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Chapter 5: Problem 41

Use the half-angle formulas to simplify the expression. $$\sqrt{\frac{1-\cos 6 x}{2}}$$

Short Answer

Expert verified
\(\sqrt{\frac{1-\cos 6 x}{2}} = \cos 3x\)

Step by step solution

01

Identify the Half-Angle Formula

Notice that the expression \( \sqrt{\frac{1-\cos 6 x}{2}} \) looks similar to the half-angle formula for cosine \( \cos \frac{x}{2} = \sqrt{\frac{1+\cos x}{2}} \) or \( \cos \frac{x}{2} = \sqrt{\frac{1-\cos x}{2}} \). The formula to choose depends on the quadrant of \( \frac{x}{2} \). In this case, the negative sign inside the square root matches the positive quadrant, thus \( \cos \frac{x}{2} = \sqrt{\frac{1-\cos x}{2}} \) is the chosen formula.
02

Apply the Half-Angle Formula

As per the previous step, \( \sqrt{\frac{1-\cos 6 x}{2}} \) can be rewritten as the half-angle of cosine. The formula is \( \cos \frac{x}{2} = \sqrt{\frac{1-\cos x}{2}} \), so if you replace \( x \) with \( 6x \), it will become \( \cos \frac{6x}{2} = \sqrt{\frac{1-\cos 6 x}{2}} \), which can further simplify to \( \cos 3x = \sqrt{\frac{1-\cos 6 x}{2}} \).

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