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Home » Using the Pythagorean theorem: \(10^2 = 8^2 + x^2\). - High Altitude Science

Using the Pythagorean theorem: \(10^2 = 8^2 + x^2\). - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Which is $ \ge \frac{5}{2} = 2.5 $ — valid.

📌 The two valid solutions are $ x = -\frac{10}{3} $ and $ x = \frac{14}{3} $. The largest is $ \frac{14}{3} $.

📌 Thus, the answer is $ \boxed{\dfrac{14}{3}} $.Question: Find the smallest positive angle $\theta$ such that $\cos(\theta + 60^\circ) + \cos(\theta - 60^\circ) = \sqrt{3}$.

📌 Solution: Using the sum-to-product identity:

📌 \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)

📌 Let $A = \theta + 60^\circ$, $B = \theta - 60^\circ$. Then:

📌 \cos(\theta + 60^\circ) + \cos(\theta - 60^\circ) = 2 \cos\theta \cos 60^\circ

📌 Since $\cos 60^\circ = \frac{1}{2}$, we have:

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