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

Precalculus

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Expressing products of sines in terms of cosine

Expressing the product of sines in terms of cosine is also derived from the sum and difference identities for cosine. In this case, we will first subtract the two cosine formulas:

\begin{array}{l} \underset{____________________________________________________}{\begin{array}{l} \begin{array}{l} \cos (α - β) = \cos α \cos β + \sin α \sin β \end{array} \\ - \cos (α + β) = - (\cos α \cos β - \sin α \sin β) \end{array}} \\ \cos (α - β) - \cos (α + β) = 2 \sin α \sin β \end{array}

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

\sin α \sin β = \frac{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.

A General Note

The product-to-sum formulas

The product-to-sum formulas are as follows:

\cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)]

\sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)]

\sin α \sin β = \frac{1}{2} [\cos (α - β) - \cos (α + β)]

\cos α \sin β = \frac{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.

\begin{array}{l} \cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)] \\ \cos (3 θ) \cos (5 θ) = \frac{1}{2} [\cos (3 θ - 5 θ) + \cos (3 θ + 5 θ)] \\ = \frac{1}{2} [\cos (2 θ) + \cos (8 θ)] & Use even-odd identity . \end{array}

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Try It

Use the product-to-sum formula to evaluate $\cos \frac{11 π}{12} \cos \frac{π}{12} .$

$\frac{- 2 - \sqrt{3}}{4}$

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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 $\frac{u + v}{2} = α$ and $\frac{u - v}{2} = β .$

Then,

\begin{array}{l} α + β = \frac{u + v}{2} + \frac{u - v}{2} \\ = \frac{2 u}{2} \\ = u \\ α - β = \frac{u + v}{2} - \frac{u - v}{2} \\ = \frac{2 v}{2} \\ = v \end{array}

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

\begin{array}{l} \sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)] \\ \sin (\frac{u + v}{2}) \cos (\frac{u - v}{2}) = \frac{1}{2} [\sin u + \sin v] & Substitute for (α + β) and (α - β) \\ 2 \sin (\frac{u + v}{2}) \cos (\frac{u - v}{2}) = \sin u + \sin v \end{array}

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

A General Note

Sum-to-product formulas

The sum-to-product formulas are as follows:

\sin α + \sin β = 2 \sin (\frac{α + β}{2}) \cos (\frac{α - β}{2})

\sin α - \sin β = 2 \sin (\frac{α - β}{2}) \cos (\frac{α + β}{2})

\cos α - \cos β = - 2 \sin (\frac{α + β}{2}) \sin (\frac{α - β}{2})

\cos α + \cos β = 2 \cos (\frac{α + β}{2}) \cos (\frac{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 (\frac{α - β}{2}) \cos (\frac{α + β}{2})

Substitute the values into the formula, and simplify.

\begin{array}{l} \sin (4 θ) - \sin (2 θ) = 2 \sin (\frac{4 θ - 2 θ}{2}) \cos (\frac{4 θ + 2 θ}{2}) \\ = 2 \sin (\frac{2 θ}{2}) \cos (\frac{6 θ}{2}) \\ = 2 \sin θ \cos (3 θ) \end{array}

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Try It

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 $\cos (15^{\circ}) - \cos (75^{\circ}) .$

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

\cos α - \cos β = - 2 \sin (\frac{α + β}{2}) \sin (\frac{α - β}{2})

Then we substitute the given angles and simplify.

\begin{array}{l} \cos (15^{\circ}) - \cos (75^{\circ}) = - 2 \sin (\frac{15^{\circ} + 75^{\circ}}{2}) \sin (\frac{15^{\circ} - 75^{\circ}}{2}) \\ = - 2 \sin (45^{\circ}) \sin (- 30^{\circ}) \\ = - 2 (\frac{\sqrt{2}}{2}) (- \frac{1}{2}) \\ = \frac{\sqrt{2}}{2} \end{array}

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Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

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BenJay

Method

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Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

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Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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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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