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Math Help - Finding the Angle in this Figure (see the figure)

  1. #1
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    Finding the Angle in this Figure (see the figure)

    I am trying to figure out the measure of the angle labeled "?" in the figure. All the labeled variables besides "?" are known. I have been staring at this for the past 2 hours and haven't found anything leading to a solution. As this figure is derived from a separate problem (approximation of the geometric apparent radius of Earth's penumbra at a given distance -- ignore if this makes things confusing), I may have forgotten a known value somewhere though it seems like there is only one unique figure for a given set of variables. Any ideas?

    a, b and "?" are angles
    r and d are distances

    Thanks.
    Attached Thumbnails Attached Thumbnails Finding the Angle in this Figure (see the figure)-penumbra.png  
    Last edited by North; December 19th 2011 at 02:33 PM.
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  2. #2
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    Re: Finding the Angle in this Figure (see the figure)

    I actually just figured it out. The right triangle containing angle a was not needed. That left a trapezoid remaining. Creating a right triangle from the top of the trapezoid and using the tangent function allowed me to find the longer height of the trapezoid. Then, it just took a simple tangent inverse to find the result.

    There, ? = atan(tan(90 - b) + r/d) where tan() uses degrees.
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