Sinüs teoremi

Sinüs teoremi

Sinüs teoremi, bir çembersel üçgende (kirişler üçgeni) bir kenar ve bu kenar karşısındaki açının sinüsleri oranı sabittir. Sinüs, dik açılı üçgenlerde dik olmayan bir açının karşısında kalan dik kenar ile hipotenüsün (dik açının karşısında kalan kenar) birbirine oranıdır.

, ve üçgenin kenar uzunlukları; , ve üçgenin iç açıları ve çevrel çemberin yarıçapı ise bunlar arasında sinüs teoremine göre aşağıdaki bağıntı mevcuttur:

İspatı

Sinüs Teoremi ispatında kullanılan şekil
  1. ABC üçgeninin çevrel çemberinin merkezi ve yarıçapı olsun. ve yarıçapları çizildiğinde aynı yayı gören çevre ve merkez açılardan dolayı olur.
  2. merkezinden kenarına noktasında yükseklik inildiğinde ikizkenar üçgen olduğundan yükseklik hem kenarortay hem de açıortay olur. O zaman üçgeni bir açısı derece olan dik üçgen olur. uzunluğu ise 'dir.
  3. Sinüsün tanımı gereği,
Bu işlem düzenlendiğinde
bulunur.

Aynı işlem diğer kenarlar için de yapıldığında sinüs teoremi bulunmuş olur.

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