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Exercises Trigonometry

Example 5 from Precalculus, 7.4 Sum-to-Product and Product-to-Sum Formulas

Evaluating Using the Sum-to-Product Formula

Evaluate $\cos (15^{\circ}) - \cos (75^{\circ}) .$

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 cosines.

$\cos α - \cos β = - 2 \sin (\frac{α + β}{2}) \sin (\frac{α - β}{2})$

Then we substitute the given angles and simplify.

$\begin{array}{l} \cos (15^{\circ}) - \cos (75^{\circ}) = - 2 \sin (\frac{15^{\circ} + 75^{\circ}}{2}) \sin (\frac{15^{\circ} - 75^{\circ}}{2}) \\ = - 2 \sin (45^{\circ}) \sin (- 30^{\circ}) \\ = - 2 (\frac{\sqrt{2}}{2}) (- \frac{1}{2}) \\ = \frac{\sqrt{2}}{2} \end{array}$

How did it go?

PreviousUsing the Double-Angle Formulas to Establish an Identity Next Solve the Trigonometric Equation in Linear Form

From Precalculus (1st edition) by OpenStax (Rice University), section 7.4, 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.