Sum-to-Product and Product-to-Sum Formulas

Learning Objectives

In this section, you will:

• Express products as sums.
• Express sums as products.

Figure 1 The UCLA marching band (credit: Eric Chan, Flickr).

A band marches down the field creating an amazing sound that bolsters the crowd. That sound travels as a wave that can be interpreted using trigonometric functions. For example, Figure 2 represents a sound wave for the musical note A. In this section, we will investigate trigonometric identities that are the foundation of everyday phenomena such as sound waves.

Figure 2

Expressing Products as Sums

We have already learned a number of formulas useful for expanding or simplifying trigonometric expressions, but sometimes we may need to express the product of cosine and sine as a sum. We can use the product-to-sum formulas, which express products of trigonometric functions as sums. Let’s investigate the cosine identity first and then the sine identity.

Expressing Products as Sums for Cosine

We can derive the product-to-sum formula from the sum and difference identities for cosine. If we add the two equations, we get:

cosαcosβ+sinαsinβ+cosαcosβ−sinαsinβ==cos(α−β)cos(α+β)___________________________________2cosαcosβ=cos(α−β)+cos(α+β)cos𝛼cos𝛽+sin𝛼sin𝛽=cos(𝛼−𝛽)+cos𝛼cos𝛽−sin𝛼sin𝛽=cos(𝛼+𝛽)___________________________________2cos𝛼cos𝛽=cos(𝛼−𝛽)+cos(𝛼+𝛽)

Then, we divide by 22 to isolate the product of cosines:

cosαcosβ=12[cos(α−β)+cos(α+β)]cos𝛼cos𝛽=12[cos(𝛼−𝛽)+cos(𝛼+𝛽)]

HOW TO

Given a product of cosines, express as a sum.

1. Write the formula for the product of cosines.
2. Substitute the given angles into the formula.
3. Simplify.

EXAMPLE 1

Writing the Product as a Sum Using the Product-to-Sum Formula for Cosine

Write the following product of cosines as a sum: 2cos(7×2)cos3x2.2cos(7𝑥2)cos3𝑥2.

Solution

We begin by writing the formula for the product of cosines:

cosαcosβ=12[cos(α−β)+cos(α+β)]cos𝛼cos𝛽=12[cos(𝛼−𝛽)+cos(𝛼+𝛽)]

We can then substitute the given angles into the formula and simplify.

2cos(7×2)cos(3×2)===(2)(12)[cos(7×2−3×2))+cos(7×2+3×2)][cos(4×2)+cos(10×2)]cos2x+cos5x2cos(7𝑥2)cos(3𝑥2)=(2)(12)[cos(7𝑥2−3𝑥2))+cos(7𝑥2+3𝑥2)]=[cos(4𝑥2)+cos(10𝑥2)]=cos2𝑥+cos5𝑥

TRY IT #1

Use the product-to-sum formula to write the product as a sum or difference: cos(2θ)cos(4θ).cos(2𝜃)cos(4𝜃).

Expressing the Product of Sine and Cosine as a Sum

Next, we will derive the product-to-sum formula for sine and cosine from the sum and difference formulas for sine. If we add the sum and difference identities, we get:

+ sin(α+β)sin(α−β)==sinαcosβ+cosαsinβsinαcosβ−cosαsinβ_________________________________________sin(α+β)+sin(α−β)=2sinαcosβ sin(𝛼+𝛽)=sin𝛼cos𝛽+cos𝛼sin𝛽+sin(𝛼−𝛽)=sin𝛼cos𝛽−cos𝛼sin𝛽_________________________________________sin(𝛼+𝛽)+sin(𝛼−𝛽)=2sin𝛼cos𝛽

Then, we divide by 2 to isolate the product of cosine and sine:

sinαcosβ=12[sin(α+β)+sin(α−β)]sin𝛼cos𝛽=12[sin(𝛼+𝛽)+sin(𝛼−𝛽)]

EXAMPLE 2

Writing the Product as a Sum Containing only Sine or Cosine

Express the following product as a sum containing only sine or cosine and no products: sin(4θ)cos(2θ).sin(4𝜃)cos(2𝜃).

