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Homework Help: Finding critical numbers of trig function

  1. May 2, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the critical numbers of the function:

    f(θ) = 2cosθ + sin^(2) θ

    2. Relevant equations

    None

    3. The attempt at a solution

    I differentiated the equation and got -2sinθ(1 - cosθ) and found the critical values to be θ = 0 degrees + 2pie * n but the correct answer was npie. Why is this?
     
  2. jcsd
  3. May 2, 2010 #2
    When is -2sinθ(1 - cosθ) equal to zero according to you? (check again I mean)
     
  4. May 8, 2010 #3
    cosθ = 1 when x = 0 + 2npie, n E Z
    -2sinθ = 0 when x = 0 + 2npie, n E Z
     
  5. May 8, 2010 #4

    Mark44

    Staff: Mentor

    Edit: Corrected an error I made.
    cosθ = 1 for x = n*2pi
    sinθ = 0 for x = ..., -2pi, - pi, 0, pi, 2pi, 3pi, ...
     
    Last edited: May 9, 2010
  6. May 9, 2010 #5
    No Mark! cosθ = 1 for θ = n*2pi, so you are correct TsAmE.
    but you indeed made an error with the sinus:

    sin(x)= 0 for x = n*pi
     
  7. May 9, 2010 #6
    Why does sinx have the same value every pie (180 degrees)? Doesnt it only repeat every 2pie since it is its period?
     
  8. May 9, 2010 #7

    Mark44

    Staff: Mentor

    Right. I don't know what I was thinking. I have edited my earlier reply.
     
  9. May 9, 2010 #8

    Mark44

    Staff: Mentor

    The graphs of y = cosx and y = sinx are periodic with period 2pi, but both cross the horizontal axis at multiples of pi. But that doesn't mean that the period of each is pi. For periodicity, f(x + p) = f(x) for all x, not just a select few values.
     
  10. May 11, 2010 #9
    What do you mean by f(x + p) = f(x) and what is your f(x). What if my critical number was t = pie/6 OR t = -pie/2, would the critical values be at pie/6 + npie OR t = -pie/2 + npie?
     
    Last edited: May 11, 2010
  11. May 11, 2010 #10

    Mark44

    Staff: Mentor

    That's the definition of periodicity. If a function f satisfies f(x + p) = f(x) for all x, f is periodic with period p.

    The cosine and sine functions are periodic with period 2pi. BTW, pie is something you eat. Pi is the name of the Greek letter [itex]\pi[/itex]. If you want to appear intelligent, don't write pie when you mean pi.

    If your critical number was t = pi/6 for the cosine or sine function, then pi/6 + n(2pi) would also be a critical number.
     
  12. May 11, 2010 #11
    Lol I didnt even notice that I wrote pie. So the only time that you add "n(pi)" to the period is when sinx or cosx = 0, and for any other values you would instead add "2n(pi)"?
     
  13. May 11, 2010 #12

    Mark44

    Staff: Mentor

    Right.
     
  14. May 12, 2010 #13
    Are there any other special cases where you wouldnt say "+ 2n(pi)"?
     
  15. May 12, 2010 #14

    Mark44

    Staff: Mentor

    No, not for the sine and cosine functions.
     
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