Z powodu braku umiejętności obsługiwania programów typu geogebra rysunek wygląda jak wygląda...
Przyjmijmy:
\(\displaystyle{ \alpha := \frac{\alpha}{2}}\) i
\(\displaystyle{ \beta := \frac{\pi}{2} - \frac{\alpha}{2}}\).
Na rysunku widać przekrój kuli i stożków.
\(\displaystyle{ \frac{a}{h} = \tg \frac{\alpha}{2} \Rightarrow h = \frac{a}{\tg \frac{\alpha}{2}} \\ \frac{a}{H} = \tg \left( \frac{\pi}{2} - \frac{\alpha}{2}\right) \Rightarrow H = \frac{a}{\tg \left( \frac{\pi}{2} - \frac{\alpha}{2}\right)} = \frac{a}{\ctg \frac{\alpha}{2} }}\)
W zależności czy badamy:
a)
\(\displaystyle{ \frac{H}{h} = \frac{\tg \frac{\alpha}{2}}{\ctg \frac{\alpha}{2}} = \tg^2 \frac{\alpha}{2}}\)
b)
\(\displaystyle{ \frac{h}{H} = \frac{\ctg \frac{\alpha}{2}}{\tg \frac{\alpha}{2}} = \ctg^2 \frac{\alpha}{2}}\)
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