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Derivative of a trigonometric function

  1. Oct 4, 2015 #1
    1. The problem statement, all variables and given/known data

    [itex]\frac{d}{dx}7.5sin(\frac{pi}{10}x) [/itex]

    3. The attempt at a solution


    Maximum: f'(x) = 0

    [itex]7.5(\frac{pi}{10})cos(\frac{pi}{10}x)[/itex] = 0

    [itex]7.5(\frac{pi}{10})cos^{-1}(0)= \frac{pi}{10}x[/itex]

    **[itex] (\frac{pi}{10}\frac{10}{pi})7.5(90) = x[/itex]

    [itex](1)(7.5)(90) = x = 675[/itex]

    To me, this doesn't seem to be nearly the correct answer because it doesn't make sense given the graph of this function:

    [itex]\frac{pi}{10}x= pi[/itex]

    x = 10

    So, the first arch is at x=0 and x = 10,

    so the the maximum of the curve can not be x = 675.

    What am I doing incorrectly in the derivative of the trigonometric function?

    Thank you
  2. jcsd
  3. Oct 4, 2015 #2
    Derivation part looks fine.
    Last edited: Oct 4, 2015
  4. Oct 4, 2015 #3


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    What, exactly, was the question? You only mention differentiating the function but then set that derivative to 0.
    Yes, this is the correct derivative.

    and so [itex]cos(\frac{\pi}{10}x)= 0[/itex]

    No. To solve [itex]Af(x)= B[/itex], you take [itex]f^{-1}(B/A)[/itex], not [itex]Af^{-1}(B)[/itex].

  5. Oct 4, 2015 #4
    Hi lep11,

    Thank you. I had a feeling I was running into the issue at solving for x.

    (1) [itex]7.5\frac{π}{10}cos(\frac{π}{10}x)=0[/itex]

    (2) [itex]7.5\frac{π}{10}cos^{-1}(0)= \frac{π}{10}x[/itex] Here is where I was getting stuck. The first thing I noticed, was that I let cos(0) = 90 instead of π/2.

    (3) [itex] 7.5\frac{π}{10}\frac{10}{π}cos(0) = x[/itex]

    (4) [itex]7.5\frac{π}{2} = x = 11.78[/itex] Which is closer to within the interval of the first arch of the function, but not quite where the maximum should be.

    Playing around with the algebra, I finally arrived at an answer that makes sense:

    If I divide the [itex]7.5\frac{π}{10}[/itex] out of the equation first at step (1):


    [itex]cos(\frac{π}{10}x) = 0[/itex]

    [itex]cos(0) = \frac{π}{10}x[/itex]

    [itex]\frac{10}{π}\frac{π}{2} = 5 [/itex] That sounds more like it.

    However, what I don't understand, is how come I have to divide [itex]7.5\frac{π}{10}[/itex] first? In a regular equation, it doesn't matter when you decide to divide both sides by a number, why does it matter in this situation?

    Thank you.
    Last edited: Oct 4, 2015
  6. Oct 4, 2015 #5

    Thank you HallsofIvy,

    [itex]7.5\frac{π }{10}Cos(\frac{π}{10}x) = 0[/itex]

    [itex]Cos(\frac{0}{(7.5\frac{π }{10})}) = \frac{π}{10}x[/itex]

    [itex]Cos(0) = \frac{π}{10}x[/itex]

    [itex]\frac{10}{π}Cos(0) = x[/itex]

    [itex]\frac{10}{π}Cos(0) = x[/itex]

    [itex]\frac{10}{π}\frac{π}{2} = x = 5[/itex]!!!

    Thank you. Now that I know how to solve this and that there is a systematic way to approach, I will be looking into why the formula provided does in fact work. Thank you for your guidance :)
  7. Oct 4, 2015 #6


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    The product of ##\ 7.5\frac{π }{10} \ ## and ##\ \cos(\frac{π}{10}x) \ ## is zero.

    The only way for a product to be zero is for one of the factors to be zero. The only one which can be zero is ##\ \cos(\frac{π}{10}x) \ ## .
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