Download Sum and Difference Formulas

CHAPTER 2.4
CHAPTER 2 ANALYTICAL TRIGONOMETRY
PART 4 –Sum and Difference Formulas
TRIGONOMETRY MATHEMATICS CONTENT STANDARDS:
 9.0 - Students compute, by hand, the values of the trigonometric functions and
the inverse trigonometric functions at various standard points.
 10.0 - Students demonstrate an understanding of the addition formulas for
sines and cosines and their proofs and can use those formulas to prove
and/or simplify other trigonometric identities.
OBJECTIVE(S):
 Students will learn the sum and difference formulas.
 Students will learn how to evaluate trigonometric expressions using
the sum and difference formulas.
 Students will learn how to solve trigonometric equations using the sum
and difference formulas.
Using Sum and Difference Formulas
In this and the following section, you will study the uses of several trigonometric
identities and formulas.
Sum and Difference Formulas
b g
sin u  v  sin u cos v  cos u sin v
b g 1tan tanu utantanvv
tan u  v 
b g
sin u  v  sin u cos v  cos u sin v
b g
cos u  v  cos u cos v  sin u sin v
b g
cos u  v  cos u cos v  sin u sin v
b g 1tan tanu utantanvv
tan u  v 
CHAPTER 2.4
EXAMPLE 1: Evaluating a Trigonometric Function
Find the exact value of cos750 .
To find the exact value of cos750 , use the fact that 750 = __________ + _________.
Consequently, the formula for cos u  v yields
b g
cos750
=
=
=
=

Try checking this result on your calculator:

cos750
EXAMPLE 2: Evaluating a Trigonometric Function

Find the exact value of sin .
12

= ___________ - _______________ together with the formula
12
for sin u  v , you obtain
Using the fact that
b g
sin

=
12
=
=
=
=
CHAPTER 2.4
EXAMPLE 3: Evaluating a Trigonometric Expression
Find the exact value of sin 420 cos 120  cos 420 sin 120
Recognizing that this expression fits the formula ___________________, you can write
sin 420 cos 120  cos 420 sin 120
=
=
=
EXAMPLE 4: An Application of a Sum Formula
Evaluate cos arctan 1 arccos x as an algebraic expression.
b
g
b g
This expression fits the formula for cos u  v . Angles u = _________________ and v =
______________.
b g
cos u  v =
=
=
CHAPTER 2.4
1.) Find the exact value:
a. sin150
c. tan1950
DAY 1
b. cos
5
12
CHAPTER 2.4
EXAMPLE 5: Proving a Cofunction Identity


Prove the cofunction identity cos  x   sin x .
2

Using the formula for cosu  v , you have


cos  x 
2

=
=
=
Sum and difference formulas can be used to rewrite expressions such as
n 

sin  
 and
2 

n 

cos 
 , where n is an integer
2 

as expressions involving only sin  or cos  . The resulting formulas are called
reduction formulas.
EXAMPLE 6: Deriving Reduction Formulas
Simplify each expression.
3 

a.) cos 

2 

Using the formula for cosu  v , you have
3 

cos 
 =
2 

=
=
CHAPTER 2.4
b.) tan   3 
Using the formula for tan u  v , you have
tan   3 
=
=
=
EXAMPLE 7: Solving a Trigonometric Equation




Find all solutions of sin  x    sin  x    1 in the interval 0,2  .
4
4


Using the sum and difference formulas, rewrite the equation as




sin  x    sin  x   = -1
4
4


Therefore, the only solutions in the interval 0,2  are:
x = ____________and x = _____________
The next example was taken from calculus. It is used to derive the formula for the
derivative of the sine function.
CHAPTER 2.4
EXAMPLE 8: An Application from Calculus
Verify that
sin x  h   sin x
 sinh 
 1  cosh 
 cos x 
  sin x 

h
 h 
 h 
where h  0 .
Using the formula for sin u  v , you have
sin  x  h   sin x
h
=
=
=
CHAPTER 2.4
2.) Rewrite in a simpler form:
a. cos 200 cos 300  sin 200 sin 300
DAY 2
b. sin
2

2

cos  cos
sin
9
10
9
10
CHAPTER 2.4
3.) Verify that the two functions are equivalent:
5
y1  cos
x
4
y2  
F
IJ
G
H K
2
bcos x  sin xg
2
4.) If sin u 
7

4
3
 v  2 , find cos u  v .
, where  u   , and cosv  , where
25
2
5
2
b g
CHAPTER 2.4
b g b g
5.) Verify sin x  y  sin x  y  sin 2 x  sin 2 y .
6.) Write as an algebraic expression:
b
cos arcsin x  arctan 2 x
g
CHAPTER 2.4
g
7.) Find all solutions on 0,2 .
I
F
G
H 2J
K 3 tanb  xg 0
2 sin x 
DAY 3