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Explain a situation where we would convert an equation from a product to a sum, and give an example.

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Algebraic

For the following exercises, rewrite the product as a sum or difference.

16 sin ( 16 x ) sin ( 11 x )

8 ( cos ( 5 x ) cos ( 27 x ) )

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20 cos ( 36 t ) cos ( 6 t )

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2 sin ( 5 x ) cos ( 3 x )

sin ( 2 x ) + sin ( 8 x )

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10 cos ( 5 x ) sin ( 10 x )

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sin ( x ) sin ( 5 x )

1 2 ( cos ( 6 x ) cos ( 4 x ) )

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For the following exercises, rewrite the sum or difference as a product.

cos ( 6 t ) + cos ( 4 t )

2 cos ( 5 t ) cos t

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sin ( 3 x ) + sin ( 7 x )

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cos ( 7 x ) + cos ( 7 x )

2 cos ( 7 x )

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sin ( 3 x ) sin ( 3 x )

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cos ( 3 x ) + cos ( 9 x )

2 cos ( 6 x ) cos ( 3 x )

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sin h sin ( 3 h )

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For the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.

cos ( 45° ) cos ( 15° )

1 4 ( 1 + 3 )

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cos ( 45° ) sin ( 15° )

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sin ( −345° ) sin ( −15° )

1 4 ( 3 2 )

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sin ( 195° ) cos ( 15° )

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sin ( −45° ) sin ( −15° )

1 4 ( 3 1 )

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For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.

cos ( 23° ) sin ( 17° )

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2 sin ( 100° ) sin ( 20° )

cos ( 80° ) cos ( 120° )

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2 sin ( −100° ) sin ( −20° )

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sin ( 213° ) cos ( )

1 2 ( sin ( 221° ) + sin ( 205° ) )

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2 cos ( 56° ) cos ( 47° )

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For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

sin ( 76° ) + sin ( 14° )

2 cos ( 31° )

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cos ( 58° ) cos ( 12° )

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sin ( 101° ) sin ( 32° )

2 cos ( 66.5 ° ) sin ( 34.5 ° )

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cos ( 100° ) + cos ( 200° )

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sin ( −1° ) + sin ( −2° )

2 sin ( −1.5° ) cos ( 0.5° )

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For the following exercises, prove the identity.

cos ( a + b ) cos ( a b ) = 1 tan a tan b 1 + tan a tan b

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4 sin ( 3 x ) cos ( 4 x ) = 2 sin ( 7 x ) 2 sin x

2 sin ( 7 x ) 2 sin x = 2 sin ( 4 x + 3 x ) 2 sin ( 4 x 3 x ) = 2 ( sin ( 4 x ) cos ( 3 x ) + sin ( 3 x ) cos ( 4 x ) ) 2 ( sin ( 4 x ) cos ( 3 x ) sin ( 3 x ) cos ( 4 x ) ) = 2 sin ( 4 x ) cos ( 3 x ) + 2 sin ( 3 x ) cos ( 4 x ) ) 2 sin ( 4 x ) cos ( 3 x ) + 2 sin ( 3 x ) cos ( 4 x ) ) = 4 sin ( 3 x ) cos ( 4 x )

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6 cos ( 8 x ) sin ( 2 x ) sin ( 6 x ) = −3 sin ( 10 x ) csc ( 6 x ) + 3

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sin x + sin ( 3 x ) = 4 sin x cos 2 x

sin x + sin ( 3 x ) = 2 sin ( 4 x 2 ) cos ( 2 x 2 ) =
2 sin ( 2 x ) cos x = 2 ( 2 sin x cos x ) cos x =
4 sin x cos 2 x

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2 ( cos 3 x cos x sin 2 x ) = cos ( 3 x ) + cos x

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2 tan x cos ( 3 x ) = sec x ( sin ( 4 x ) sin ( 2 x ) )

2 tan x cos ( 3 x ) = 2 sin x cos ( 3 x ) cos x = 2 ( .5 ( sin ( 4 x ) sin ( 2 x ) ) ) cos x
= 1 cos x ( sin ( 4 x ) sin ( 2 x ) ) = sec x ( sin ( 4 x ) sin ( 2 x ) )

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cos ( a + b ) + cos ( a b ) = 2 cos a cos b

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Numeric

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.

cos ( 58 ) + cos ( 12 )

2 cos ( 35 ) cos ( 23 ) ,  1 .5081

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sin ( 2 ) sin ( 3 )

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cos ( 44 ) cos ( 22 )

2 sin ( 33 ) sin ( 11 ) ,   0.2078

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cos ( 176 ) sin ( 9 )

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sin ( 14 ) sin ( 85 )

1 2 ( cos ( 99 ) cos ( 71 ) ) ,   0.2410

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Technology

For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.

