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Hi purplec16,
The horizon makes a 90 degree angle with the Earth's centre.
This is because the astronaut's line of sight makes a tangent with the planet
(if you balance a perfectly round ball on a table, it's centre is directly above the point of contact or tangency).
The triangle has the same sides and angles,
so the trigonometry works exactly as skeeter originally outlined.
$\displaystyle Sin(65.8^o)=\frac{opposite}{hypotenuse}=\frac{R}{R +380}$
$\displaystyle (R+380)Sin(65.8^o)=R$
$\displaystyle RSin(65.8^o)+380Sin(65.8^o)=R$
$\displaystyle 380Sin(65.8^o)=RRSin(65.8^o)=R[1Sin(65.8^o)]$
$\displaystyle R=\frac{380Sin(65.8^o)}{1Sin(65.8^o)}$

Sorry Skeeter, my mistake 380 miles is correct

Thank you so much, I get it now and I completely understand, Thank you Skeeter and Archie