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Using the Double-Angle Formulas to Establish an Identity · Physics Playground

Exercises Trigonometry

Example 3 from Precalculus, 7.3 Double-Angle, Half-Angle, and Reduction Formulas

Using the Double-Angle Formulas to Establish an Identity

Establish the following identity using double-angle formulas:

1 + \sin (2 θ) = {(\sin θ + \cos θ)}^{2}

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 the right side of the equal sign and rewrite the expression until it matches the left side.

$\begin{array}{l} {(\sin θ + \cos θ)}^{2} = \sin^{2} θ + 2 \sin θ \cos θ + \cos^{2} θ \\ = (\sin^{2} θ + \cos^{2} θ) + 2 \sin θ \cos θ \\ = 1 + 2 \sin θ \cos θ \\ = 1 + \sin (2 θ) \end{array}$

Your turn

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

Establish the identity: $\cos^{4} θ - \sin^{4} θ = \cos (2 θ) .$

How did it go?

PreviousFinding the Exact Value of an Expression Involving an Inverse Trigonometric Function Next Evaluating Using the Sum-to-Product Formula

From Precalculus (1st edition) by OpenStax (Rice University), section 7.3, 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.

Solution

Your turn