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Math Help - Showing how pi/2+theta = cos using unit circle.

  1. #1
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    Showing how pi/2+theta = cos using unit circle.

    Hey all,

    Been stuck with my classwork lately, I can not visualize why pi/2+theta = cos theta. This is similar to my last post except i was able to understand it when its in the first quadrant.

    Thanks.
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  2. #2
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    \sin{\left(\frac{\pi}{2}+\theta\right)}=\cos{\thet  a}

    You mean this correct?
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    dwsmith, yes.
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    Quote Originally Posted by Oiler View Post
    I can not visualize why pi/2+theta = cos theta. Thanks.
    Draw \sin{\left(\frac{\pi}{2}+\theta\right)} and \cos{\theta} separately on the same set of axes.

    What do you notice?
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  5. #5
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    The angles are symmetrical at pi/2 ?

    \ | /
    \ | /
    __\|/___
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    Quote Originally Posted by Oiler View Post
    The angles are symmetrical at pi/2 ?

    \ | /
    \ | /
    __\|/___
    At \theta = \frac{\pi}{2} they are the same.

    Did you plot them together yet?
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    This is all i can imagine the angle to be xx - Slimber.com: Drawing and Painting Online
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    If you put the \theta between the vertical axis instead, then where you currently have \theta will be \pi-(\theta+\frac{\pi}2) and we know sin ({\pi-(\theta+\frac{\pi}2})) = sin( \theta+\frac{\pi}2) and by alternate angles, the other angle in the triangle will be \theta.
    see diagram

    cos \theta = \frac{y}1
    sin (\pi-(\theta+\frac{\pi}2}))=\frac{y}1
    hence sin (\theta+\frac{\pi}2)=cos \theta

    Attachment 20466
    Attached Thumbnails Attached Thumbnails Showing how pi/2+theta = cos using unit circle.-trigforum.png  
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  9. #9
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    Thanks jacs. i can clearly see this now.
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    This in theory should also hold true to cos(pi/2+theta)=sinx ?..
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    Quote Originally Posted by Oiler View Post
    This in theory should also hold true to cos(pi/2+theta)=sinx ?..
    Nope, it is \cos \left( \theta -\frac{\pi}{2}\right) = \sin \theta
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  12. #12
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    hmm for cos(pi/2+theta) i get cos(pi/2+theta) = sin(theta), I am guessing it should be -sin(theta)
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    Quote Originally Posted by Oiler View Post
    hmm for cos(pi/2+theta) i get cos(pi/2+theta) = sin(theta), I am guessing it should be -sin(theta)
    Looking at the unit circle, adding pi/2 to an angle in the first quadrant gives and angle in the second quadrant where the "x" coordinate is negative. You should also be able to see that the vertical and horizontal sides of the right triangles formed (|x| and |y|) are swapped. That is (x, y), in the first quadrant, with both x and y positive, is rotated to (-y, x).

    Since, on the unit circle, the first coordinate is [itex]cos(\theta)[/itex] and the second coordinate is [itex]sin(theta)[/itex], [itex]cos(\theta+ \pi/2)= -y= -sin(\theta)[/itex].
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