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Home » What These ‘Payphone’ Lyrics Are Trying to Tell You — Spoiler Alert! - Professionaloutdoormedia

What These ‘Payphone’ Lyrics Are Trying to Tell You — Spoiler Alert! - Professionaloutdoormedia

📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 r^2 \sin(2\theta) = 100.

📌 Find the maximum possible value of $r$ along this path.

📌 We are given the polar equation:

📌 We want to find the **maximum value of $r$** for which this equation has a solution.

📌 Step 1: Solve for $r^2$:

📌 r^2 = \frac{100}{\sin(2\theta)}.

📌 Since $r^2 \geq 0$, we must have $\sin(2\theta) > 0$. The maximum value of $r^2$ occurs when $\sin(2\theta)$ is minimized (but still positive). The minimum positive value of $\sin(2\theta)$ is approaching 0, but we must consider the domain where the expression is defined and real.

📌 However, to **maximize $r$**, we must **minimize the denominator** $\sin(2\theta)$, but not go below 0. So we consider the **maximum of $r^2$** under the constraint that $\sin(2\theta) > 0$. The maximum occurs when $\sin(2\theta)$ is at its **minimum positive value**, but that would make $r^2$ large — but we must find the **actual maximum possible finite value**.

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