2.4 Sum and Difference Identities.
Sum and Difference Identities for Sine
Sine of a Sum or Difference
sin  A  B   ____________________________________
sin  A  B   ____________________________________
Cosine of a Sum or Difference
cos  A  B   ____________________________________
cos  A  B   ____________________________________
Tangent of a Sum or Difference
tan  A  B  
tan  A  B  
Proof: see page 276
Applying the Sum and Difference Identities
Find Exact Values (Text: see Ex 1, 2 page 252)
Find the exact value of each expression.
13
(a) sin (−15°)
(b) tan
12
CLASSROOM EXAMPLE 1
(c)
tan100  tan 70
1  tan100 tan 70
Answers: a)
2- 6
; b) 2 4
3 ; c)
3
3
Find Function Values and the Quadrant. (See Ex 3 p 253)
7
3
, and
Suppose that A and B are angles in standard position, with cos A   ,   A 
25
2
3 3
sin B   ,
 B  2 . Find each of the following.
5
2
CLASSROOM EXAMPLE 2
(a) sin (A − B)
(b) tan (A − B)
Answers: a) -
(c) the quadrant of A − B
117
;
125
b) -
117
; c) quadrant IV
44
Prooving a Cofunction Identities
Cofunction Identities
The following identities hold for any angle  for which the functions are defined.
cos  90     ___________
cot  90     ___________
sin  90     ___________
sec  90     ___________
tan  90     ___________
csc  90     ___________
The same identities can be obtained for a real number domain by replacing 90° with
CLASSROOM EXAMPLE 5
Verifying an Identity
sec   x    sec x
2
.
Verify that the equation is an identity. (See Ex. 5 page 254)


tan   t   cot t
2 
CLASSROOM EXAMPLE 6

Verify that the following equation is an identity.
2.5 Multiple-Angle Identities
■ Double-Angle Identities ■ Product-to-Sum and Sum-to-Product Identities
Double-Angle Identities
Double-Angle Identities
sin 2 A  ___________
cos 2 A  ___________
tan 2 A 
CLASSROOM EXAMPLE 1
Given sin  
Find Function Values of 2 (see Ex. 3 page 260)
8
and cos  0, find sin 2 , cos 2 , and tan 2 .
17
CLASSROOM EXAMPLE 2
Deriving a Multiple-Angle Identity (See Ex. 4 page 261)
Write cos 3x in terms of cos x.
Product-to-Sum and Sum-to-Product Identities
Product-to-Sum Identities
cos A cos B  ____________________________________
sin A sin B  ____________________________________
sin A cos B  ____________________________________
cos A sin B  ____________________________________
CLASSROOM EXAMPLE 3
Using a Product-to-Sum Identity (See Ex. 8 page264)
Write 6sin 40sin15 as the sum or difference of two functions.
Sum-to-Product Identities
sin A  sin B  ____________________________________
sin A  sin B  ____________________________________
cos A  cos B  ____________________________________
cos A  cos B  ____________________________________
CLASSROOM EXAMPLE 4
Using a Sum-to-Product Identity (See Ex. 9 page 264)
Write cos3  cos7 as a product of two functions.
CLASSROOM EXAMPLE 5
Solving a Trigonometric Equation
(B)Solve 2cos x  sin 2x  0 (Ex. 1 page 259)
(B)Solve sin5x  sin3x  0 (Ex. 10 page 265)
Half-Angle Identities
In the following identities, the symbol _____________ indicates that the sign is chosen based
A
on the function under consideration and the _____________ of .
2
sin
A

2
cos
A

2
tan
A

2
tan
A

2
CLASSROOM EXAMPLE 6
Using a Half-Angle Identity to Find an Exact Value
Find the exact value of sin 22.5° using the half-angle identity for sine. (See ex. 6 page 262)
CLASSROOM EXAMPLE 7
Using a Half-Angle Identity to Find an Exact Value
Find the exact value of tan 75° using the identity tan
A
sin A

.
2 1  cos A