Solution

Write the formula for the product of sine and cosine. Then substitute the given values into the formula and simplify.

sinαcosβsin(4θ)cos(2θ)===12[sin(α+β)+sin(α−β)]12[sin(4θ+2θ)+sin(4θ−2θ)]12[sin(6θ)+sin(2θ)]sin𝛼cos𝛽=12[sin(𝛼+𝛽)+sin(𝛼−𝛽)]sin(4𝜃)cos(2𝜃)=12[sin(4𝜃+2𝜃)+sin(4𝜃−2𝜃)]=12[sin(6𝜃)+sin(2𝜃)]

TRY IT #2

Use the product-to-sum formula to write the product as a sum: sin(x+y)cos(x−y).sin(𝑥+𝑦)cos(𝑥−𝑦).

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:

cos(α−β)=cosαcosβ+sinαsinβ−cos(α+β)=−(cosαcosβ−sinαsinβ)____________________________________________________cos(α−β)−cos(α+β)=2sinαsinβcos(𝛼−𝛽)=cos𝛼cos𝛽+sin𝛼sin𝛽−cos(𝛼+𝛽)=−(cos𝛼cos𝛽−sin𝛼sin𝛽)____________________________________________________cos(𝛼−𝛽)−cos(𝛼+𝛽)=2sin𝛼sin𝛽

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

sinαsinβ=12[cos(α−β)−cos(α+β)]sin𝛼sin𝛽=12[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:

cosαcosβ=12[cos(α−β)+cos(α+β)]cos𝛼cos𝛽=12[cos(𝛼−𝛽)+cos(𝛼+𝛽)]

sinαcosβ=12[sin(α+β)+sin(α−β)]sin𝛼cos𝛽=12[sin(𝛼+𝛽)+sin(𝛼−𝛽)]

sinαsinβ=12[cos(α−β)−cos(α+β)]sin𝛼sin𝛽=12[cos(𝛼−𝛽)−cos(𝛼+𝛽)]

cosαsinβ=12[sin(α+β)−sin(α−β)]cos𝛼sin𝛽=12[sin(𝛼+𝛽)−sin(𝛼−𝛽)]

EXAMPLE 3

Express the Product as a Sum or Difference

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

Solution

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

cosαcosβcos(3θ)cos(5θ)===12[cos(α−β)+cos(α+β)]12[cos(3θ−5θ)+cos(3θ+5θ)]12[cos(2θ)+cos(8θ)]Use even-odd identity.cos𝛼cos𝛽=12[cos(𝛼−𝛽)+cos(𝛼+𝛽)]cos(3𝜃)cos(5𝜃)=12[cos(3𝜃−5𝜃)+cos(3𝜃+5𝜃)]=12[cos(2𝜃)+cos(8𝜃)]Use even-odd identity.

TRY IT #3

Use the product-to-sum formula to evaluate cos11π12cosπ12.cos11𝜋12cos𝜋12.

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+v2=α𝑢+𝑣2=𝛼 and u−v2=β.𝑢−𝑣2=𝛽.

Then,

α+βα−β======u+v2+u−v22u2uu+v2−u−v22v2v𝛼+𝛽=𝑢+𝑣2+𝑢−𝑣2=2𝑢2=𝑢𝛼−𝛽=𝑢+𝑣2−𝑢−𝑣2=2𝑣2=𝑣

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

sinαcosβsin(u+v2)cos(u−v2)2sin(u+v2)cos(u−v2)===12[sin(α+β)+sin(α−β)]12[sinu+sinv]sinu+sinvSubstitute for(α+β) and (α−β)sin𝛼cos𝛽=12[sin(𝛼+𝛽)+sin(𝛼−𝛽)]sin(𝑢+𝑣2)cos(𝑢−𝑣2)=12[sin𝑢+sin𝑣]Substitute for(𝛼+𝛽) and (𝛼−𝛽)2sin(𝑢+𝑣2)cos(𝑢−𝑣2)=sin𝑢+sin𝑣

