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Thread: Complementary Angle Theorem

  1. #1
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    Complementary Angle Theorem

    There is an effect of the complementary angle theorom that I don't understand. For example:

    sin squared(40) + sin squared(50) = sin squared(40) + cos squared(40) = 1

    This previous statement's purpose was to show how sin(50) = cos(40), but what got me was the answer being equal to 1. Doing some experimenting, to see if this is a rule:

    sin squared(10) + sin squared(80) = sin squared(10) + cos squared(10) = 1

    and so on:

    cos squared(10) + cos squared(80) = 1
    sin squared(30) + sin squared(60) = 1

    Why is it, when these functions are squared, that the answer = 1?
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  2. #2
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    Quote Originally Posted by spiritualfields
    There is an effect of the complementary angle theorom that I don't understand. For example:

    sin squared(40) + sin squared(50) = sin squared(40) + cos squared(40) = 1

    This previous statement's purpose was to show how sin(50) = cos(40), but what got me was the answer being equal to 1. Doing some experimenting, to see if this is a rule:

    sin squared(10) + sin squared(80) = sin squared(10) + cos squared(10) = 1

    and so on:

    cos squared(10) + cos squared(80) = 1
    sin squared(30) + sin squared(60) = 1

    Why is it, when these functions are squared, that the answer = 1?
    Pythagoras' theorem in terms of trig functions is:

    $\displaystyle
    \sin^2(x)+\cos^2(x)=1
    $

    RonL
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  3. #3
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    Okay, I see it now. Thanks.
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  4. #4
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    Quote Originally Posted by spiritualfields
    There is an effect of the complementary angle theorom that I don't understand. For example:

    sin squared(40) + sin squared(50) = sin squared(40) + cos squared(40) = 1
    ...
    Hello,

    I've attached a diagram to show you what I've calculated.

    If the sum of the 2 angles is 90 you are dealing with a right triangle.

    $\displaystyle \alpha+\beta=90^\circ \Longrightarrow \beta=90^\circ-\alpha$

    The red line corresponds to $\displaystyle \sin(\alpha)$ and the blue line corrsponds to $\displaystyle \sin(\beta)=sin(90^\circ-\alpha)$

    According to the diagram is: $\displaystyle (\mbox{red})^2+(\mbox{blue})^2=1$

    Plug in the sine values and you'll get the property you have detected.

    The blue line corresponds to $\displaystyle \cos(\alpha)$. So now you can easily complete the Pythagoran rule of the Sine and Cosine function.

    Greetings

    EB
    Attached Thumbnails Attached Thumbnails Complementary Angle Theorem-pyth_theorem1.gif  
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  5. #5
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    spiritualfields

    earboth, thanks. What program did you use to draw that?
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  6. #6
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    Quote Originally Posted by spiritualfields
    earboth, thanks. What program did you use to draw that?
    Hello,

    have a look here: http://www.dynageo.de

    Greetings

    EB
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