Ii
MATU • Trigonometria
MATU_TRI_488
Práctica de Identidades
Enunciado
Si $A + B + C = \pi$, demostrar que:
$$ \sin^2 A + \sin^2 B + \sin^2 C = 2(1 + \cos A \cos B \cos C) $$
$$ \sin^2 A + \sin^2 B + \sin^2 C = 2(1 + \cos A \cos B \cos C) $$
Solución Paso a Paso
1. Desarrollo:
Usamos $\sin^2 x = 1 - \cos^2 x$:
$$ E = (1 - \cos^2 A) + (1 - \cos^2 B) + (1 - \cos^2 C) $$
$$ E = 3 - (\cos^2 A + \cos^2 B + \cos^2 C) $$
Del ejercicio 006, sabemos que $\cos^2 A + \cos^2 B + \cos^2 C = 1 - 2 \cos A \cos B \cos C$:
$$ E = 3 - (1 - 2 \cos A \cos B \cos C) $$
$$ E = 2 + 2 \cos A \cos B \cos C $$
$$ \boxed{\sin^2 A + \sin^2 B + \sin^2 C = 2(1 + \cos A \cos B \cos C)} $$
Usamos $\sin^2 x = 1 - \cos^2 x$:
$$ E = (1 - \cos^2 A) + (1 - \cos^2 B) + (1 - \cos^2 C) $$
$$ E = 3 - (\cos^2 A + \cos^2 B + \cos^2 C) $$
Del ejercicio 006, sabemos que $\cos^2 A + \cos^2 B + \cos^2 C = 1 - 2 \cos A \cos B \cos C$:
$$ E = 3 - (1 - 2 \cos A \cos B \cos C) $$
$$ E = 2 + 2 \cos A \cos B \cos C $$
$$ \boxed{\sin^2 A + \sin^2 B + \sin^2 C = 2(1 + \cos A \cos B \cos C)} $$