Question: Find the maximum value of $ (\sin x + 2\cos x)^2 + \sin^2 x $.
["Title: How to Find the Maximum Value of ( (\sin x + 2\cos x)^2 + \sin^2 x ) – A Step-by-Step Guide", "---", "Introduction", "Mathematical optimization problems often appear deceptively simple yet require clever techniques to solve efficiently. One commonly encountered challenge is finding the maximum value of expressions involving trigonometric functions—such as ( (\sin x + 2\cos x)^2 + \sin^2 x ). In this article, we’ll explore how to determine the maximum value of this expression using algebraic manipulation and trigonometric identities. This approach not only solves the problem but enhances your analytical skills in trigonometry and calculus.", "---", "### Step 1: Expand the Expression", "We begin by expanding ( (\sin x + 2\cos x)^2 + \sin^2 x ):", "[\n(\sin x + 2\cos x)^2 = \sin^2 x + 4\sin x \cos x + 4\cos^2 x\n]", "Add ( \sin^2 x ) to this:", "[\n\sin^2 x + 4\sin x \cos x + 4\cos^2 x + \sin^2 x = 2\sin^2 x + 4\sin x \cos x + 4\cos^2 x\n]", "So the expression simplifies to:", "[\nE(x) = 2\sin^2 x + 4\sin x \cos x + 4\cos^2 x\n]", "---", "### Step 2: Use Trigonometric Identities", "Recall the Pythagorean identity:", "[\n\sin^2 x + \cos^2 x = 1\n]", "We rewrite parts of (E(x)) using this:", "- ( 4\sin^2 x + 4\cos^2 x = 4(\sin^2 x + \cos^2 x) = 4 )", "So,", "[\nE(x) = 4 + 4\sin x \cos x + (\sin^2 x + \cos^2 x) = 4 + 4\sin x \cos x + 1 = 5 + 4\sin x \cos x\n]", "Now apply the double-angle identity:", "[\n\sin 2x = 2\sin x \cos x \implies \sin x \cos x = \frac{1}{2} \sin 2x\n]", "Substitute:", "[\nE(x) = 5 + 4 \cdot \frac{1}{2} \sin 2x = 5 + 2\sin 2x\n]", "---", "### Step 3: Maximize the Simplified Expression", "We now aim to maximize:", "[\nE(x) = 5 + 2\sin 2x\n]", "Since the sine function satisfies ( -1 \leq \sin 2x \leq 1 ), the maximum value occurs when ( \sin 2x = 1 ):", "[\nE_{\ ext{max}} = 5 + 2(1) = 7\n]", "---", "### Conclusion", "Thus, the maximum value of ( (\sin x + 2\cos x)^2 + \sin^2 x ) is 7, achieved when ( \sin 2x = 1 ), or equivalently, ( x = \frac{\pi}{4} + k\pi ) for any integer ( k ).", "This method—expanding, applying identities, and using symmetry properties of trigonometric functions—proves powerful and efficient. Understanding this pattern helps solve a wider class of trigonometric optimization problems with confidence.", "---", "Keywords: maximize ((\sin x + 2\cos x)^2 + \sin^2 x), trigonometric optimization, find maximum value, sine and cosine identities, double angle formula, apply (\sin 2x), maximum of (5 + 2\sin 2x), step-by-step trigonometry", "---", "Explore more: Master integrals involving trig functions or practice with alternates forms to deepen your analytical math toolkit."]
