SUM AND DIFFERENCES OF
PERIODIC FUNCTIONS
Dr. Shildneck
Spring, 2015
Using the Unit Circle to
DERIVE THE COSINE OF A DIFFERENCE
Given two angles, u and v, we want to find a formula for the cosine of the difference
between u and v.
B(cos v ,sinv)
u
θ=u-v
A(cos u ,sinu)
v
A '(cos ,sin )
B '(1,0)
A '(cos ,sin )
B(cos v ,sinv)
θ
B '(1,0)
θ=u-v
A(cos u ,sinu)
Since AB  A ' B ' , we can write an equivalence relation using the distance formula
for the lengths of the two segments.
Now…
we can use the previous identity and the even/odd
identities to
DERIVE THE COSINE OF A SUM
And…
Then…
we can use the previous identities, co-function
identities, and even/odd identities to
DERIVE THE SINE OF A SUM
AND THE SINE OF A DIFFERENCE
And…
And then…
we can use the previous identities, quotient
identities, and even/odd identities to
DERIVE THE TANGENT OF A SUM
AND THE TANGENT OF A DIFFERENCE
But… we aren’t going to… So, here are the rest…
SUM and DIFFERENCE IDENTITIES
sin(x  y)  sin x cos y  sin y cos x
sin(1st)cos(2nd) [same operation] sin(2nd)cos(1st)
cos(x  y)  cos x cos y sin x sin y
cos(1st)cos(2nd) [opposite operation] sin(1st)sin(2nd)
sin(x  y) sin x cos y  sin y cos x tan x  tan y
tan(x  y) 


cos(x  y) cos x cos y sin x sin y 1 tan x tan y
Example 1
A) Find the exact value of sin75
B) Find the exact value of cos75
C) Find the exact value of tan75
Example 2
7
Find the exact value of cos
12
Example 3
Simplify the expression:
cos12 cos52  sin12 sin52
Example 4
Simplify the expression:
tan5x  tan3 x
1  tan5x tan3 x
Example 5
Write cos(arctan1  arccos x) as an expression of x.
Example 6
4
Find the exact value of sin(u  v) if sinu  ,
5
5
in Quadrant 1 and tanv  
in Quadrant 2.
12
ASSIGNMENT
Assignment 2 WS