2 sin ( 2 x ) sin ( 3 x ) = cos x cos ( 5 x )

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cos ( 10 θ ) + cos ( 6 θ ) cos ( 6 θ ) cos ( 10 θ ) = cot ( 2 θ ) cot ( 8 θ )

It is and identity.

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sin ( 3 x ) sin ( 5 x ) cos ( 3 x ) + cos ( 5 x ) = tan x

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2 cos ( 2 x ) cos x + sin ( 2 x ) sin x = 2 sin x

It is not an identity, but 2 cos 3 x is.

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sin ( 2 x ) + sin ( 4 x ) sin ( 2 x ) sin ( 4 x ) = tan ( 3 x ) cot x

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For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

sin ( 9 t ) sin ( 3 t ) cos ( 9 t ) + cos ( 3 t )

tan ( 3 t )

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2 sin ( 8 x ) cos ( 6 x ) sin ( 2 x )

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sin ( 3 x ) sin x sin x

2 cos ( 2 x )

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cos ( 5 x ) + cos ( 3 x ) sin ( 5 x ) + sin ( 3 x )

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sin x cos ( 15 x ) cos x sin ( 15 x )

sin ( 14 x )

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Extensions

For the following exercises, prove the following sum-to-product formulas.

sin x sin y = 2 sin ( x y 2 ) cos ( x + y 2 )

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cos x + cos y = 2 cos ( x + y 2 ) cos ( x y 2 )

Start with cos x + cos y . Make a substitution and let x = α + β and let y = α β , so cos x + cos y becomes
cos ( α + β ) + cos ( α β ) = cos α cos β sin α sin β + cos α cos β + sin α sin β = 2 cos α cos β

Since x = α + β and y = α β , we can solve for α and β in terms of x and y and substitute in for 2 cos α cos β and get 2 cos ( x + y 2 ) cos ( x y 2 ) .

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For the following exercises, prove the identity.

sin ( 6 x ) + sin ( 4 x ) sin ( 6 x ) sin ( 4 x ) = tan ( 5 x ) cot x

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cos ( 3 x ) + cos x cos ( 3 x ) cos x = cot ( 2 x ) cot x

cos ( 3 x ) + cos x cos ( 3 x ) cos x = 2 cos ( 2 x ) cos x 2 sin ( 2 x ) sin x = cot ( 2 x ) cot x

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cos ( 6 y ) + cos ( 8 y ) sin ( 6 y ) sin ( 4 y ) = cot y cos ( 7 y ) sec ( 5 y )

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

cos ( 2 y ) cos ( 4 y ) sin ( 2 y ) + sin ( 4 y ) = 2 sin ( 3 y ) sin ( y ) 2 sin ( 3 y ) cos y = 2 sin ( 3 y ) sin ( y ) 2 sin ( 3 y ) cos y = tan y

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sin ( 10 x ) sin ( 2 x ) cos ( 10 x ) + cos ( 2 x ) = tan ( 4 x )

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cos x cos ( 3 x ) = 4 sin 2 x cos x

cos x cos ( 3 x ) = 2 sin ( 2 x ) sin ( x ) = 2 ( 2 sin x cos x ) sin x = 4 sin 2 x cos x

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( cos ( 2 x ) cos ( 4 x ) ) 2 + ( sin ( 4 x ) + sin ( 2 x ) ) 2 = 4 sin 2 ( 3 x )

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tan ( π 4 t ) = 1 tan t 1 + tan t

tan ( π 4 t ) = tan ( π 4 ) tan t 1 + tan ( π 4 ) tan ( t ) = 1 tan t 1 + tan t

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Practice Key Terms 2

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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