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                     cos ( α β ) = cos α cos β + sin α sin β                   cos ( α + β ) = ( cos α cos β sin α sin β ) ____________________________________________________ cos ( α β ) cos ( α + β ) = 2 sin α sin β

Then, we divide by 2 to isolate the product of sines:

sin α sin β = 1 2 [ cos ( α β ) cos ( α + β ) ]

Similarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas.

The product-to-sum formulas

The product-to-sum formulas are as follows:

cos α cos β = 1 2 [ cos ( α β ) + cos ( α + β ) ]
sin α cos β = 1 2 [ sin ( α + β ) + sin ( α β ) ]
sin α sin β = 1 2 [ cos ( α β ) cos ( α + β ) ]
cos α sin β = 1 2 [ sin ( α + β ) sin ( α β ) ]

Express the product as a sum or difference

Write cos ( 3 θ ) cos ( 5 θ ) as a sum or difference.

We have the product of cosines, so we begin by writing the related formula. Then we substitute the given angles and simplify.

cos α cos β = 1 2 [ cos ( α β ) + cos ( α + β ) ] cos ( 3 θ ) cos ( 5 θ ) = 1 2 [ cos ( 3 θ 5 θ ) + cos ( 3 θ + 5 θ ) ] = 1 2 [ cos ( 2 θ ) + cos ( 8 θ ) ] Use even-odd identity .
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Use the product-to-sum formula to evaluate cos 11 π 12 cos π 12 .

2 3 4

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Expressing sums as products

Some problems require the reverse of the process we just used. The sum-to-product formulas allow us to express sums of sine or cosine as products. These formulas can be derived from the product-to-sum identities. For example, with a few substitutions, we can derive the sum-to-product identity for sine . Let u + v 2 = α and u v 2 = β .

Then,

α + β = u + v 2 + u v 2 = 2 u 2 = u α β = u + v 2 u v 2 = 2 v 2 = v

Thus, replacing α and β in the product-to-sum formula with the substitute expressions, we have

sin α cos β = 1 2 [ sin ( α + β ) + sin ( α β ) ] sin ( u + v 2 ) cos ( u v 2 ) = 1 2 [ sin u + sin v ] Substitute for ( α + β )  and  ( α β ) 2 sin ( u + v 2 ) cos ( u v 2 ) = sin u + sin v

The other sum-to-product identities are derived similarly.

Sum-to-product formulas

The sum-to-product formulas are as follows:

sin α + sin β = 2 sin ( α + β 2 ) cos ( α β 2 )
sin α sin β = 2 sin ( α β 2 ) cos ( α + β 2 )
cos α cos β = −2 sin ( α + β 2 ) sin ( α β 2 )
cos α + cos β = 2 cos ( α + β 2 ) cos ( α β 2 )

Writing the difference of sines as a product

Write the following difference of sines expression as a product: sin ( 4 θ ) sin ( 2 θ ) .

We begin by writing the formula for the difference of sines.

sin α sin β = 2 sin ( α β 2 ) cos ( α + β 2 )

Substitute the values into the formula, and simplify.

sin ( 4 θ ) sin ( 2 θ ) = 2 sin ( 4 θ 2 θ 2 ) cos ( 4 θ + 2 θ 2 ) = 2 sin ( 2 θ 2 ) cos ( 6 θ 2 ) = 2 sin θ cos ( 3 θ )
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Use the sum-to-product formula to write the sum as a product: sin ( 3 θ ) + sin ( θ ) .

2 sin ( 2 θ ) cos ( θ )

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Evaluating using the sum-to-product formula

Evaluate cos ( 15° ) cos ( 75° ) . Check the answer with a graphing calculator.

We begin by writing the formula for the difference of cosines.

cos α cos β = 2 sin ( α + β 2 ) sin ( α β 2 )

Then we substitute the given angles and simplify.

cos ( 15° ) cos ( 75° ) = −2 sin ( 15° + 75° 2 ) sin ( 15° 75° 2 ) = −2 sin ( 45° ) sin ( −30° ) = −2 ( 2 2 ) ( 1 2 ) = 2 2
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Proving an identity

Prove the identity:

cos ( 4 t ) cos ( 2 t ) sin ( 4 t ) + sin ( 2 t ) = tan t

We will start with the left side, the more complicated side of the equation, and rewrite the expression until it matches the right side.

cos ( 4 t ) cos ( 2 t ) sin ( 4 t ) + sin ( 2 t ) = 2 sin ( 4 t + 2 t 2 ) sin ( 4 t 2 t 2 ) 2 sin ( 4 t + 2 t 2 ) cos ( 4 t 2 t 2 ) = 2 sin ( 3 t ) sin t 2 sin ( 3 t ) cos t = 2 sin ( 3 t ) sin t 2 sin ( 3 t ) cos t = sin t cos t = tan t
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Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
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ranges
EDWIN
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Oliver
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find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
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factoring polynomial
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find general solution of the Tanx=-1/root3,secx=2/root3
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the value of 2 sin square 60 Cos 60
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0.75
Lynne
0.75
Inkoom
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depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
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where can I get indices
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I need matrices
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Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
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What do you mean by a
Cromwell
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Cromwell
Solve the x? x=18+(24-3)=72
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x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
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b=p-4a-2c
Suddhen
b= p - 4a - 2c
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p=2(2a+C)+b
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b=p-2(2a+c)
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P=4a+b+2C
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b=P-4a-2c
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John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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The sequence is {1,-1,1-1.....} has
amit Reply
Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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