Download Sum and Difference Formulas

Sum and Difference
Formulas
Sum and Difference Formulas for
Cosines and Sines
cos(   )  cos  cos   sin  sin 
cos(   )  cos  cos   sin  sin 
sin(    )  sin  cos   cos  sin 
sin(    )  sin  cos   cos  sin 
The Cosine of the Difference of
Two Angles
cos(   )  cos  cos   sin  sin 
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 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( )  cos  cos   sin  sin 
Substitute exact values from
memory or use special triangles.
Multiply.
Add.
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( - )
with   80° and  = 20°.
cos( )  cos  cos   sin  sin 
cos 80° cos 20°  sin 80° sin 20°  cos (80°  20°)  cos 60°  1/2
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
Example
• Find the exact value of sin(30º+45º)
Solution
sin(    )  sin  cos   cos  sin 
sin( 30  45)  sin 30 cos 45  cos 30 sin 45
1
2
3
2
 


2 2
2
2
2 6

4
Sum and Difference Formulas for
Tangents
The tangent of the sum of two angles equals the tangent of the first angle
plus the tangent of the second angle divided by 1 minus their product.
tan   tan 
tan(   ) 
1  tan  tan 
tan   tan 
tan(   ) 
1  tan  tan 
The tangent of the difference of two angles equals the tangent of the first
angle minus the tangent of the second angle divided by 1 plus their
product.
Example
• Find the exact value of tan(105º)
Solution
•tan(105º)=tan(60º+45º)
tan   tan 
tan(   ) 
1  tan  tan 
tan 60  tan 45

1  tan 60 tan 45
3 1 1 3


1 3 1 3
Example
• Write the following expression as the sine, cosine, or
tangent of an angle. Then find the exact value of the
expression.
7

7

sin
cos  cos
sin
12
12
12
12
Solution
7

7

sin
cos  cos
sin
12
12
12
12
6
 7  
 sin 
   sin
12
 12 12 
 sin

2
1
Example
• Verify the following identity:
5 
2

cos x    
(cos x  sin x)
4 
2

Solution
5 

cos x 

4 

 5
 cos x cos
 4

 5 
  sin x sin  

 4 
2
2
cos x  
sin x
2
2
2

(cos x  sin x)
2
