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Exercises Trigonometry

Example 6 from Algebra and Trigonometry, 9.3 Double-Angle, Half-Angle, and Reduction Formulas

Using the Power-Reducing Formulas to Prove an Identity

Use the power-reducing formulas to prove

\sin^{3} (2 x) = [\frac{1}{2} \sin (2 x)] [1 - \cos (4 x)]

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 will work on simplifying the left side of the equation:

$\begin{matrix} \sin^{3} (2 x) & = & [\sin (2 x)] [\sin^{2} (2 x)] \\ = & \sin (2 x) [\frac{1 - \cos (4 x)}{2}] & Substitute the power-reduction formula . \\ = & \sin (2 x) (\frac{1}{2}) [1 - \cos (4 x)] \\ = & \frac{1}{2} [\sin (2 x)] [1 - \cos (4 x)] \end{matrix}$

Your turn

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

Use the power-reducing formulas to prove that $10 \cos^{4} x = \frac{15}{4} + 5 \cos (2 x) + \frac{5}{4} \cos (4 x) .$

How did it go?

PreviousWriting an Equivalent Expression Not Containing Powers Greater Than 1 Next Using a Half-Angle Formula to Find the Exact Value of a Sine Function

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 9.3, 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.