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Home » Let $ y = \sin^2 x $, where $ y > 0 $. Then: - Appfinity Technologies

Let $ y = \sin^2 x $, where $ y > 0 $. Then: - Appfinity Technologies

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Thus, the area increases by \(\boxed{11\sqrt{3}}\) square centimeters.

📌 Compute \(\cos 60^\circ\) using the unit circle and verify using the properties of an equilateral triangle.

📌 On the unit circle, the coordinates of the point corresponding to \(60^\circ\) are \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\). Thus, \(\cos 60^\circ = \frac{1}{2}\).

📌 Alternatively, consider an equilateral triangle with side length 2. By splitting it into two 30-60-90 right triangles, the side opposite \(60^\circ\) is \(\sqrt{3}\), and the hypotenuse is 2. Therefore:

📌 \cos 60^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{2}

📌 Thus, \(\cos 60^\circ\) is \(\boxed{\frac{1}{2}}\).Question: Find $\tan \theta$ where $\theta$ is the angle between the line of sight from a weather balloon at $(2, 3)$ to a cloud at $(5, 11)$ and the horizontal axis.

📌 Solution: The slope of the line from $(2, 3)$ to $(5, 11)$ is $\frac{11 - 3}{5 - 2} = \frac{8}{3}$. Since $\tan \theta$ equals the slope, $\tan \theta = \frac{8}{3}$. \boxed{\dfrac{8}{3}}

📌 Question: Find the sum of all angles $z \in [0^\circ, 360^\circ]$ that satisfy $\sin(2z) = \frac{\sqrt{3}}{2}$.

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