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.

$\displaystyle Sin(65.8^o)=\frac{opposite}{hypotenuse}=\frac{R}{R +380}$

$\displaystyle (R+380)Sin(65.8^o)=R$

$\displaystyle RSin(65.8^o)+380Sin(65.8^o)=R$

$\displaystyle 380Sin(65.8^o)=R-RSin(65.8^o)=R[1-Sin(65.8^o)]$

$\displaystyle R=\frac{380Sin(65.8^o)}{1-Sin(65.8^o)}$