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Thread: How do you convert from cos(pi/number)

  1. #1
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    How do you convert from cos(pi/number)

    Let's say I have cos(5pi/3). How do I know this equals 0.5 without using a calculator?
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  2. #2
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by jayshizwiz View Post
    Let's say I have cos(5pi/3). How do I know this equals 0.5 without using a calculator?

    draw your self circle ... with that angle ... and you'll see


    P.S. you should know by heart this

    $\displaystyle \displaystyle
    \begin{matrix}
    \alpha & 0 & \frac {\pi}{6} == 30 & \frac {\pi}{4} == 45 & \frac {\pi}{3} == 60 & \frac {\pi}{2} == 90 & \pi == 180 & \frac {3\cdot \pi}{2} == 270 &2\pi == 360 \\
    \sin{\alpha} &0 & \frac {1}{2} & \frac {\sqrt{2}}{2} & \frac {\sqrt{3}}{2} & 1 &0 & -1& 0\\
    \cos{\alpha} & 1 & \frac {\sqrt{3}}{2} & \frac {\sqrt{2}}{2} & \frac {1}{2} & 0 &-1 &0 &1 \\
    \tan{\alpha}& 0& \frac {\sqrt{3}}{3} & 1 & \sqrt{3} & \pm \infty & 0 & \mp \infty & 0\\
    \csc{\alpha} & \pm \infty & \sqrt{3} & 1 & \frac {\sqrt{3}}{3} & 0 & \mp \infty &0 & \pm \infty
    \end{matrix}
    $



    sorry for bad table ... lol forgot how to draw table
    Last edited by yeKciM; Sep 15th 2010 at 09:55 AM.
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    thanks but hopefully someone has a better explanation (:

    and you wrote pi/6 twice...

    I know how to do it I guess by converting it from radian to degrees: cos(5pi/3) = cos300=cos60=0.5

    but isn't there a way to measure it by radian?
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  4. #4
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by jayshizwiz View Post
    thanks but hopefully someone has a better explanation (:

    and you wrote pi/6 twice...

    I know how to do it I guess by converting it from radian to degrees: cos(5pi/3) = cos300=cos60=0.5

    but isn't there a way to measure it by radian?
    hehehehe... while pasting forgot to change that pi/3

    but if you draw your self that angle ... you'll realize that angle of $\displaystyle \displaystyle \frac {5\pi}{3} $ is the same as the angle of $\displaystyle \displaystyle -\frac{\pi}{3} $ and you know that angle that is $\displaystyle \displaystyle \cos {-\frac {\pi}{3} } = \frac {1}{2} $
    because cos is positive in first and 4th quadrant get it ?

    when drawn circle, draw $\displaystyle \frac {\pi}{3} $ and add it 5 times and you'll see that is the angle $\displaystyle -\frac {\pi}{3} $




    P.S. you realize that $\displaystyle \displaystyle \frac {6\pi}{3} == 2\pi $ so $\displaystyle \displaystyle \frac {5\pi}{3} $ is just for the $\displaystyle \frac {\pi}{3} $ smaller than $\displaystyle 2\pi$ so it have to be $\displaystyle -\frac {\pi}{3} $
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    MHF Contributor undefined's Avatar
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    Quote Originally Posted by jayshizwiz View Post
    Let's say I have cos(5pi/3). How do I know this equals 0.5 without using a calculator?
    yeKciM's answer is good, also you can use identities, for example cos(5pi/3) = cos(-5pi/3) = cos(-5pi/3 + 2pi) = cos(pi/3), then you can draw a 30-60-90 triangle if you don't have it memorised.
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    MHF Contributor ebaines's Avatar
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    Quote Originally Posted by jayshizwiz View Post
    I know how to do it I guess by converting it from radian to degrees: cos(5pi/3) = cos300=cos60=0.5

    but isn't there a way to measure it by radian?
    The key concept her is to realize that 1 radian is the angle that is covered if you wrap a string of length 1 x radius around the perimeter of a circle. By convention we usually just consider a circle of radius 1, which would have a total perimeter length 2 pi. Hence a string of length 2 pi would wrap precisely once around this unit circle, and so the angle 2 pi radians is eqivalent to 360 degrees. Similarly, pi/2 radians is 180 degreees, pi/3 radians = 120 degrees, etc. Now, when first learning trig there is a natural tendency for some students to resist thinking in radians, and try to convert everything to degrees in their head. But trust me - you'd be much beter off getting to a point where you can "visualize" what pi/4 radians means without first thinking about degrees. Just consider that each quadrant of the circle is pi/2 radians, and so 2/3 pi is in the second quadrant, 5/3 pi is in the fourth, etc. So get used to visualizing how it is that tan(pi/4) = 1, and you'll be well on your way.
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  7. #7
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    Quote Originally Posted by jayshizwiz View Post
    Let's say I have cos(5pi/3). How do I know this equals 0.5 without using a calculator ?
    Without using a calculator...

    $\displaystyle \displaystyle\frac{5{\pi}}{3}=\frac{5(180^o)}{3}=3 00^o$

    That's $\displaystyle -60^o$

    Hence, if you draw a unit-circle centred at the origin, you can draw a right-angled triangle
    with hypotenuse going off at 60 degrees to the x-axis,
    since Cos(angle) gives the x co-ordinate, and Cos(-A)=CosA.

    This allows us to draw a regular hexagon, made of equilateral triangles as shown in the attachment.

    From that, we can see that $\displaystyle Cos60^o=0.5$
    Attached Thumbnails Attached Thumbnails How do you convert from cos(pi/number)-hexagon.jpg  
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