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Exercises Trigonometry

Example 1 from Algebra and Trigonometry, 9.4 Sum-to-Product and Product-to-Sum Formulas

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

Write the following product of cosines as a sum: $2 \cos (\frac{7 x}{2}) \cos \frac{3 x}{2} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

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

$\cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)]$
We can then substitute the given angles into the formula and simplify.

$\begin{matrix} 2 \cos (\frac{7 x}{2}) \cos (\frac{3 x}{2}) & = & (2) (\frac{1}{2}) [\cos (\frac{7 x}{2} - \frac{3 x}{2})) + \cos (\frac{7 x}{2} + \frac{3 x}{2})] \\ = & [\cos (\frac{4 x}{2}) + \cos (\frac{10 x}{2})] \\ = & \cos 2 x + \cos 5 x \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

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

How did it go?

PreviousFinding the Measurement of a Half Angle Next Writing the Product as a Sum Containing only Sine or Cosine

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 9.4, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.