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Thread: [SOLVED] Determing absolute minimum and maximum values

  1. #1
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    [SOLVED] Determing absolute minimum and maximum values

    $\displaystyle f(x)=sin(x)+cos(x)$ on the integral $\displaystyle [0, \frac{\pi}{3}]$
    $\displaystyle f'(x) = cos(x) - sin(x) = 0$
    $\displaystyle \frac{sin(x)}{cos(x)}=tan(x)=1=\frac{\pi}{4}$

    $\displaystyle 0, (\frac{\pi}{4}), (\frac{\pi}{3})$

    $\displaystyle f(0)=sin(1)+cos(1)=0+1=1$
    But how do I find out these two calculations below without using a calculator?
    $\displaystyle f(\frac{\pi}{4}) = sin(\frac{\pi}{4}) + cos(\frac{\pi}{4})=\sqrt{2} = 1.44$
    and
    $\displaystyle f(\frac{\pi}{3}) = sin(\frac{\pi}{3}) + cos(\frac{\pi}{3}) = \sqrt{3} + 1 / 2 = 1.37$
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    Quote Originally Posted by !!! View Post
    $\displaystyle f(x)=sin(x)+cos(x)$ on the integral $\displaystyle [0, \frac{\pi}{3}]$
    $\displaystyle f'(x) = cos(x) - sin(x) = 0$
    $\displaystyle \frac{sin(x)}{cos(x)}=tan(x)=1=\frac{\pi}{4}$

    $\displaystyle 0, (\frac{\pi}{4}), (\frac{\pi}{3})$

    $\displaystyle f(0)=sin(1)+cos(1)=0+1=1$
    But how do I find out these two calculations below without using a calculator?
    $\displaystyle f(\frac{\pi}{4}) = sin(\frac{\pi}{4}) + cos(\frac{\pi}{4})=\sqrt{2} = 1.44$
    and
    $\displaystyle f(\frac{\pi}{3}) = sin(\frac{\pi}{3}) + cos(\frac{\pi}{3}) = \sqrt{3} + 1 / 2 = 1.37$
    The points you have to check to know which are largest and smallest are: $\displaystyle 1,\sqrt{2}, (\sqrt{3}+1)/2$. To determine which is largest perform the same operations on this triple. First multiple each by 2: $\displaystyle 2,2\sqrt{2}, \sqrt{3}+1$. Now square each one: $\displaystyle 4,8,4+\sqrt{3}$. Subtract 4 from each one: $\displaystyle 0,4,\sqrt{3}$. And square again: $\displaystyle 0,16,3$. We see that that 1st point is the smallest, 3rd point is in between, and 2nd point is largest.
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    Ahh, thank you.
    But how do I determine $\displaystyle \sqrt{2}$ from $\displaystyle sin(\frac{\pi}{4}) + cos(\frac{\pi}{4})$ and $\displaystyle \sqrt{3} + 1 / 2$ from $\displaystyle sin(\frac{\pi}{3}) + cos(\frac{\pi}{3})$?
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    Quote Originally Posted by !!! View Post
    Ahh, thank you.
    But how do I determine $\displaystyle \sqrt{2}$ from $\displaystyle sin(\frac{\pi}{4}) + cos(\frac{\pi}{4})$ and $\displaystyle \sqrt{3} + 1 / 2$ from $\displaystyle sin(\frac{\pi}{3}) + cos(\frac{\pi}{3})$?
    Because (you are expected to know) $\displaystyle \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}2$. And $\displaystyle \cos \frac{\pi}{3} = \frac12$ and $\displaystyle \sin \frac{\pi}{3} = \frac{\sqrt{3}}2$
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    Re: [SOLVED] Determing absolute minimum and maximum values

    Quote Originally Posted by ThePerfectHacker View Post
    Because (you are expected to know) $\displaystyle \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}2$. And $\displaystyle \cos \frac{\pi}{3} = \frac12$ and $\displaystyle \sin \frac{\pi}{3} = \frac{\sqrt{3}}2$
    Construct a right triangle with legs each of length 1. Since this is an isosceles triangle, the two acute angles are congruent. Since they must add to $\displaystyle \pi/2$, they are each $\displaystyle \pi/4$. The hypotenuse has, by the Pythagorean theorem, length $\displaystyle \sqrt{2}$ so that $\displaystyle sin(\pi/4)$ equals "opposite side over hypotenuse" which is $\displaystyle \frac{1}{\sqrt{2}}= \frac{\sqrt{2}}{2}$ and the same for the cosine.

    For $\displaystyle \pi/6$ and $\displaystyle \pi/3$ construct an equilateral triangle (all three side of equal length so all three angles congruent so each angle $\displaystyle \frac{\pi}{3}$) with sides of length 2 and draw a line from one vertex perpendicular to the opposites side. By symmetry (that line divides the equilateral triangle into two congruent right triangles) we have a right triangle with one angle $\displaystyle (\pi/3)/2= \pi/6$, another angle $\displaystyle \pi/3$, hypotenuse of length 2 and side opposite the [tex]\pi/6[itex] angle of length 1. By the Pythagorean theorem the other leg (opposite the $\displaystyle \pi/3$ angle and the perpendicular constructed) has length $\displaystyle \sqrt{4- 1}= \sqrt{3}$.

    From that, we have $\displaystyle sin(\pi/6)= \frac{1}{2}$, $\displaystyle cos(\pi/6)= \frac{\sqrt{3}}{2}$ and $\displaystyle sin(\pi/3)= \frac{\sqrt{3}}{2}$, $\displaystyle cos(\pi/3)= \frac{1}{2}$.
    Last edited by HallsofIvy; Aug 7th 2014 at 05:06 AM.
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