Download 5_4_ Sum and Difference Formulas NOTES

LESSON 5.4. Intro to Calculus Topics.
Sum and Difference Formulas for Trigonometric Functions.
Warm-up:
The following statements are false. Provide enough evidence (for a deductive argument) to prove these
equations false.
Formulas for Cos(a + b) and sin(a + b)
The two main purposes for these formulas are:
 finding exact values of other trigonometric functions/angles
(those not defined on the unit circle)

simplifying expressions to find other identities
Sum and Difference Formulas for Sine And Cosine
cos(a + b) = cos a cos b - sin a sin b
cos(a - b) = cos a cos b +sin a sin b
sin ( + ) = sin  cos  + sin  cos 
sin ( - ) = sin  cos  - sin  cos 
How do we use them?
EXAMPLE 1) Find the exact value of
cos

12

Solution:
We know that cos 12 = cos 15o
Think of two angles that either add or subtract and give you
15°. There are many, but the best pair is a = 45 and b = 30 .
WHY???
Using the formula for the difference of the cosine
cos 15 = cos(45 - 30) = cos 45 cos 30 + sin 45 sin 30
Translate into radians:
cos

  
   
   
 cos    cos  cos   sin   sin   
12
4 6
4 6
4 6
EXAMPLE 2) Find the exact value of
7
sin
12
Solution:
7
sin
o
We know that
=
sin
105
12
Think of two angles that either add or subtract and give you 15°.
We could use fact that
105°= (60°+ 45°) so
7  
 
12 3 4
Using the formula for the difference of the sine
sin
7
  
   
   
 sin     sin   cos   sin   cos  
12
3 4
3 4
4 3
3 2
2 1
6
2




 
2  2  2  2 
4
4
6 2
exact answer
4
EXAMPLE 3) Find the exact value of
5


5
sin
cos  sin cos
18
18
18
18
Solution:
Match the problem to one of the formulas.
 it is the sum formula for the sine function
5


5
 5  
sin
cos  sin cos
 sin 
 
18
18
18
18
 18 18 

3
 6 
sin    sin 
3
2
 18 
EXAMPLE 4) Suppose sin  = 4/5 and sin = 5/13, where both a
and b are in the first quadrant. Find cos ( - )
Solution:
We need the difference formula for cosine
Cos( - ) = cos cos + sin sin
GIVEN: sin = 4/5 and sin  = 5/13
We can use basic trig to find cos and sin .
cos  = 3/5
cos  = 12/13
Substitute the values in the formula:
Cos( - ) = (3/5)(12/13) + (4/5)(5/13) = (36/65) + (20/65) = 56/65
EXAMPLE 5) Verify that cos(x - ) = - cos x
Solution:
Use the difference formula for cosine:
cos(x - ) = cos x cos  + sin x sin 
We know that cos  = -1 and sin  = 0,
Therefore
cos(x - ) = cos x cos  + sin x sin 
= -1(cos x) + 0(sin x)
= - cos x
EXAMPLE 6) Simplify:
Solution:
Formulas for tan(a + b)
The sum and difference formulas for tangent are valid for values
in which tan a, tan b, and tan(a +b) are defined.
Sum and Difference formulas for
Tangent
EXAMPLE 7) Find the exact value of tan 105o
Solution:
Use one of the above formulas. Find a pair of numbers that you
know the exact value of that add up to 105.
Try 45 and 60! Use the addition formula!
EXAMPLE 8) If the tan x = -7/24 and cot y = 3/4, x is in
quadrant II and y is in quadrant III, find each of the following:
a) tan(x + y)
b) tan(x - y)
Solution:
a) Since the tangent and cotangent functions are reciprocals, tan
y = 4/3
b)
EXAMPLE 9) Verify tan(x - /2) = -cot x
Solution:
Since the tangent is undefined at /2, we must change to
sine and cosine.
EXAMPLE 10) Let sin x = 3/5 and sin y = 5/13 and both angles
are in quadrant I, find tan(x + y).
Solution:
Since sin x = 3/5, cos x = 4/5 and tan x = 3/4.
Since sin y = 5/13, cos y = 12/13 and tan y = 5/12