1. **State the problem:** Simplify or analyze the expression $4\sin x + 2\cos x$. 2. **Formula and rules:** We can express a linear combination of sine and cosine functions as a single sinusoidal function using the formula: $$a\sin x + b\cos x = R\sin(x + \phi)$$ where $$R = \sqrt{a^2 + b^2}$$ and $$\phi = \arctan\left(\frac{b}{a}\right)$$ This helps to rewrite the expression in a simpler form. 3. **Calculate $R$:** $$R = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$$ 4. **Calculate $\phi$:** $$\phi = \arctan\left(\frac{2}{4}\right) = \arctan\left(\frac{1}{2}\right)$$ 5. **Rewrite the expression:** $$4\sin x + 2\cos x = 2\sqrt{5} \sin\left(x + \arctan\left(\frac{1}{2}\right)\right)$$ 6. **Explanation:** This means the original expression can be seen as a sinusoidal wave with amplitude $2\sqrt{5}$ and phase shift $\arctan(\frac{1}{2})$. This form is often easier to analyze or graph. **Final answer:** $$4\sin x + 2\cos x = 2\sqrt{5} \sin\left(x + \arctan\left(\frac{1}{2}\right)\right)$$