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Thread: 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 $\displaystyle \sin x+p\cos x=2p$ has a solution
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  2. #2
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    Hello great_math
    Quote Originally Posted by great_math View Post
    Find the values of p for which the equation $\displaystyle \sin x+p\cos x=2p$ has a solution
    Let $\displaystyle \sin x + p\cos x = r\sin(x+\alpha)=r\sin x\cos\alpha + r\cos x \sin \alpha$

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

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

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

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

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

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

    Can you complete it from here?

    Grandad
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