Algebra & Trig, Sullivan & Sullivan
Fourth Edition
Notes:§7.4
Page 1 / 2
§7.4 – Sum and Differences Formulas
Basic game plan:
Introduce the sum/difference formula for cosine by having them use it.
 Use it to find exact values (degrees and radians)
 Use it to establish new trig identities
Cover the remaining trig identities
 Mostly via proofs
Have them practice using all the identities
 Both for proving new things,
 And for finding exact values.
Sum and Difference Formulas for Cosine
We'd like a way to figure out exact answers, even when we don't have a special triangle.
Also, we're looking for more tools for our Identity-Proving Toolbox
First, let's take the given formulas, and try using them:
cos     cos  cos   sin  sin 
(Note that these work for both degrees, and radians)
cos     cos  cos   sin  sin 
Problem:
We'll be given an angle, that isn't on a special triangle, and asked to find the exact value of
some trig function.
Basic strategy:
Find a combination of special triangles' angles that add up to the given angle, then use the
sum/difference formula to reduce this problem to the 'trig function of an angle of a special
triangle' problem, which we do know how to do.
Example:
cos75°
 75 = 30 + 45
= cos 30   45
 cos     cos  cos   sin  sin  ; α = 30 ; β = 45
= cos 30 cos 45  sin 30 sin 45
 <draw special triangles, pick the trig functions off of them>
3 1 1 1
=

 
2
2 2
2
 Simplify
3
1

=
2 2 2 2
 Combine (Common denom)
3 1
=
2 2
 Move Sqrt(2) up by multiplying top & bottom by Sqrt(2)
2 3 1
=
22
 Move Sqrt(2) up by multiplying top & bottom by Sqrt(2)




Algebra & Trig, Sullivan & Sullivan
Fourth Edition
Notes:§7.4
Page 2 / 2
6 2
4
Example: This also works w/ radians:
=
cos

12
 1/12 = 3/12 – 2 /12 ; 3/12 is ¼ (a special angle-45°) ; 2/12 is 1/6 (a special angle - 30°)
 3 2 
 
= cos
 12 12 
 etc.
A Slight Variation
Once we've done that, we can (clearly) work backwards from there, as well:
   
   
cos 80  cos 20   sin 80  sin 20 
 cos     cos  cos   sin  sin  ; α = 80 β = 20
= cos 80   20 
 Subtraction
= cos 60 
 <Draw special triangle, get cos60 by definition>
1
=
2


 
<ICE time>
Establishing Identities With This
Use this both as a way to demonstrate how to use this, as well as a way to demonstrate how to
use the 'Presentation Format'
Establishing The Section's Identities
Let's assume that cos     cos  cos   sin  sin 
(Quick outline of proof:
On the unit circle, draw out α & β angles (ending at P2 =(cos β, sin β ) & P1=(cos α, sin α))
We're interested in the difference between them (α - β),
Point out that we could rotate that angle so that one side is on the X axis
(A = (1,0), P3=(cos (α –β), sin (α –β) )
The distance between A & P3 is the same as the dist between P2 and P1.
From there, we'll algebraically isolate)
cos     cos  cos   sin  sin 
cos     cos  cos   sin  sin 
(Note that these work for both degrees, and radians)
sin      sin  cos   sin  cos 
sin      sin  cos   sin  cos 
(Note that these work for both degrees, and radians)