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Math Help - Use a double angle to rewrite each trigonometric ratio

  1. #1
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    Use a double angle to rewrite each trigonometric ratio

    cos 3x

    The answer at the back of the book says

    2 sinē (1.5x) - 1

    but how do you get that?
    Isn't the "Double Angle Formula" for Cosine supposed to follow
    cos 2Θ = 2 cosēΘ - 1 ??? Why does the answer say "sin"? And how do they know to use 1.5x? (the book is known to have many typos, if that helps...)
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by lanvin View Post
    cos 3x

    The answer at the back of the book says

    2 sinē (1.5x) - 1

    but how do you get that?
    Isn't the "Double Angle Formula" for Cosine supposed to follow
    cos 2Θ = 2 cosēΘ - 1 ??? Why does the answer say "sin"? And how do they know to use 1.5x? (the book is known to have many typos, if that helps...)
    We know that \cos(nx)=\cos^2\left(\frac{n}{2}\right)-\sin^2\left(\frac{n}{2}x\right)={\color{red}2\cos^  2\left(\frac{n}{2}x\right)}=1-2\sin^2\left(\frac{n}{2}x\right)

    So we may do the following

    \begin{aligned}\cos(3x)&=\cos\left(2\cdot\frac{3}{  2}x\right)\\<br />
&=\cos^2\left(\frac{3}{2}x\right)-\sin^2\left(\frac{3}{2}x\right)\\<br />
&={\color{red}2\cos^2\left(\frac{3}{2}x\right)-1}\\<br />
&=1-2\sin^2\left(\frac{3}{2}x\right)\end{aligned}

    It just goes to show the fallibility of books.
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  3. #3
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    thanx a lot for your help,
    but where did you get (n/2)x from?

    the Double Angle Formulas for Cosine we were given say
    cos 2Θ = cosēΘ - sinēΘ = 2cosēΘ -1 = 1- 2sinēΘ

    is it just accepted that (n/2)x = Θ?
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by lanvin View Post
    thanx a lot for your help,
    but where did you get (n/2)x from?

    the Double Angle Formulas for Cosine we were given say
    cos 2Θ = cosēΘ - sinēΘ = 2cosēΘ -1 = 1- 2sinēΘ

    is it just accepted that (n/2)x = Θ?
    Yes
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  5. #5
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    Quote Originally Posted by Mathstud28 View Post
    Yes
    Ok... that helps. Thank you
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