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Math Help - Rsin(x+a) problem

  1. #1
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    Rsin(x+a) problem

    Hi
    I am stuck on the last bit were I am asked to state the values of x, their maximum and where the occur

    do i just set the other side to -1 and multiply out to solve?
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  2. #2
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    Hi

    a \sin x + b \cos x = \sqrt{a^2+b^2} \cdot \left(\frac{a}{\sqrt{a^2+b^2}} \sin x + \frac{b}{\sqrt{a^2+b^2}} \cos x \right)

    And since \left(\frac{a}{\sqrt{a^2+b^2}}\right)^2 + \left(\frac{b}{\sqrt{a^2+b^2}}\right)^2 = 1

    there exists \phi such that

    \frac{a}{\sqrt{a^2+b^2}} = \cos \phi

    \frac{b}{\sqrt{a^2+b^2}} = \sin \phi

    Therefore a \sin x + b \cos x = \sqrt{a^2+b^2} \cdot  \left(\sin x \cos \phi + \cos x \sin \phi \right) = \sqrt{a^2+b^2} \cdot \sin\left(x+\phi\right)

    For instance

    5 \sin x + 12 \cos x = 13  \sin\left(x+\phi\right)

    The minimum is -13 and the maximum is 13
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  3. #3
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    Thanks for the lengthy response
    Im cool with that but not where that 30 has come from at the end?
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  4. #4
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    Well the minimum of 5 \sin x + 12 \cos x is -13 and the maximum is 13

    Therefore
    the minimum of 5 \sin x + 12 \cos x +17 is -13+17=4 and the maximum is 13+17=30

    the minimum of \frac{1}{5 \sin x + 12 \cos x +17} is 1/30 and the maximum is 1/4

    the minimum of \frac{30}{5 \sin x + 12 \cos x +17} is 30/30 = 1 and the maximum is 30/4 = 7.5
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  5. #5
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    great! makes sense now
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