7.4 - Sum and Difference Identities

L
Bysimilartrianglesweseethat
Hence we have
sin(α)
= cos(β)
1 andthereforeL = sin(α)cos(β).
sin(α+β) = cos(α)sin(β)+sin(α)cos(β).
Sum and Difference Identities
We have established the following result.
Theorem 1.
cos(a±b) = cos(a)cos(b)∓sin(a)sin(b)
sin(a±b) = sin(a)cos(b)±cos(a)sin(b)
tan(a±b) = tan(a)±tan(b)
1∓tan(a)tan(b)
Example 1. Findcos(α+β),giventhatαandβ areinquadrantIII,cos(α) =
−3 12 , and tan(β) = 5 5 .
We have cos(α) = − 3
5
−12
−3
−4
α
5
−5
13
5
, sin(α) = −4,
cos(β) = − 5 13
cos(α+β) = cos(α)cos(β)−sin(α)sin(β) =
β
− 3
5
− 5
13
and sin(β) = −12
13 ,
− − 4
−
5
12
13
Example 2. Use a trig formula to find the exact value for the expression
tan
π . 12
2
.