Download Trigonometry Section 5.3 Sum and Difference Identities for Cosine

Trigonometry
Section 5.3
Sum and Difference Identities
for Cosine
Page 202
Objectives:
1. Difference Identity for Cosine.
2. Sum Identity for Cosine.
3. Cofunction identities.
4. Applying the Sum and Difference
identities.
The figure below gives a graphical
representation of the cosine identity.
The Cosine of the Difference of
Two Angles
cos(α − β ) = cos α cos β + sin α sin β
The cosine of the difference of two angles
equals the cosine of the first angle times
the cosine of the second angle plus the
sine of the first angle times the sine of
the second angle.
Text Example
Find the exact value of cos 80° cos 20° + sin 80° sin 20°.
Solution
The given expression is the right side of the formula for cos(α - β)
with α = 80° and β = 20°.
cos(α
α −β) = cos α cos β + sin α sin β
cos 80° cos 20° + sin 80° sin 20° = cos (80° − 20°) = cos 60° = 1/2
Example
(cos 114°)(cos 54°) + (sin 114°)(sin 54°) =
Notice that this is the subtraction formula for
1 cosine!
(cos 114°)(cos 54°) + (sin 114°)(sin 54°) = cos
2 (114° - 54°)
= cos (60°) = 1/2
Example
cos 17π/12 =
First, we'll rewrite 17π/12 as 2π/3 + 3π/4
So we get:
cos 17π/12 = cos (2π/3 + 3π/4)
= (cos 2π/3)(cos 3π/4) − (sin 2π/3)(sin 3π/4)
2   3  2 
 1 
=  −   −
 − 



 2   2   2  2 
2
6
=
−
4
4
2
6
=
−
4
4
Sum Identity for Cosine
cos( A + B ) = cos A cos B − sin A sin B
Example: Find the exact value of
o
cos(−15 ) without using a calculator!
cos(−15o ) = cos(30o − 45o )
o
o
o
o
= cos 30 cos 45 + sin 30 sin 45
 3  2   1  2 
= 
+



  


 2  2   2  2 
6
2
=
+
4
4
6+ 2
=
4
Example
• Find the exact value of cos(180º-30º)
Solution
cos(180 − 30)
= cos180 cos 30 + sin 180 sin 30
3
1
= −1*
+ 0*
2
2
3
=−
2
cos 23π/12 =
First, we'll rewrite 23π/12 as 5π/3 + π/4
So we get:
cos 23π/12 = cos (5π/3 + π/4) = (cos 5π/3)(cos π/4) − (sin 5π/3)(sin π/4)
3  2 
 1  2  
=   
 −  −
 

 2  2   2  2 
2
6
=
−−
4
4
2
6
=
+
4
4
Example
(cos 81°)(cos 54°) - (sin 81°)(sin 54°) =
Notice that this is the addition formula for cosine!
= (cos 81°)(cos 54°) - (sin 81°)(sin 54°)
= cos (81° + 54°)
= cos (135°) = √2/2
(1) Find the exact value of
Text Example
• Find the exact value of cos 15°
Solution
We know exact values for trigonometric functions of 60° and 45°.
Thus, we write 15° as 60° − 45° and use the difference formula for cosines.
cos l5° = cos(60° − 45°)
= cos 60° cos 45° + sin 60° sin 45°
1
2
3
2
•
+
•
2 2
2
2
=
2
6
+
4
4
=
2+ 6
4
cos(α
α −β) = cos α cos β + sin α sin β
Substitute exact values from
memory or use special triangles.
Multiply.
Add.
Example
• Verify the following identity:
5π 
2

cos x −
(cos x + sin x)
=−
4 
2

Solution
5π 

cos x −

4


 5π 
 5π 
= cos x cos  + sin x sin  
 4 
 4 
2
2
sin x
cos x + −
2
2
2
(cos x + sin x)
=−
2
=−
π
Example: Find cos 12
To do this we think about
π
how to get 12 . We notice:
π
12
= π4 − π6
Apply the sum of angles
for cosine. (This is really
15°=45°-30°)
cos(12π ) = cos(π4 − π6 )
= cos π4 cos π6 + sin π4 sin π6
=
=
2
2
3
2
6+ 2
4
+
2 1
2 2
Cofunction Identities
Example 1
Example 2
Example 3
Reducing cos (A – B) to a Function
of a Single Variable
Finding cos (s + t) Given
Information about s and t
Verification of an Identity
Homework
Assignment
Section 5.3:
Pages 207 - 209:
4, 6,12, 14, 15 - 18all, 30, 32, 34,
38, 46, 48, 50, 54, 56, 58, 60, 62.