lat03 05

(1)

(2)

Chapter 5

(3)

5.1

(4)

Fundamental Identities



Reciprocal Identities



Quotient Identities

1

1 cot

sec

csc

tan

cos

sin





sin

cos

tan

cot

cos

sin



(5)

More Identities



Pythagorean Identities



Negative-Angle Identities

2

2 sin

cos

1 tan

1 sec

1 cot

csc







 





(6)

Example: If and



is in quadrant II,

find each function value.



a) sec



Look for an identity that

relates tangent and

secant.



5 tan

3 

 

2

2 tan

1 sec

5 1 sec

3

25 1 sec

9

34 sec

9

34 sec

9 



 



_



_{ }









 







2

(7)

Example: If and



is in quadrant II,

find each function value continued



b) sin





c) cot (





)

sin

tan

cos

tan

sin

tan

sin

3 34

5 s

cos

1 sec

in

3

4

3 









_











_

_







_

_









 

5

3

(8)

Example: Express One Function in

Terms of Another



Express cot

x

in

terms of sin

x

.

2

2 1 cot

csc

(9)

Example: Rewriting an Expression in

Terms of Sine and Cosine



Rewrite cot





tan



in terms of sin



and cos



.



2

2 cos

sin

cot

tan

sin

cos

sin

sin cos

cos

sin

sin cos













(10)

5.2

(11)

Hints for Verifying Identities



1. Learn the fundamental identities given in the last

section. Whenever you see either side of a

fundamental identity, the other side should come to

mind. Also, be aware of equivalent forms of the

fundamental identities. For example

is an alternative form of the identity



2. Try to rewrite the more complicated side of the

equation so that it is identical to the simpler side.

2

2 sin



 

1 cos



2

(12)

Hints for Verifying Identities continued



3. It is sometimes helpful to express all

trigonometric functions in the equation in terms

of sine and cosine and them simplify the result.



4. Usually, any factoring or indicated algebraic

operations should be performed. For example,

the expression can be factored as

The sum of difference of two

trigonometric expressions such as can

be added or subtracted in the same way as any

other rational expression.

2 sin

x



2sin

x



1

2 (sin

x



1) .

1

(13)

Hints for Verifying Identities continued



5. As you select substitutions, keep in mind the

side you are changing, because it represents

your goal. For example, to verify the identity

try to think of an identity that relates tan

x

to

cos

x

. In this case, since and

the secant function is the best

link between the two sides.

2

1 tan

1 cos

x

 

1 sec

cos

x



2

(14)

Hints for Verifying Identities continued



6. If an expression contains 1 + sin

x

,

multiplying both the numerator and denominator

by 1



sin

x

would give 1



sin

2 x

, which could be

replaced with cos

2 x

. Similar results for 1



sin

x

,

1 + cos

x

, and 1



cos

x

may be useful.



Remember that verifying identities is NOT the

(15)

Example: Working with One Side



Prove the identity



Solution: Start with the left side.

2

2 (tan

x



1)(cos

x

  

1)

tan

x

2

2 (tan

1)(cos

1)

tan

sin

1 (cos

1)

tan

cos

sin

cos

1 tan

cos

sin

x



  







  













  

2

2 sin

1

1 tan

(16)

Example: Working with One Side



Prove the identity



Solution—start with the

right side



continued



continued

1 csc

sin

sec tan

x



x



x

1 csc

sin

sec tan

1 sin

sin

x









2

1 1 sin

sec tan

sin

cos

sin

cos

sin

1 cot cos

1

1 tan

sec

x

(17)

Example: Working with One Side



Prove the identity



Start with the left side.

tan

cot

tan

cot

tan cot

x

y

x

y



_

_

tan

cot

tan

cot

tan cot

tan

cot

tan cot

1

1 x

y

x

y

x

y

x

y

x

y







(18)

Example: Working with Both Sides



Verify that the following equation is an identity.



Solution: Since both sides appear complex, verify the identity by changing each side into a common

third expression.

2

2 sec

tan

1 2sin

sin

sec

tan

cos





_



(19)

Example: Working with Both Sides

continued



Left side: Multiply numerator and denominator by

cos .



sin

cos

sin

1 cos

1 c

o

s



















sec

tan

sec

tan

sec

tan

sec

tan

sec cos

tan cos

sec cos

tan cos

(20)

Example: Working with Both Sides

continued



Right Side:

Begin by factoring.



We have shown that















 



2

2 cos

1 1 sin

1 2sin

sin

cos

1 sin

1 s

1 sin

1 in

sin























2

2 1 sin

1 sin

sec

tan

1 2sin

sin

sec

tan

cos

(21)

5.3

(22)

Cosine of a Sum or Difference



Find the

exact

value of cos 15



.



cos(

) cos cos

sin sin

cos(

) cos cos

sin sin

A B

A

B

A

B

A B

A

B

A

B











30 cos30

c

sin 30

3

1

2

2 os15

co

45 cos 45

sin

s(

)

6

2

45

2

2 





















(23)

More Examples



cos

5 

12 

cos87 cos93





sin87 sin 93



cos cos

sin sin

6

4

6

4

3

2

1

2

2 2 2

6

2

4 











 





cos(87

93 )

cos180

1 





 



(24)

Cofunction Identities



Similar identities can be obtained for a real

number domain by replacing 90



with



/2.

