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9.4 Sum-to-product and product-to-sum formulas  (Page 2/6)

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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

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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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