6.4 Sum & Difference Formulas

6.4
Sum and difference formula
Objective: Use formulas to find
exact values and establish
identities.
Title: Apr 23­12:10 PM (1 of 6)

Sum and Difference Formulas
sin(α + β) = sin( α ) cos( β ) + cos( α ) sin( β )
sin (α ­ β) = sin( α ) cos( β ) ­ cos( α ) sin( β )
cos(α + β) = cos( α ) cos( β ) ­ sin( α ) sin( β )
cos(α ­ β) = cos( α ) cos( β ) + sin( α ) sin( β )
tan(α + β) =
tan( α ) + tan( β )
1 ­ tan( α )tan( β )
tan(α ­ β) =
tan( α ) ­ tan( β )
1 + tan( α )tan( β )
Title: Apr 23­12:10 PM (2 of 6)

Lets look at the differenece between π/4 and π/6.
120
90
60
π/4 ­ π/6 = 3π/12­ 2π/12= π/12Wow
5
45
How about the difference between π/3 and π/4.
30
π/3 ­ π/4 = 4π/12­ 3π/12= π/12 Hmm a pattern
360
330
5
315
240
300
270
Title: Apr 23­12:10 PM (3 of 6)

Ex 1
Find the exact value of
cos(π/12)
cos(3π/12­ 2π/12)
cos(π/4 ­ π/6)
cos(π/4) cos(π/6) + sin(π/4) sin(π/6)
Break into known angles
Reduce
Use formula
Use unit circle
( √2/2 )( √3/2 ) + ( √2/2 )( 1/2 ) Multiply
√6/4 + √2/4 Add
(√6 + √2) /4 Done!!
Title: Apr 23­12:10 PM (4 of 6)

Ex 2
Find the exact value
sin(5π/12)
sin(9π/12­ 4π/12)
sin(3π/4 ­ π/3)
sin(3π/4)cos(π/3) ­ cos(3π/4)sin(π/3)
(√2/2 )( 1/2 ) ­ (­√2/2)(√3/2)
(√2/4) ­ (­√6/4)
Break into known angles
Reduce
Use Formula
Use Unit Circle
Multiply
Add
(√2 + √6) /4 Done
Title: Apr 23­12:10 PM (5 of 6)

Home Work:
Page 481
# 10 ­ 30 Even
Title: Apr 23­12:10 PM (6 of 6)