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Home » \cos 2\theta = 1 - 2\sin^2\theta - Dyverse

\cos 2\theta = 1 - 2\sin^2\theta - Dyverse

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Simplifying: \( 60t + 320 - 80t = 280 \) → \( -20t = -40 \) → \( t = 2 \).

📌 The sum of the squares of two consecutive integers is 365. What are the integers?

📌 \( n^2 + (n+1)^2 = 365 \).

📌 Simplifying: \( 2n^2 + 2n - 364 = 0 \) → \( n^2 + n - 182 = 0 \).

📌 Solving using quadratic formula: \( n = \frac{-1 \pm \sqrt{1 + 728}}{2} = \frac{-1 \pm 27}{2} \).

📌 Positive solution: \( n = 13 \), so integers are 13 and 14.

📌 #### 13, 14

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$ p(2) = 8 + 2 = 10 $,
$ p(3) = 18 + 3 = 21 $,
$ p(4) = 32 + 4 = 36 $.
All match. Thus, even if not strictly cubic, $ p(0) = 0 $.
But perhaps we misread—maybe the problem says cubic but the data force...
Alternatively, suppose we assume $ p(x) $ is cubic: $ p(x) = ax^3 + 2x...
Thus, $ p(0) = 0 $.
Question: A behavioral health researcher studies patient engagement ov...
Solution: We are given the functional equation:
f(x + y) + f(x - y) = 2f(x) + 2f(y) \quad \text{for all } x, y \in \ma...

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