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Math Help - trigo problem no.6

  1. #1
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    trigo problem no.6

    trigo problem no.6
    p181 q11 ex8b
    question:
    show that the equation acosx + bsinx +c = 0 has only one root in the range of 0 =< x < 360 if and only if a^2 + b^2 + c^2 .

    though i know the discrimination D = 0 i don't know how to do the proving as it is not a quadratic equation.
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  2. #2
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    Quote Originally Posted by afeasfaerw23231233 View Post
    trigo problem no.6
    p181 q11 ex8b
    question:
    show that the equation acosx + bsinx +c = 0 has only one root in the range of 0 =< x < 360 if and only if a^2 + b^2 = c^2 . Mr F edit correction red.

    though i know the discrimination D = 0 i don't know how to do the proving as it is not a quadratic equation.
    You should be comfortable with the fact that a \cos x + b \sin x can be expressed in the form \sqrt{a^2 + b^2} \cos(x + \phi), say.

    So the equation can be re-written as \sqrt{a^2 + b^2} \cos (x + \phi) = -c \Rightarrow \cos (x + \phi) = - \frac{c}{\sqrt{a^2 + b^2}}.

    Now note that this equation will only have one solution over the given domain if \cos(x + \phi) = \pm 1.

    So -\frac{c}{\sqrt{a^2 + b^2}} = \pm 1.

    Square both sides and re-arrange: c^2 = a^2 + b^2.
    Last edited by mr fantastic; January 20th 2008 at 12:19 AM. Reason: Fixing latex - hold on ...... Fixed!
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