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Chapter 5: Problem 73

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

Short Answer

Expert verified
\(\sin 3x\)

Step by step solution

01

Identify and apply the half-angle identity

The given equation is \(\sqrt{\frac{1-\cos 6 x}{2}}\). This might remind one of the half-angle identity for cosine which is \(\cos(\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos x}{2}}\). Here, the sign is negative inside the square root instead of positive, and the angle is doubled. With this recognition, the expression can be rewritten into a simpler form.
02

Rewrite the equation

When the sign inside the square root is negative, this is also equivalent to \(\sin(\frac{x}{2}) = \pm \sqrt{\frac{1 - \cos x}{2}}\). Apply this to the equation, replacing the \(x\) in the identity with \(6x\), the expression can be rewritten as \(\sin 3x\).
03

Simplify the equation

Substitute the value into the expression. So, the simplified form of the equation is \(\sin 3x\).

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