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Math Help - Minimum value of trigonometric expression?

  1. #1
    Super Member fardeen_gen's Avatar
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    Minimum value of trigonometric expression?

    The minimum value of 27^cos x + 81^sin x is?
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  2. #2
    Super Member fardeen_gen's Avatar
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    Does using AM-GM inequality help?
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  3. #3
    Moo
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    Hello,
    Quote Originally Posted by fardeen_gen View Post
    Does using AM-GM inequality help?
    Yes !

    27=3^3 and 81=3^4

    Therefore 27^{\cos(x)}+81^{\sin(x)}=3^{3 \cos(x)}+3^{4 \sin(x)}

    By AM-GM :
    3^{3 \cos(x)}+3^{4 \sin(x)} \ge 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}}

    So you just have to find the minimum of 3 \cos(x)+4 \sin(x)
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by fardeen_gen View Post
    The minimum value of 27^cos x + 81^sin x is?
    This is periodic with perion 2 \pi , so sketch the curve. This has a minimum near x=4. The minimum is a root of:


    f(x)=\ln(81) \cos(x) e^{\ln(81) \sin(x)}-\ln(27) \sin(x) e^{\ln(27) \cos(x)}

    Running the bisection method over this gives the minimum occurs at x \approx 3.86147, where the function is very close to 0.141494 .


    RonL
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by Moo View Post
    Hello,

    Yes !

    27=3^3 and 81=3^4

    Therefore 27^{\cos(x)}+81^{\sin(x)}=3^{3 \cos(x)}+3^{4 \sin(x)}

    By AM-GM :
    3^{3 \cos(x)}+3^{4 \sin(x)} \ge 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}}

    So you just have to find the minimum of 3 \cos(x)+4 \sin(x)
    And how exactly do you show that the minimum of a lower bound is equal to the minimum of the function?

    Note a numerical solution to the first problem is not equal to the function value at a minimum of 3 \cos(x)+4 \sin(x)


    RonL
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  6. #6
    Super Member fardeen_gen's Avatar
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    @Moo: Minimum of 3 cos x + 4 sin x = - 5. But that means putting a negative quantity under root!

    @CaptainBlack: The answer is 1/4. Doesn't match.

    Still in the dark about this one.
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  7. #7
    Moo
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    Quote Originally Posted by CaptainBlack View Post
    And how exactly do you show that the minimum of a lower bound is equal to the minimum of the function?

    Note a numerical solution to the first problem is not equal to the function value at a minimum of 3 \cos(x)+4 \sin(x)


    RonL
    Then we can say that the minimum of the function will always be superior to the minimum of 3cos(x)+4sin(x)
    But with the large inequality, I don't know, maybe it can be equal.

    @Moo: Minimum of 3 cos x + 4 sin x = - 5.
    No, because it's 3 to the power -5.

    The answer is 1/4.
    Gotta think about that then
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  8. #8
    Grand Panjandrum
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    Quote Originally Posted by fardeen_gen View Post
    @Moo: Minimum of 3 cos x + 4 sin x = - 5. But that means putting a negative quantity under root!

    @CaptainBlack: The answer is 1/4. Doesn't match.

    Still in the dark about this one.

    Code:
    >27^(cos(4))+81^(sin(4))
          0.15193 
    >
    There comes a point where numerical calculation over rules the answer at the back of the book.

    RonL
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  9. #9
    Super Member fardeen_gen's Avatar
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    If a cos x + b sin x = c, then - sqrt(a^2 + b^2) <= a cos x + b sin x <= + sqrt (a^2 + b^2). Therefore, for 3 cos x + 4 sin x, minimum seems to be -5.?? I am muddled up!

    EDIT: My bad!!!! Its 3^(3 cos x + 4 sin x)
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  10. #10
    Moo
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    It's ok, I figured it out !

    There is equality in AM GM if 3^{3 \cos(x)}=3^{4 \sin(x)}

    So let m be the minimum value of 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}}, that is \frac 23 \sqrt{\frac 13} (substituting -5).

    3^{3 \cos(x)}+3^{4 \sin(x)} \ge 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}} \ge m

    If 3^{3 \cos(x)}=3^{4 \sin(x)}, then 3^{3 \cos(x)}+3^{4 \sin(x)}=2 \sqrt{3^{3 \cos(x)+4 \sin(x)}} \ge m

    Thus m is the minimum, in particular when 3^{3 \cos(x)}=3^{4 \sin(x)}


    Sounds more correct ?


    Edit : I have to work it out deeper, because I assumed that 3cos(x)=4sin(x)
    Last edited by Moo; September 6th 2008 at 10:20 AM.
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  11. #11
    Grand Panjandrum
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    Quote Originally Posted by Moo View Post
    It's ok, I figured it out !

    There is equality in AM GM if 3^{3 \cos(x)}=3^{4 \sin(x)}

    So let m be the minimum value of 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}}, that is \frac 23 \sqrt{\frac 13} (substituting -5).

    3^{3 \cos(x)}+3^{4 \sin(x)} \ge 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}} \ge m

    If 3^{3 \cos(x)}=3^{4 \sin(x)}, then 3^{3 \cos(x)}+3^{4 \sin(x)}=2 \sqrt{3^{3 \cos(x)+4 \sin(x)}} \ge m

    Thus m is the minimum, in particular when 3^{3 \cos(x)}=3^{4 \sin(x)}


    Sounds more correct ?


    Edit : I have to work it out deeper, because I assumed that 3cos(x)=4sin(x)
    Well it is true that the minimum of 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}} is less than the minimum of 27^{\cos (x)} + 81^{\sin (x)}, but the minimums do not occur at the same x, so what is this telling us?


    RonL
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