Pertanyaan

Diketahui A , B , dan C merupakan sudut-sudut dalam sebuah segitiga. Buktikan bahwa: sin 2 ( A ) + sin 2 ( B ) + sin 2 ( C ) − 2 cos ( A ) cos ( B ) cos ( C ) = 2

Diketahui $A, B, dan C$ merupakan sudut-sudut dalam sebuah segitiga. Buktikan bahwa:

$sin^{2} (A) + sin^{2} (B) + sin^{2} (C) - 2 cos (A) cos (B) cos (C) = 2$

H. Firmansyah

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Jawaban

terbukti bahwa sin 2 ( A ) + sin 2 ( B ) + sin 2 ( C ) − 2 cos ( A ) cos ( B ) cos ( C ) = 2

terbukti bahwa

sin^{2} (A) + sin^{2} (B) + sin^{2} (C) - 2 cos (A) cos (B) cos (C) = 2

Pembahasan

Diketahui A , B , dan C merupakan sudut-sudut dalam sebuah segitiga. Akan dibuktikan sin 2 ( A ) + sin 2 ( B ) + sin 2 ( C ) − 2 cos ( A ) cos ( B ) cos ( C ) = 2 Ingat bahwa cos ( 2 A ) = 1 − 2 sin 2 A ⇒ sin 2 A = 2 1 − cos ( 2 A ) cos ( A − B ) = cos A cos B + sin A sin B cos ( 18 0 ∘ − A ) = − cos A cos A + cos B = 2 cos ( 2 A + B ) cos ( 2 A − B ) Diperhatikan = = = = = = = = = = = = = = = sin 2 ( A ) + sin 2 ( B ) + sin 2 ( C ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 1 − c o s ( 2 A ) + 2 1 − c o s ( 2 A ) + 2 1 − c o s ( 2 A ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 1 − 2 c o s ( 2 A ) + 2 1 − 2 c o s ( 2 B ) + 2 1 − 2 c o s ( 2 C ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 3 − 2 1 ( cos ( 2 A ) + cos ( 2 B ) + cos ( 2 C ) ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 3 − 2 1 ( 2 cos ( A + B ) cos ( A − B ) + cos ( 2 C ) ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 3 − 2 1 ( 2 cos ( π − C ) cos ( A − B ) + cos ( 2 C ) ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 3 − 2 1 ( − 2 cos ( C ) cos ( A − B ) + 2 cos 2 ( C ) − 1 ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 3 + cos ( C ) cos ( A − B ) − cos 2 ( C ) + 2 1 − 2 cos ( A ) cos ( B ) cos ( C ) 2 + cos ( C ) ( cos ( A − B ) − cos ( C ) ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 + cos ( C ) ( cos ( A − B ) − cos ( π − ( A + B ) ) ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 + cos ( C ) ( cos ( A − B ) + cos (( A + B ) ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 + cos ( C ) ( cos A cos B + sin A sin B + cos A cos B − sin A sin B ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 + cos ( C ) ( cos A cos B + cos A cos B ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 + cos ( C ) ( 2 cos A cos B ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 + 2 cos ( A ) cos ( B ) cos ( C ) − 2 cos ( A ) cos ( B ) cos ( C ) 2 Dengan demikian, terbukti bahwa sin 2 ( A ) + sin 2 ( B ) + sin 2 ( C ) − 2 cos ( A ) cos ( B ) cos ( C ) = 2

Diketahui $A, B, dan C$ merupakan sudut-sudut dalam sebuah segitiga. Akan dibuktikan

$sin^{2} (A) + sin^{2} (B) + sin^{2} (C) - 2 cos (A) cos (B) cos (C) = 2$

Ingat bahwa

$cos (2 A) = 1 - 2 sin^{2} A \Rightarrow sin^{2} A = \frac{1 - cos ( 2 A )}{2} cos (A - B) = cos A cos B + sin A sin B cos (18 0^{\circ} - A) = - cos A cos A + cos B = 2 cos (\frac{A + B}{2}) cos (\frac{A - B}{2})$

Diperhatikan

$= = = = = = = = = = = = = = = sin^{2} (A) + sin^{2} (B) + sin^{2} (C) - 2 cos (A) cos (B) cos (C) \frac{1 - c o s ( 2 A )}{2} + \frac{1 - c o s ( 2 A )}{2} + \frac{1 - c o s ( 2 A )}{2} - 2 cos (A) cos (B) cos (C) \frac{1}{2} - \frac{c o s ( 2 A )}{2} + \frac{1}{2} - \frac{c o s ( 2 B )}{2} + \frac{1}{2} - \frac{c o s ( 2 C )}{2} - 2 cos (A) cos (B) cos (C) \frac{3}{2} - \frac{1}{2} (cos (2 A) + cos (2 B) + cos (2 C)) - 2 cos (A) cos (B) cos (C) \frac{3}{2} - \frac{1}{2} (2 cos (A + B) cos (A - B) + cos (2 C)) - 2 cos (A) cos (B) cos (C) \frac{3}{2} - \frac{1}{2} (2 cos (π - C) cos (A - B) + cos (2 C)) - 2 cos (A) cos (B) cos (C) \frac{3}{2} - \frac{1}{2} (- 2 cos (C) cos (A - B) + 2 cos^{2} (C) - 1) - 2 cos (A) cos (B) cos (C) \frac{3}{2} + cos (C) cos (A - B) - cos^{2} (C) + \frac{1}{2} - 2 cos (A) cos (B) cos (C) 2 + cos (C) (cos (A - B) - cos (C)) - 2 cos (A) cos (B) cos (C) 2 + cos (C) (cos (A - B) - cos (π - (A + B))) - 2 cos (A) cos (B) cos (C) 2 + cos (C) (cos (A - B) + cos ((A + B)) - 2 cos (A) cos (B) cos (C) 2 + cos (C) (cos A cos B + sin A sin B + cos A cos B - sin A sin B) - 2 cos (A) cos (B) cos (C) 2 + cos (C) (cos A cos B + cos A cos B) - 2 cos (A) cos (B) cos (C) 2 + cos (C) (2 cos A cos B) - 2 cos (A) cos (B) cos (C) 2 + 2 cos (A) cos (B) cos (C) - 2 cos (A) cos (B) cos (C) 2$