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Exercises Trigonometry

Example 8 from Algebra and Trigonometry, 9.2 Sum and Difference Identities

Verifying an Identity Involving Sine

Verify the identity $\sin (α + β) + \sin (α - β) = 2 \sin α \cos β .$

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 see that the left side of the equation includes the sines of the sum and the difference of angles.

$\begin{matrix} \sin (α + β) & = & \sin α \cos β + \cos α \sin β \\ \sin (α - β) & = & \sin α \cos β - \cos α \sin β \end{matrix}$
We can rewrite each using the sum and difference formulas.

$\begin{matrix} \sin (α + β) + \sin (α - β) & = & \sin α \cos β + \cos α \sin β + \sin α \cos β - \cos α \sin β \\ = & 2 \sin α \cos β \end{matrix}$
We see that the identity is verified.

How did it go?

PreviousFinding a Cofunction with the Same Value as the Given Expression Next Verifying an Identity Involving Tangent

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 9.2, 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.