9.4 Sum-to-product and product-to-sum formulas

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In this section, you will:
• Express products as sums.
• Express sums as products.

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, [link] 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.

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:

$\begin{array}{l}\underset{___________________________________}{\begin{array}{ccc}\hfill \mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta +\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta & =& \mathrm{cos}\left(\alpha -\beta \right)\hfill \\ \hfill +\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta -\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta & =& \mathrm{cos}\left(\alpha +\beta \right)\hfill \end{array}}\\ \begin{array}{ccc}\hfill \phantom{\rule{5.7em}{0ex}}2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta & =& \mathrm{cos}\left(\alpha -\beta \right)+\mathrm{cos}\left(\alpha +\beta \right)\hfill \end{array}\end{array}$

Then, we divide by $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ to isolate the product of cosines:

$\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta =\frac{1}{2}\left[\mathrm{cos}\left(\alpha -\beta \right)+\mathrm{cos}\left(\alpha +\beta \right)\right]$

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.

Writing the product as a sum using the product-to-sum formula for cosine

Write the following product of cosines as a sum: $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{7x}{2}\right)\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\frac{3x}{2}.$

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

$\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta =\frac{1}{2}\left[\mathrm{cos}\left(\alpha -\beta \right)+\mathrm{cos}\left(\alpha +\beta \right)\right]$

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

$\begin{array}{ccc}\hfill 2\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{7x}{2}\right)\mathrm{cos}\left(\frac{3x}{2}\right)& =& \left(2\right)\left(\frac{1}{2}\right)\left[\mathrm{cos}\left(\frac{7x}{2}-\frac{3x}{2}\right)\right)+\mathrm{cos}\left(\frac{7x}{2}+\frac{3x}{2}\right)\right]\hfill \\ & =& \left[\mathrm{cos}\left(\frac{4x}{2}\right)+\mathrm{cos}\left(\frac{10x}{2}\right)\right]\hfill \\ & =& \mathrm{cos}\text{\hspace{0.17em}}2x+\mathrm{cos}\text{\hspace{0.17em}}5x\hfill \end{array}$

Use the product-to-sum formula to write the product as a sum or difference: $\text{\hspace{0.17em}}\mathrm{cos}\left(2\theta \right)\mathrm{cos}\left(4\theta \right).$

$\frac{1}{2}\left(\mathrm{cos}6\theta +\mathrm{cos}2\theta \right)$

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:

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

$\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta =\frac{1}{2}\left[\mathrm{sin}\left(\alpha +\beta \right)+\mathrm{sin}\left(\alpha -\beta \right)\right]$

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: $\text{\hspace{0.17em}}\mathrm{sin}\left(4\theta \right)\mathrm{cos}\left(2\theta \right).$

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

$\begin{array}{ccc}\hfill \mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta & =& \frac{1}{2}\left[\mathrm{sin}\left(\alpha +\beta \right)+\mathrm{sin}\left(\alpha -\beta \right)\right]\hfill \\ \hfill \mathrm{sin}\left(4\theta \right)\mathrm{cos}\left(2\theta \right)& =& \frac{1}{2}\left[\mathrm{sin}\left(4\theta +2\theta \right)+\mathrm{sin}\left(4\theta -2\theta \right)\right]\hfill \\ & =& \frac{1}{2}\left[\mathrm{sin}\left(6\theta \right)+\mathrm{sin}\left(2\theta \right)\right]\hfill \end{array}$

Use the product-to-sum formula to write the product as a sum: $\text{\hspace{0.17em}}\mathrm{sin}\left(x+y\right)\mathrm{cos}\left(x-y\right).$

$\frac{1}{2}\left(\mathrm{sin}2x+\mathrm{sin}2y\right)$

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:

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