Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(-\sin\left(\frac{x-1}{2}\right)\)

Step 1: Identify Half-angle Formula

The first step is to identify the relevant half-angle formula. Looking at the given expression, it's clear that it matches the half-angle formula for sine, up to a sign and the variable inside cosine, which is \(x-1\) instead of \(x\). The formula is: \(\sin(\theta/2) = \pm \sqrt{\frac{1 - \cos \theta}{2}}\).

Step 2: Apply the Half-angle Formula

Next, apply the half-angle formula to simplify the given expression. The expression looks exactly like the formula for the half-angle of sine. However, instead of \(\theta\), we have \(x-1\). Therefore, considering the negative sign outside the square root, this whole expression can be rewritten as: \(-\sin((x-1)/2)\).

Step 3: Check Your Work

Given the half-angle formula for sine, the expression has been simplified correctly. Always remember to double-check your work for possible mistakes.