1. Stated problem: Simplify or analyze the expression $$\sin(x) - \sqrt{\pi} \cos(x).$$ 2. Understanding the expression: This is a linear combination of sine and cosine functions with coefficients 1 and $$-\sqrt{\pi}$$ respectively. 3. One way to express this combination is in a single trigonometric function form: $$A\sin(x + \phi),$$ where $$A$$ is the amplitude and $$\phi$$ the phase shift. 4. Calculate amplitude $$A$$: $$A = \sqrt{1^2 + (-\sqrt{\pi})^2} = \sqrt{1 + \pi}.$$ 5. Calculate phase shift $$\phi$$ using $$\tan(\phi) = \frac{b}{a} = \frac{-\sqrt{\pi}}{1} = -\sqrt{\pi}$$, so $$\phi = \arctan(-\sqrt{\pi}).$$ 6. Final single-function form: $$\sin(x) - \sqrt{\pi} \cos(x) = \sqrt{1+\pi} \sin\left(x + \arctan(-\sqrt{\pi})\right).$$ This expresses the original expression as a single sinusoid with amplitude $$\sqrt{1+\pi}$$ and phase shift $$\arctan(-\sqrt{\pi})$$ for easier interpretation and further analysis.