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Math Help - [SOLVED] Deriving a Formula

  1. #1
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    [SOLVED] Deriving a Formula

    How do you derive a formula for both \cos {3\Theta} and \cos {4\Theta}? Do you just use the sum and difference formulas?
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  2. #2
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    What do you mean by "deriving" the "formulas"? Are you supposed to restate the expressions in terms of \sin{(\theta)} and \cos{(\theta)}?

    Thank you!
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  3. #3
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    Quote Originally Posted by stapel View Post
    What do you mean by "deriving" the "formulas"? Are you supposed to restate the expressions in terms of \sin{(\theta)} and \cos{(\theta)}?

    Thank you!
    It doesn't say specifically but I think so. So would you use the sum and difference formulas to derive?
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  4. #4
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    Assuming you mean what stapel suggested, that is writing them in terms of cos(\theta), sin(\theta), then, yes. You use the sum formulas.

    cos(2\theta)= sin(\theta+ \theta)= cos(\theta)cos(\theta)- sin(\theta)sin(\theta)= cos^2(\theta)- sin^2(\theta)
    sin(2\theta)= cos(\theta)sin(\theta)+ sin(\theta)cos(\theta)= 2 cos(\theta)sin(\theta)

    So
    sin(3\theta)= sin(2\theta+ \theta)= cos(2\theta)sin(\theta)+ sin(2\theta)cos(\theta)
    cos(3\theta)= cos(2\theta+ \theta)= cos(2\theta)cos(\theta)- sin(2\theta)sin(\theta)
    and use the formulas above.

    Similarly for
    cos(4\theta)= cos(3\theta+ \theta) and sin(4\theta)= sin(3\theta+ \theta).
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  5. #5
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    Quote Originally Posted by HallsofIvy View Post
    Assuming you mean what stapel suggested, that is writing them in terms of cos(\theta), sin(\theta), then, yes. You use the sum formulas.

    cos(2\theta)= sin(\theta+ \theta)= cos(\theta)cos(\theta)- sin(\theta)sin(\theta)= cos^2(\theta)- sin^2(\theta)
    sin(2\theta)= cos(\theta)sin(\theta)+ sin(\theta)cos(\theta)= 2 cos(\theta)sin(\theta)

    So
    sin(3\theta)= sin(2\theta+ \theta)= cos(2\theta)sin(\theta)+ sin(2\theta)cos(\theta)
    cos(3\theta)= cos(2\theta+ \theta)= cos(2\theta)cos(\theta)- sin(2\theta)sin(\theta)
    and use the formulas above.

    Similarly for
    cos(4\theta)= cos(3\theta+ \theta) and sin(4\theta)= sin(3\theta+ \theta).
    Ok thanx alot.
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