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Math Help - number of solutions

  1. #1
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    number of solutions

    let a,b,c be non-zero real numbers such that



    Then find the number of roots of the equation



    ???



    i was getting no roots as the given expression (on the LHS) according to me satisfies:



    which is a totally different set than that of the values of c.

    so how can they ever be equal? ???

    and the answer is not correct according to the book. :'(
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  2. #2
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    Quote Originally Posted by adhyeta View Post
    let a,b,c be non-zero real numbers such that



    Then find the number of roots of the equation



    ???



    i was getting no roots as the given expression (on the LHS) according to me satisfies:



    which is a totally different set than that of the values of c.

    so how can they ever be equal? ???

    and the answer is not correct according to the book. :'(
    if the question is exactly what you gave us here, then your answer is correct and the book is certainly wrong!
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  3. #3
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    Hello, adhyeta!

    Let a,b,c be non-zero real numbers such that: . |c| \;{\color{red}>} \;\sqrt{a^2+b^2} . Is this correct?

    Then find the number of roots of the equation: . a\sin x + b\cos x \:=\:c
    Divide by \sqrt{a^2+b^2}

    . . \frac{a}{\sqrt{a^2+b^2}}\,\sin x + \frac{b}{\sqrt{a^2+b^2}}\cos x \;=\;\frac{c}{\sqrt{a^2+b^2}}


    Let \tan\theta \:=\:\frac{b}{a}
    . . Then: . \cos\theta \:=\:\frac{a}{\sqrt{a^2+b^2}},\;\;\sin\theta \:=\:\frac{b}{\sqrt{a^2+b^2}}


    Then we have: . \cos\theta\sin x + \sin\theta\cos x \:=\:\frac{c}{\sqrt{a^2+b^2}} \quad\Rightarrow\quad \sin(x + \theta) \:=\:\frac{c}{\sqrt{a^2+b^2}} .[1]


    But if |c| \:>\:\sqrt{a^2+b^2}, then: . \frac{|c|}{\sqrt{a^2+b^2}} \:>\:1

    Then [1] becomes: . \sin(x + \theta) \:> \:1 \quad\hdots .which is impossible.


    Therefore, there are no solutions.

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  4. #4
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    is its certainly correct.
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