cos(90

) sin

cot(90

) tan

sin(90

) cos

sec(90

) csc

tan(90

) cot

csc(90

) sec





























(25)

Example: Using Cofunction Identities



Find an angle that satisfies sin (



30 

) = cos





sin( 30 ) cos

sin( 30 ) sin(90

)

30

90

120 





















(26)

Example: Reducing



Write cos (180







) as a trigonometric function

of



.



cos(180

)

cos180

cos

sin180

si

1 n

( )cos

( )sin

cos

0 













 



(27)

5.4

(28)

Sine of a Sum of Difference



Tangent of a Sum or Difference

sin(

) sin cos

cos sin

sin(

) sin cos

cos sin

A B

A

B

A

B

A B

A

B

A

B













tan

tan(

)

tan(

)

1 tan tan

A

B

A

B

A B

A

B

A

B













(29)

Example: Finding Exact Values



Find an exact value for sin 105



.

sin105

sin 60

sin 45

sin 60 cos 45

cos60 sin 45

3

2

1

2

2 2 2

6

2

4

4 











 







(30)

Example: Finding Exact Values

continued



Find an exact value for

sin 90



cos 135





cos 90



sin 135



sin(90

135 )

sin( 45 )

sin 45

2

2 







 



(31)

Example: Write each function as an

expression involving function of



.



sin (30



+



)



tan (45



+



)

cos

cos30

sin

sin 30

3

2 cos

si

1

2 n













tan 45

tan

1 tan 45 tan

1 tan

















(32)

Example: Finding Function Values and

the Quadrant of

A

+

B



Suppose that

A

and

B

are angles in standard

position, with sin

A

= 4/5,



/2 <

A

<



,

and cos

b

=



5/13,



<

B

< 3



/2. Find sin (

A

+

B

).



2

16

25

3 sin

cos

1 cos

1

9 cos

2

5 A

A









4

3

5

5 sin(

)

20 26 16

(33)

5.5

(34)

Double-Angle Identities



2

2 cos 2

cos

sin

cos 2

1 2sin

cos 2

2cos

1 sin 2

2sin cos

2 tan

tan 2

1 tan

A





 







(35)

Example: Given cos



= 3/5 and sin



<

0,

find sin 2



,

cos 2



,

and tan 2



.



Find the value of sin



.



Use the double-angle

identity for sine,

2

3 sin

1

5

16 sin

25

4 sin

5 



 



_{ }



 



 

sin 2

2sin co

(36)

Example: Given cos



= 3/5 and sin



<

0,

find sin 2



,

cos 2



,

and tan 2



continued



Use any of the forms for

cos to find cos 2



.



Find tan 2



.

2

2 cos 2

cos

si

7 n

9

16

2

25 5 25















sin 2

tan 2

co

s 2

(37)

Example: Multiple-Angle Identity



Find an equivalent expression for cos 3

x

.



Solution

2

2 cos3

cos(2

)

cos 2 cos

sin 2 sin

(1 2sin )cos

2sin cos sin

cos

2sin

cos

2sin

cos

x

x x

x









 



(38)

Product-to-Sum Identities



















1 cos cos

cos(

) cos(

2

1 sin sin

cos(

) cos(

)

2

1 sin cos

sin(

) sin(

)

2

1 cos sin

sin(

) sin(

)

2 A

B

A B

A

B

A B

A

B

A B

A

B

A B



















(39)

Example



Write sin 2



cos



as the sum or difference of two

functions.











1 sin 2 cos

sin(2

) sin(2

)

2

1 sin(3 ) sin

2

1

1 sin 3

sin



 













(40)

Sum-to-Product Identities



sin

2sin

cos

2

2 sin

sin

2cos

sin

2

2 cos

cos

2cos

cos

2

2 cos

cos

2sin

sin

2

2 A B

A B

A

B

A B

A

B

A B

A

B

A B

A

B

















_

_

_

_

























_

_

_

_

























_

_

_

_























 

_

_

_

_

(41)

Example



Write cos 2





cos 4



as a product of two

functions.



cos 2

cos 4

2sin

2

4 sin

2

4

2

6

2 2sin

sin

2

2 2sin(3 )sin(

)

















 

_

_

_

_



















 

_

_

_

_









(42)

5.6

(43)

Half-Angle Identities



cos

1 cos

sin

1 cos

2

2 1 cos

sin

tan

2 1 cos

1 cos

tan

2 sin

A





 



 





(44)

Example: Finding an Exact Value



Find the sin (



/8) exactly.

1 cos

4

4 sin

2

1

2

2 





 



 

(45)

Example: Finding an Exact Value



Find the exact value of tan 22.5



using the

identity



Sinc 22.5



= replace

A

with 45



.

sin

tan

.

2 1 cos

A





1

2 (45 ),



2 sin 45

45

22.5

2 tan

tan

1 cos 45

2

1 







(46)

Example: Finding an Exact Value

continued



Multiply the numerator and denominator by 2,

than rationalize the denominator.

2

2 tan 22.5

2

2 2 2 2

2( 2 1

1

2 )

2

2 



















(47)

Example: Simplifying Expressions



Simplify:

tan tan

1 

Solution:

2 x

x



1 cos

sin

tan tan

1

2 sin

cos

sin (1 cos )

1 sin cos

1 cos

cos

x



 



