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

SUM-TO-PRODUCT FORMULAS

The sum-to-product formulas are as follows:

sinα+sinβ=2sin(α+β2)cos(α−β2)sin𝛼+sin𝛽=2sin(𝛼+𝛽2)cos(𝛼−𝛽2)

sinα−sinβ=2sin(α−β2)cos(α+β2)sin𝛼−sin𝛽=2sin(𝛼−𝛽2)cos(𝛼+𝛽2)

cosα−cosβ=−2sin(α+β2)sin(α−β2)cos𝛼−cos𝛽=−2sin(𝛼+𝛽2)sin(𝛼−𝛽2)

cosα+cosβ=2cos(α+β2)cos(α−β2)cos𝛼+cos𝛽=2cos(𝛼+𝛽2)cos(𝛼−𝛽2)

EXAMPLE 4

Writing the Difference of Sines as a Product

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

Solution

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

sinα−sinβ=2sin(α−β2)cos(α+β2)sin𝛼−sin𝛽=2sin(𝛼−𝛽2)cos(𝛼+𝛽2)

Substitute the values into the formula, and simplify.

sin(4θ)−sin(2θ)===2sin(4θ−2θ2)cos(4θ+2θ2)2sin(2θ2)cos(6θ2)2sinθcos(3θ)sin(4𝜃)−sin(2𝜃)=2sin(4𝜃−2𝜃2)cos(4𝜃+2𝜃2)=2sin(2𝜃2)cos(6𝜃2)=2sin𝜃cos(3𝜃)

TRY IT #4

Use the sum-to-product formula to write the sum as a product: sin(3θ)+sin(θ).sin(3𝜃)+sin(𝜃).

EXAMPLE 5

Evaluating Using the Sum-to-Product Formula

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

Solution

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

cosα−cosβ=−2sin(α+β2)sin(α−β2)cos𝛼−cos𝛽=−2sin(𝛼+𝛽2)sin(𝛼−𝛽2)

Then we substitute the given angles and simplify.

cos(15°)−cos(75°)====−2sin(15°+75°2)sin(15°−75°2)−2sin(45°)sin(−30°)−2(2√2)(−12)2√2cos(15°)−cos(75°)=−2sin(15°+75°2)sin(15°−75°2)=−2sin(45°)sin(−30°)=−2(22)(−12)=22

EXAMPLE 6

Proving an Identity

Prove the identity:

cos(4t)−cos(2t)sin(4t)+sin(2t)=−tantcos(4𝑡)−cos(2𝑡)sin(4𝑡)+sin(2𝑡)=−tan𝑡

Solution

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(4t)−cos(2t)sin(4t)+sin(2t)=====−2sin(4t+2t2)sin(4t−2t2)2sin(4t+2t2)cos(4t−2t2)−2sin(3t)sint2sin(3t)cost−2sin(3t)sint2sin(3t)cost−sintcost−tantcos(4𝑡)−cos(2𝑡)sin(4𝑡)+sin(2𝑡)=−2sin(4𝑡+2𝑡2)sin(4𝑡−2𝑡2)2sin(4𝑡+2𝑡2)cos(4𝑡−2𝑡2)=−2sin(3𝑡)sin𝑡2sin(3𝑡)cos𝑡=−2sin(3𝑡)sin𝑡2sin(3𝑡)cos𝑡=−sin𝑡cos𝑡=−tan𝑡

Analysis

Recall that verifying trigonometric identities has its own set of rules. The procedures for solving an equation are not the same as the procedures for verifying an identity. When we prove an identity, we pick one side to work on and make substitutions until that side is transformed into the other side.

EXAMPLE 7

Verifying the Identity Using Double-Angle Formulas and Reciprocal Identities

Verify the identity csc2θ−2=cos(2θ)sin2θ.csc2𝜃−2=cos(2𝜃)sin2𝜃.

Solution

For verifying this equation, we are bringing together several of the identities. We will use the double-angle formula and the reciprocal identities. We will work with the right side of the equation and rewrite it until it matches the left side.

cos(2θ)sin2θ===1−2sin2θsin2θ1sin2θ−2sin2θsin2θcsc2θ−2cos(2𝜃)sin2𝜃=1−2sin2𝜃sin2𝜃=1sin2𝜃−2sin2𝜃sin2𝜃=csc2𝜃−2

TRY IT #5

Verify the identity tanθcotθ−cos2θ=sin2θ.