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Long story short what I don't understand is underlined here in red: enter image description here

So, they somehow seem to assume the angle on the triangle on the right has an angle $\frac{\theta}{2}$.

How do they know that? How can they assume it is exactly half of the angle of theta?

Thanks

EDIT: to give a bit more of a context, it has to do with the mapping of a rocket's position in the ground-fixed coordinates to the spherical earth's coordinates. When going from one coordinate to the other the altitude varies.

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  • $\begingroup$ How was the triangle on the left constructed? $\endgroup$
    – Mike
    Jan 5, 2019 at 16:22
  • $\begingroup$ Is this something related to physics? $\endgroup$
    – cqfd
    Jan 5, 2019 at 16:24
  • $\begingroup$ @ThomasShelby yes this is related to physics $\endgroup$
    – traducerad
    Jan 5, 2019 at 16:32

1 Answer 1

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There is an isoscales triangle between the circle center $C$ and the 2 points $A$ and $B$ where the lines through the center meet the circle. The angle at $C$ is $\theta$, so each base angle is $90^\circ-\frac\theta2 = \angle CAB$. Since the tangent has a right angle with the radius ($\angle CAD=90^\circ$), the angle under consideration ($\angle BAD)$ is $\frac\theta2$.

enter image description here

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  • $\begingroup$ If you could provide a small ugly drawing to clarify what you mean, I'll accept your answer $\endgroup$
    – traducerad
    Jan 5, 2019 at 16:27
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    $\begingroup$ @traducerad Added the picture $\endgroup$
    – Ingix
    Jan 5, 2019 at 16:41
  • $\begingroup$ @ThomasShelby You think the assumption that (in my notation) $CA$ is a radius and $AD$ a tangent is strange? $\endgroup$
    – Ingix
    Jan 5, 2019 at 16:43
  • $\begingroup$ Great answer @Ingix $\endgroup$
    – Mike
    Jan 5, 2019 at 16:46
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    $\begingroup$ Great question @traducerad btw! $\endgroup$
    – Mike
    Jan 5, 2019 at 16:54

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