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Home » Is Sliggoo the Next Big Thing? Discover the Mind-Blowing Truth Today! - Carbonext

Is Sliggoo the Next Big Thing? Discover the Mind-Blowing Truth Today! - Carbonext

📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 \( A \approx \pi \times 5^2 = 25\pi \).

📌 \( A \approx 25 \times 3.14159 = 78.53975 \).

📌 #### 78.54 cm²Question: Compute the minimum value of $ |\cos x + i\sin x + 1| $ as $ x $ ranges over all real numbers.

📌 Solution: The expression $ \cos x + i\sin x $ corresponds to $ e^{ix} $ on the unit circle in the complex plane. Therefore, $ \cos x + i\sin x + 1 = e^{ix} + 1 $. The modulus is $ |e^{ix} + 1| $. Using the identity $ |e^{ix} + 1| = \sqrt{(\cos x + 1)^2 + \sin^2 x} $, we simplify:

📌 |e^{ix} + 1| = \sqrt{(\cos x + 1)^2 + \sin^2 x} = \sqrt{\cos^2 x + 2\cos x + 1 + \sin^2 x} = \sqrt{2 + 2\cos x}

📌 The minimum value occurs when $ \cos x $ is minimized, i.e., $ \cos x = -1 $:

📌 \sqrt{2 + 2(-1)} = \sqrt{0} = 0

📌 Thus, the minimum value is $ \boxed{0} $.

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