class of 8.29

    • express \(\sin(3x)\) in term of \(\sin(x)\).
    • express \(\cos(4x)\) in terms of \(\cos(x)\).
    • simplify
      \[
      \sum_{k=-20}^{k=20} e^{k\phi}.
      \]
  1. solve

    • \[
      y’=\frac{2t}{1+t^2}, y(0)=0;
      \]
    • \[
      y’=\frac{2t}{1-t^2}, y(0)=0;
      \]
    • \[
      y’=\frac{2t}{1-t^2}, y(2)=0.
      \]
  2. solve
    • \[
      y’=\frac{2x^2-4x+3}{x-1}, y(0)=2;
      \]
    • \[
      y’=\frac{x^2-4x+3}{x-2}, y(0)=2;
      \]
    • \[
      y’=\frac{x-2}{x^2-4x+3}, y(0)=2;
      \]
  3. Find with precision 5%
    \[
    \int_{-\pi}^\pi\cos(x)^{100} dx.
    \]

    (hint a: \(\cos(x)=(e^{ix}+e^{-ix})/2\); hint b: \(\cos(x)\approx 1-x^2/2\) for small \(x\)).

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