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 does the planets on our solar system orbit

cheten Reply

how many Messier objects are there in space

satish Reply

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Richard Reply

what are astronomy

Issan Reply

Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.

Rafael

vjuvu

Elgoog

what is big bang theory?

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what type of activity astronomer do?

Rosemary

Richard

the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen

Roaul

I want to be an astronomer. That's my dream

Astrit

Who named the the whole galaxy?

Shola Reply

solar Univers

GPOWER

what is space

Richard

what is the dark matter

Richard

what are the factors upon which the atmosphere is stratified

Nicholas Reply

is the big bang the sun

Folakemi Reply

Sokak

bigbang is the beginning of the universe

Sokak

but thats just a theory

Sokak

nothing will happen, don't worry brother.

Vansh

what does comet means

GANGAIN Reply

these are Rocky substances between mars and jupiter

GANGAIN

Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.

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John

hey can anyone guide me abt international astronomy olympiad

sahil

how can we learn right and true ?

Govinda Reply

why the moon is always appear in an elliptical shape

Gatjuol Reply

Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.

Hemen

what's see level?

lidiya Reply

Did you mean eye sight or sea level

Minal

oh sorry it's sea level

lidiya

according to the theory of astronomers why the moon is always appear in an elliptical orbit?

Gatjuol

hi !!! I am new in astronomy.... I have so many questions in mind .... all of scientists of the word they just give opinion only. but they never think true or false ... i respect all of them... I believes whole universe depending on true ...থিউরি

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Elyana

we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe

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another theory of universe except big ban

Albash Reply

how was universe born

Asmit Reply

there many theory to born universe but what is the reality of big bang theory to born universe

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what is the exact value of π?

Nagalakshmi

by big bang

universal

there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.

Aarya

I think after Big Bang!

Michele

from where on earth could u observe all the stars during the during the course of an year

Karuna Reply

I think it couldn't possible on earth

Nagalakshmi

in this time i don't Know

Michele

is that so. the question was in the end of this chapter

Karuna

in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.

Christopher

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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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