Transcript Section 5.3 - University of South Florida

Chapter 5
Analytic
Trigonometry
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All rights reserved
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1
SECTION 5.3
Sum and Difference Formulas
OBJECTIVES
1
2
3
4
Use the sum and difference formulas for
cosine.
Know and use cofunction identities.
Use the sum and difference formulas for sine.
Use the sum and difference formulas for
tangent.
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SUM AND DIFFERENCE FORMULAS
FOR COSINE
cos  u  v   cos u cos v  sin u sin v
cos  u  v   cos u cos v  sin u sin v
You will be provided with a total of three
sum and difference equations. They each
represent a pair of equations
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EXAMPLE 1
Using the Difference Formula for Cosine

Find the exact value of cos
by using
12
  
  .
12 3 4
Solution





  
cos  cos     cos cos  sin sin
12
3
4
3
4
3 4
1 2
3 2
 


2 2
2 2
2
6


4
4

2 6
4
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Compare this with Example
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1.
7
BASIC COFUNCTION IDENTITIES
If v is any real number or angle measured in
radians, then


cos   v   sin v
2



sin   v   cos v
2


If angle v is measured in degrees, then replace
2
by 90º in these identities.
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EXAMPLE 3
Using Cofunction Identities


Prove that for any real number x, tan   x   cot x.
2

Solution


sin   x 
2



tan   x  
2



cos   x 
2

cos x

sin x
 cot x
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SUM AND DIFFERENCE FORMULAS
FOR SINE
sin  u  v   sin u cos v  cos u sin v
sin  u  v   sin u cos v  cos u sin v
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EXAMPLE 5
Using the Sum Formula for Sine
Find the exact value of
sin 63º cos27º  cos63º sin 27º
without using a calculator.
Solution
This expression is the right side of the sum
formula for sine (u + v), where u = 63º and v = 27º.
sin 63º cos 27º  cos63ºsin 27º  sin  63º 27º 
 sin  90º 
1
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EXAMPLE 6
Finding the Exact Value of a Sum
12
3
3
Let sin u =  and cos v = , with π < u <
13
5
2
3
and
< v < 2π. Find the exact value of
2
sin (u + v).
Solution
cos u   1  sin 2 u
Find cos u.
In QIII, cos < 0.
9
16
  1

25
25
4
cos u  
5
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EXAMPLE 6
Finding the Exact Value of a Sum
Solution continued
Find sin v.
In QIV, sin < 0.
sin v   1  cos v
2
144
25
  1

169
169
5
sin v  
13
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EXAMPLE 6
Finding the Exact Value of a Sum
Solution continued
sin (u + v) = sin u cos v + cos u sin v
 3  12   4  5 
          
 5  13   5  13 
36 20
16
 

65 65
65
16
The exact value of sin (u + v) is  .
65
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Omit per Dept: REDUCTION FORMULA
If (a, b) is any point on the terminal side of an
angle  (radians) in standard position, then
a sin x  b cos x  a 2  b 2 sin  x   
for any real number x.
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SUM AND DIFFERENCE FORMULAS
FOR TANGENT
tan u  tan v
tan  u  v  
1  tan u tan v
tan u  tan v
tan  u  v  
1  tan u tan v
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EXAMPLE 11 Verifying an Identity
Verify the identity: tan   x    tan x
Solution
Apply the difference formula.
tan   tan x
tan   x  
1  tan  tan x
0  tan x
tan   x  
1  0  tan x
tan   x    tan x
Therefore the given equation is an identity.
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