# How can people assume this angle is exactly half of the other angle?

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

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.

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

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 $$\frac\theta2$$.
• @ThomasShelby You think the assumption that (in my notation) $CA$ is a radius and $AD$ a tangent is strange? Jan 5, 2019 at 16:43