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Math Help - find values of p:

  1. #1
    Member great_math's Avatar
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    find values of p:

    Find the values of p for which the equation \sin x+p\cos x=2p has a solution
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    Hello great_math
    Quote Originally Posted by great_math View Post
    Find the values of p for which the equation \sin x+p\cos x=2p has a solution
    Let \sin x + p\cos x = r\sin(x+\alpha)=r\sin x\cos\alpha + r\cos x \sin \alpha

    Then r\cos\alpha = 1 and r\sin\alpha = p

    Square and add: r^2(\cos^2\alpha + \sin^2\alpha) = 1+p^2

    \Rightarrow r = \sqrt{1+p^2}

    So \sqrt{1+p^2}\sin(x+\alpha) = 2p

    \Rightarrow \sin(x+\alpha) = \frac{2p}{\sqrt{1+p^2}}

    And this has solutions provided -1 \le \frac{2p}{\sqrt{1+p^2}} \le +1

    Can you complete it from here?

    Grandad
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