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Home » Discover the Surprising Plural of ‘Axis’—You Won’t Believe What It Is! - Portal da AcÃºstica

Discover the Surprising Plural of ‘Axis’—You Won’t Believe What It Is! - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Use \( \pi \approx 3.14159 \): \( r \approx \frac{31.4}{6.28318} \approx 5 \).

📌 Area formula: \( A = \pi r^2 \).

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

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

📌 #### 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 $:

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