Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The simplified form of the given expression using half-angle formulas is \( \cos 3x \).

Step 1: Identify the Half-Angle Identity

The half-angle identity for cosine is given as \( \cos{\frac{\theta}{2}} = \sqrt{\frac{1 + \cos \theta}{2}} \). Notice the similarity of the given expression with the right-hand side of this identity.

Step 2: Recognize the Angle

It's important to see that \( \theta \) in the half-angle identity is twice as large as the angle inside the cosine function of the given expression. In this case, \( \theta = 2(3x) = 6x \).

Step 3: Apply the Half-Angle Identity

Apply the cosine half-angle identity to the expression and simplify. Given that the provided identity \( \cos{\frac{\theta}{2}} = \sqrt{\frac{1 + \cos \theta}{2}} \) resembles the given expression, one can simply write this as \( \cos{\frac{\theta}{2}} = \cos 3x \).