math pre-calc

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find an expression for sin (5 theta) as a fifth degree polynomial in the variable sin(theta) ?

  • math pre-calc -

    One way -- do the algebra involved in the following:

    sin 5x = sin 3x cos 2x + sin 2x cos 3x
    where
    sin 3x = sin 2x cos x + cos 2x sin x
    cos 3x = cos 2x cos x - sin 2x sin x
    cos 2x = 1 - 2 sin^2 x
    sin 2x = 2 sin x cos x
    and cos^2 x = 1 - sin^2 x

    substitute these up the line, simplify and get:

    sin 5 x =16 sin^5 x - 20sin^3 x +5sin x

    Another way -- combine Demoivre and the binomial theorem :

    sin^5 theta is the imaginary part of ( cos theta + i sin theta)^5

    (a+bi)^5 = a^5 +5a^4bi -10a^3b^2 -10a^2b^3i +5ab^4 +b^5i etc

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