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Exercises Trigonometry

Example 3 from Algebra and Trigonometry, 9.4 Sum-to-Product and Product-to-Sum Formulas

Express the Product as a Sum or Difference

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

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

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

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 evaluate $\cos \frac{11 π}{12} \cos \frac{π}{12} .$

How did it go?

PreviousWriting the Product as a Sum Containing only Sine or Cosine Next Writing the Difference of Sines as a Product

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.