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Generalized Trigonometric Intervals

  1. May 25, 2015 #1
    1. The problem statement, all variables and given/known data

    Identify the intervals of increase/decrease of ##f(x) = \sin x + \cos x##

    2. Relevant equations

    ##f(x) = \sin x + \cos x##

    ##f'(x) = \cos x - \sin x = \sqrt 2 \cos(x+\frac {\pi}{4})##

    3. The attempt at a solution

    ##f## is increasing when ##f'(x) > 0##

    ##\sqrt 2 \cos(x+\frac {\pi}{4}) > 0 \to \cos(x + \frac {\pi}{4}) > 0##

    So, ##f## is increasing on the intervals ##(0, \frac {\pi}{4})## and ##(\frac {5 \pi}{4}, 2\pi)##, and is decreasing on the interval ##(\frac {\pi}{4}, \frac {5 \pi}{4})##.

    I know that in order to generalize this, you would add ##2 \pi n## to all intervals. I simply would like to know how this would be mathematically written. Thanks!
     
  2. jcsd
  3. May 25, 2015 #2

    micromass

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    Like this: ##f## is decreasing on all intervals of the form ##(\frac{\pi}{4} + 2\pi n, \frac{5\pi}{4} + 2\pi n)## for each ##n\in \mathbb{Z}##.

    A more compact notation is available from set theory as

    [tex]\bigcup_{n\in \mathbb{Z}} (\frac{\pi}{4} + 2\pi n, \frac{5\pi}{4} + 2\pi n)[/tex]
     
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