5.2 Sum and Difference Formulas

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5.2 Sum and Difference Formulas
Objective To develop and use formulas for the
trigonometric functions of a sum or difference of
two angle measures
Yep, they sure are!
Aint they a silly bunch?
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Whats the point?

•  

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For example

• sin(15) sin(45 - 30)
• cos(75) cos(30 45)
• Now we just need to know what this means when we
  use the sum and difference formulas

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Sum and Difference Formulas for Cosines
Sum and Difference Formulas for Sines
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Sum and Difference Formulas for Sines
 
 
 
 
sin
sin
cos
cos
 
 
sin
cos
 
 
 
 
cos
sin
 
 

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Sum and Difference Formulas for Cosines
 
 
 
cos
 
sin
cos
sin
 
 
sin
sin
 
 
 
 
cos
cos
 
 


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• Can you memorize these formulas? You will have
  to if you take college trigonometry. Here is a
  love story to help introduce the trigonometry sum
  and difference formulas in an interesting way

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• As we all know, some of the people to whom we are
  attracted are not attracted to us. And it is not
  unusual for a person who has shown interest in us
  to later lose interest in us. Maybe that is a
  good thing, because it forces us to date a lot of
  people and to become more experienced in
  maintaining relationships.

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• Anyway, this is the story of Sinbad and Cosette.
  Sinbad loved Cosette, but Cosette did not feel
  the same way about Sinbad.

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• Naturally, when Sinbad was in charge of their
  double date, he put himself with Cosette, and he
  put her sister with his brother
• sin(A B) sin A cosB cosA sinB.
• sin(A - B) sin A cosB - cosA sinB.
• Sinbad loved to tell people that his and
  Cosette's signs were the same.

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• However, when Cosette was in charge of the double
  date she placed herself with her sister and put
  Sinbad with his brother. She made sure everyone
  knew that their signs were NOT the same
• cos(A B) cosA cosB - sinA sinB.
• cos(A - B) cosA cosB sinA sinB.
• Also, notice that Cosette placed herself and her
  sister BEFORE Sinbad and his brother. This detail
  was important to Cosette. She was very snobby,
  you know.

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Finding exact values of trig expressions

1. Split the given number into the sum/difference of
   unit circle values we know
2. Change the problem using the correct formula
3. Simplify by replacing in trig values

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1. Split 75o into 30o and 45o
 
 
 
2. Use the cosine formula
cos(A B) cosA cosB - sinA sinB.
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cos(A B) cosA cosB - sinA sinB.
3. Replace with Trig values
cos
sin
sin
cos
 
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cos(A B) cosA cosB - sinA sinB.
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Look at the formulas.
?
Which one does it match?
 
 
 
 
 
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Find the exact value of
 
 
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You will need to know these formulas so let's
study them a minute to see the best way to
memorize them.
opposite
cos has same trig functions in first term and in
last term, but opposite signs between terms.
same
sin has opposite trig functions in each term but
same signs between terms.
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Verifying Identities

• These three steps are key in verifying identities
  that require the sum and difference formulas
• 1. Write in expanded form
• 2. Substitute known values
• 3. Simplify

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Verifying Identities

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We will work with the left side.
 
 
 
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negative
 
 
What is the formula for cosine?
 
 
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positive
 
 
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Sum and Difference Formulas for Tangent
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Find tan 105
tan 105 tan ( 60 45)
tan 60 tan 45 1 tan 60 tan 45

 

 
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Find tan 105
 
 

 
 
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