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Is this a valid approach to finding critical values of a trig function

  1. Jul 6, 2014 #1
    I was stuck for an hour trying to do this calculus 1 problem. Think I figured it out but it's a even problm.

    Find the absolute maximum and absolute minimum values of f on the given interval.

    f(t)=t+cot (t/2), [pie/4,7pie/4]

    f'=1-(1/2) csc^2 (t/2)

    So 1=1/2*csc^2 (t/2)

    2=csc^2 (t/2)

    For some reason I didn't know what to to do with cc squared so I applied an identity.

    +/_ 1=cot (t/2)

    Took arc cot on both side.

    Arccot (+-1) =t/2

    The arc cot gives me pie/4 and 3pie/4

    So now I multiply both by 2.

    So I get pie/2=t or 3pie/2=t.


    Then I just plug the values in f and solve.

    To see which ones are global max n min including the endpoints.

    Also my question was.


    Are we allowed to take the inverse cot on both sides of

    Arccot (1)=arccot (cot^2 (t/2)) ?

    If so how would we work with it?

    I was thinking arccot1=1 and on the right side. Arccot will cancel one cot so I'm left with cot(t/2)

    Now 1=cot (t/2)


    Then I use pie/4 +pie (k)=t/2 and solve for solutions in the restricted interval? Is this correct
     
  2. jcsd
  3. Jul 6, 2014 #2

    verty

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    Homework Helper

    At this step: 2=csc^2 (t/2), I see two ways to proceed.

    1) Use the identity for csc^2, remember that it is very similar to the sec^2 formula: csc^2 = 1 + cot^2. This gives 1 = cot^2(t/2).
    OR
    2) Take reciprocals. This gives 1/2 = sin^2(t/2).

    Then if you are careful to include the minus sign when taking square roots, you should get 4 candidate values for t/2.
     
  4. Jul 6, 2014 #3
    Aha. Thank you I over thought the problem.

    I forgot that say sin^2 (x)=(sin x)^2.

    Thanks verty.


    Not will say arcsin (sin ^2 (x)) that's the same as sin x? I think I need to relearn trig equations lol
     
  5. Jul 6, 2014 #4

    verty

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    Homework Helper

    Sorry, I didn't read the whole of your first post earlier. Arccot or arcsin won't work, the square messes them up.
     
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