Transcript Introduction to Database Systems

Sum and Difference Formulas
Dr .Hayk Melikyan
Departmen of Mathematics and CS
[email protected]
H.Melikyan/1200
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The Cosine of the Difference of Two Angles
cos(   )  cos cos   sin  sin 
The cosine of the difference of two angles equals the
cosine of the first angle times the cosine of the
second angle plus the sine of the first angle times the
sine of the second angle.
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2
Text Example

Find the exact value of cos 15°
Solution
We know exact values for trigonometric functions of 60° and 45°.
Thus, we write 15° as 60°  45° and use the difference formula for cosines.
cos l5°  cos(60°  45°)
 cos 60° cos 45°  sin 60° sin 45°
1
2
3
2



2 2
2
2


2
6

4
4
2 6
4
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cos( )  cos  cos   sin  sin 
Substitute exact values from
memory or use special triangles.
Multiply.
Add.
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Text Example
Find the exact value of ( cos 80° cos 20°  sin 80° sin 20°) .
Solution
The given expression is the right side of the formula for cos( - )
with   80° and  = 20°.
cos( )  cos  cos   sin  sin 
cos 80° cos 20°  sin 80° sin 20°  cos (80°  20°)  cos 60°  1/2
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Example

Find the exact value of cos(180º-30º)
Solution
cos(180 30)
 cos180cos30  sin 180sin 30
3
1
 1*
 0*
2
2
3

2
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Example

Verify the following identity:
5 
2

cos x    
(cosx  sin x)
4 
2

Solution
5 

cos x 

4


 5 
 5 
 cos x cos   sin x sin  
 4 
 4 
2
2
cos x  
sin x
2
2
2

(cos x  sin x )
2

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Sum and Difference Formulas for Cosines and Sines
cos(   )  cos cos   sin  sin 
cos(   )  cos cos   sin  sin 
sin(   )  sin  cos   cos sin 
sin(   )  sin  cos   cos sin 
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Example

Find the exact value of sin(30º+45º)
Solution
sin(   )  sin  cos   cos sin 
sin(30  45)  sin 30 cos 45  cos30sin 45
1
2
3
2
 


2 2
2
2
2 6

4
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Sum and Difference Formulas for Tangents
The tangent of the sum of two angles equals the tangent of the first angle
plus the tangent of the second angle divided by 1 minus their product.
t an  t an 
t an(   ) 
1  t an t an 
t an  t an 
t an(   ) 
1  t an t an 
The tangent of the difference of two angles equals the tangent of the first angle
minus the tangent of the second angle divided by 1 plus their product.
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Example

Find the exact value of tan(105º)
Solution
•tan(105º)=tan(60º+45º)
tan  tan 
tan(   ) 
1  tan tan 
tan60  tan 45

1  tan60 tan 45
3 1 1 3


1 3 1 3
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Example

Write the following expression as the sine, cosine, or tangent of
an angle. Then find the exact value of the expression.
7

7 
sin cos  cos sin
12 12
12 12
Solution
7

7

sin
cos  cos sin
12
12
12
12
6
 7  
 sin 
   sin
12
 12 12 
 sin
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
2
1
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