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Home » Standard result: In spherical coordinates, \( - Groen Casting

Standard result: In spherical coordinates, \( - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 \begin{pmatrix} \sin x \\ \cos x \end{pmatrix} \cdot \begin{pmatrix} \cos x \\ -\sin x \end{pmatrix} = \sin x \cos x + \cos x (-\sin x) = \sin x \cos x - \sin x \cos x = 0

📌 The dot product is always zero, regardless of \(x\). Hence, the vectors are orthogonal for all \(x\).

📌 Thus, the solution is all real numbers \(x\), and the answer can be expressed as \(\boxed{\text{All } x \in \mathbb{R}}\).

📌 Find the sum of all angles \(z \in [0^\circ, 360^\circ]\) that satisfy \(\sin z + \cos z = 1\).

📌 We solve \(\sin z + \cos z = 1\) for \(z \in [0^\circ, 360^\circ]\).

📌 Square both sides:

📌 + 2\sin z \cos z = 1 \Rightarrow 2\sin z \cos z = 0 \Rightarrow \sin 2z = 0

📌 So, \(2z = 0^\circ, 180^\circ, 360^\circ, 540^\circ, \ldots\), giving:

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