin 
Reciprocal Identities:
1
sin  
csc 
1
cot  
tan 
1
cos  
sec 
1
sec 
cos
1
tan  
cot 
1
csc 
sin 
Pythagorean Identities:
sin2 + cos2 = 1
tan2 + 1 = sec2
cot2 + 1 = csc2
Sin2 = 1 - cos2
tan2 = sec2 - 1
cot2 = csc2 - 1
cos2 = 1 - sin2
Trigonometric Identities
(write these down on an index card)
Negative-Angle Identities:
sin( )   sin 
csc( )   csc
cos( )  cos
sec( )  sec
tan( )   tan 
cot( )   cot 
Where did our Pythagorean Identities come from?
Do you remember the Unit Circle?
What is the equation for the unit circle?
x2 + y2 = 1
What does x = ? What does y = ?
(in terms of trig functions)
Sin2 θ + cos2 θ = 1
Pythagorean
Identity!
Take the Pythagorean Identity and
discover a new one!
Hint: Try dividing everything by cos2θ
sin2θ + cos2θ = 1 .
cos2θ cos2θ cos2θ
tan2θ + 1 = sec2θ
Quotient
Identity
another
Pythagorean
Identity
Reciprocal
Identity
Take the Pythagorean Identity and
discover a new one!
Hint: Try dividing everything by sin2θ
sin2θ + cos2θ = 1 .
sin2θ sin2θ sin2θ
1 + cot2θ = csc2θ
Quotient
Identity
a third
Pythagorean
Identity
Reciprocal
Identity
Using the identities you now know, find the trig
value
:
5
If tan   
3
and  is in quadrant II, find sec .
tan 2   1  sec 2 
Look for an identity that
relates tangent and secant.
tan 2   1  sec 2 
2
 5
2


1

sec

 3 
25
 1  sec 2 
9
34
 sec 2 
9
34
sec   
9
34
sec   
3
Using the identities you now know, find the trig
value :
5
If
tan   
3 and  is in quadrant II, find cot ()
1
cot( ) 
tan( )
1
cot( ) 
 tan 
1
3
cot( ) 

5
 3  5
Using the identities you now know,
find the trig value.
1.) If cosθ = 3/5, find csc θ.
sin 2   cos 2   1
2


3
sin 2      1
 5
25 9
sin 2  

25 25
16
2
sin  
25
4
sin   
5
csc  
1
1
5


sin   4
4
5
Using the identities you now know, find
the trig value.
2.) If cosθ = 3/4, find secθ
1
1
4
sec  


cos 3
3
4