Question

Solve the equation for \(x\). $$ \arcsin \sqrt{2 x}=\arccos \sqrt{x} $$

Short answer

The solution to the equation \(\arcsin \sqrt{2 x}=\arccos \sqrt{x}\) is \(x = 0\).

Step-by-step solution

Identification

Identify the properties of the arcsine and arccosine functions, specifically, \(\arcsin(a) + \arccos(a) = \frac{\pi}{2}\), for \(a \in [-1, 1]\). Since \(\sqrt{2 x}\) and \(\sqrt{x}\) are both within this range, we can use this property on the given equation.

Application of trigonometric properties

Applying the property \(\arcsin(a) + \arccos(a) = \frac{\pi}{2}\), the equation \(\arcsin \sqrt{2 x}=\arccos \sqrt{x}\) can become \(\arcsin \sqrt{2x} + \arccos \sqrt{2x} = \frac{\pi}{2}\). Therefore, using these equivalent terms, we have \(\arccos \sqrt{x} + \arccos \sqrt{2x} = \frac{\pi}{2}\). By isolating \(\arccos \sqrt{x}\), we get \(\arccos \sqrt{x} = \frac{\pi}{2} - \arccos \sqrt{2x}\).

Solving for x

Noting that \(\arccos a = \frac{\pi}{2} - \arcsin a\), we can substitute and get \(\arccos \sqrt{x} = \arcsin \sqrt{2x}\) which simplifies to \(\sqrt{x} = \sqrt{2x}\). Squaring both sides, we get \(x = 2x\), which simplifies to \(x = 0\).