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Home » ¡Descubre el Secretos Ocultos de Feliz Día de Acción de Gracias! - Groen Casting

¡Descubre el Secretos Ocultos de Feliz Día de Acción de Gracias! - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 \boxed{(0, 1, 1)}

📌 Question: Find all angles $z \in [0^\circ, 360^\circ]$ satisfying $2\sin^2 z - 3\sin z + 1 = 0$.

📌 Solution: Let $u = \sin z$. The equation becomes $2u^2 - 3u + 1 = 0$. Solving:

📌 u = \frac{3 \pm \sqrt{9 - 8}}{4} = \frac{3 \pm 1}{4} \implies u = 1 \text{ or } u = \frac{1}{2}.

📌 For $\sin z = 1$: $z = 90^\circ$.

📌 For $\sin z = \frac{1}{2}$: $z = 30^\circ, 150^\circ$.

📌 Thus, solutions are $30^\circ, 90^\circ, 150^\circ$.

📌 \boxed{30^\circ}, \boxed{90^\circ}, \boxed{150^\circ}

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To find $\cos \theta$, we use the formula for the dot product:
\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \thet...
First, compute the dot product $\mathbf{u} \cdot \mathbf{v}$:
\mathbf{u} \cdot \mathbf{v} = 3(-1) + 4(2) = -3 + 8 = 5
Next, find the magnitudes of $\mathbf{u}$ and $\mathbf{v}$:
\|\mathbf{u}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\|\mathbf{v}\| = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}
Now, substitute these into the formula:
= 5 \cdot \sqrt{5} \cdot \cos \theta
Solve for $\cos \theta$:

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#### 116
Actually: 2⁸⁰ = 120892581961462417476233206292340255376
2⁶⁴ = 18446744073709551616
Difference = 120892581961462417476233206292340255376 − 184467440737095...
But for compact:
≈ 1.2089 × 10¹⁹ → but per format, use exact expression? But answer exp...
#### 120892059414852865724713604639740173760
But this is too long. Instead, recognize:
Factor: 2⁶⁴(2¹⁶ − 1) = 18446744073709551616 × (65536 − 1) = 1844674407...
Now compute:

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