Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 7.4
Sum and Differences Formulas
Sum & Difference Formulas for Cosine
Fill in these formulas
1)
cos    
2)
cos    
Using Sum & Difference Formula for Cosine
Find the exact value of each given expression
 
3) Cos 15
5 
4) Cos  
 12 
   
   
5) cos 5 cos 25  sin 5 sin 25
 3
6) cos
 4
  
 3
 cos   sin 
 2
 4
  
 sin  
 2
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 7.4
Page 2 / 9
Using Sum & Difference Formula for Cosine
Establish each of the given identities, using the formulas you've just seen
7) cos     cos     2 cos cos 
First, get a separate piece of paper, and do a 'rough draft' of your work, so you know how to solve
it.
Next, fill in this Presentation Table with your proof:
Establish that
(Rewrite the entire identity here. It should look like an equation)
Starting Point:
(Rewrite JUST ONE side of the identity here. From here on out, you'll work with ONLY that side)

(The explanation for the first step you're doing goes here)
=
(Take whatever you've got in the Starting Point box, and rewrite it here, so as to show how the first step has changed it)

=

=

=

=

=

=
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
8)
§ 7.4
Page 3 / 9
cos    1  tan  tan 

cos    1  tan  tan 
First, get a separate piece of paper, and do a 'rough draft' of your work, so you know how to solve
it.
Next, fill in this Presentation Table with your proof:
Establish that
(Rewrite the entire identity here. It should look like an equation)
Starting Point:
(Rewrite JUST ONE side of the identity here. From here on out, you'll work with ONLY that side)

(The explanation for the first step you're doing goes here)
=
(Take whatever you've got in the Starting Point box, and rewrite it here, so as to show how the first step has changed it)

=

=

=

=

=

=

=
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 7.4
Page 4 / 9
Proving the Sum & Difference Formulas for (Co)Sine
Establish each of the given identities, using the formulas you've just seen
9) As it so happens, cos     cos  cos   sin  sin  (If you're interested in why it's true, there's
a proof in your book you can look at). We'd like to use this fact to show that cos    must be
cos  cos   sin  sin  . The key is to observe that cos    is the same thing as cos     .
The next hint is to remember to look at the Even-Odd Properties when working through this proof.
Using those hints, fill out the following Presentation Format proof:
     cos cos   sin  sin 
Establish that cos
Starting Point:
cos   

(The explanation for the first step you're doing goes here)
=
(Take whatever you've got in the Starting Point box, and rewrite it here, so as to show how the first step has changed it)

=

=

=
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 7.4
Page 5 / 9
10) As it so happens, sin      sin  cos   sin  cos (If you're interested in why it's true, there's
a proof in your book you can look at).
We'd like to use this fact to show that sin     must be sin  cos   sin  cos  . The key is to
observe that sin     is the same thing as sin      . The next hint is to remember to look at
the Even-Odd Properties when working through this proof. Using those hints, fill out the following
Presentation Format proof:
Establish that sin
Starting Point:
     sin  cos   sin  cos
sin    

(The explanation for the first step you're doing goes here)
=
(Take whatever you've got in the Starting Point box, and rewrite it here, so as to show how the first step has changed it)

=

=

=
11) Establish that


cos     sin 
2

Starting Point:


cos   
2


=

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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 7.4
=

=

=
12) Find a formula (identity) for tan     , using the hints provided below
Establish that
tan     
tan   tan 
1  tan  tan 
Starting Point:
tan    
sin 
 tan  
cos
=
 Divide both sides by cos α cos β
=

=

=
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 7.4
Page 7 / 9
13) Find a formula (identity) for tan     , using the hints provided below
Establish that
tan     
tan   tan 
1  tan  tan 
Starting Point:
tan    

=

=

=

=
Using Sum & Difference Formulas in Establishing New Trig Identities
Establish each of the given identities, using any or the formulas you've seen
3


14) sin       cos  (If you don't have enough space here, write your answer out on a separate sheet of paper)
2


(You do not need to do a 'presentation format' style proof for this problem)
15)
sin     
sin  cos 
 1  cot  tan 
(You do not need to do a 'presentation format' style proof for this problem)
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 7.4
Using Sum & Difference Formulas
Find the exact value of each given expression
5 

 12 
16) sin 
17) sin(15 )

7

7
 
 
 
 
18) sin   cos    cos   sin  
 12 
 12 
 12 
 12 
19) cos 70  cos 20  sin70  sin20
7


20) tan  
 12 
21) tan(15 )
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 7.4
Toolbox
Fill in these formulas for future reference
cos    
cos    
sin     
sin     
tan     
tan     
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