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Exercises Trigonometry

Example 4 from Algebra and Trigonometry, 9.4 Sum-to-Product and Product-to-Sum Formulas

Writing the Difference of Sines as a Product

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

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 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.

$\begin{matrix} \sin (4 θ) - \sin (2 θ) & = & 2 \sin (\frac{4 θ - 2 θ}{2}) \cos (\frac{4 θ + 2 θ}{2}) \\ = & 2 \sin (\frac{2 θ}{2}) \cos (\frac{6 θ}{2}) \\ = & 2 \sin θ \cos (3 θ) \end{matrix}$

Your turn

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

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

How did it go?

PreviousExpress the Product as a Sum or Difference Next Evaluating Using the Sum-to-Product Formula

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.