9. Jika A + B + C = 18 0 ∘ , tunjukkan bahwa: a. sin 2 A + sin 2 B − sin 2 C = 4 cos A ⋅ cos B ⋅ sin C

Pembahasan

Ingat kembali: r umus sinus untuk sudut ganda: sin 2 α = 2 ⋅ sin α ⋅ cos α sudut berelasi: sin ( 180 − α ) = sin α rumus sinus untuk penjumlahan dua sudut: sin ( α + β ) = sin α ⋅ cos β + cos α ⋅ sin β Olehkarena itu,dapat diperoleh: ​ = = = = = = = = = ​ sin 2 A + sin 2 B − sin 2 C 2 ⋅ sin A ⋅ cos A + 2 ⋅ sin B ⋅ cos B − 2 ⋅ sin C ⋅ cos C 2 ⋅ sin ( 18 0 ∘ − ( B + C ) ) ⋅ cos A + 2 ⋅ sin ( 18 0 ∘ − ( A + C ) ) ⋅ cos B − 2 ⋅ sin ( 18 0 ∘ − ( A + B ) ) ⋅ cos C 2 ⋅ sin ( B + C ) ⋅ cos A + 2 ⋅ sin ( A + C ) ⋅ cos B − 2 ⋅ sin ( A + B ) ⋅ cos C 2 ⋅ ( sin B ⋅ cos C + cos B ⋅ sin C ) ⋅ cos A + 2 ⋅ ( sin A ⋅ cos C + cos A ⋅ sin C ) ⋅ cos B − 2 ⋅ ( sin A ⋅ cos B + cos A ⋅ sin B ) ⋅ cos C ( 2 ⋅ cos A ⋅ sin B ⋅ cos C + 2 ⋅ cos A ⋅ cos B ⋅ sin C ) + ( 2 ⋅ sin A ⋅ cos B ⋅ cos C + 2 ⋅ cos A ⋅ cos B ⋅ sin C ) − ( 2 ⋅ sin A ⋅ cos B ⋅ cos C + 2 ⋅ cos A ⋅ sin B ⋅ cos C ) 2 ⋅ cos A ⋅ sin B ⋅ cos C + 2 ⋅ cos A ⋅ cos B ⋅ sin C + 2 ⋅ sin A ⋅ cos B ⋅ cos C + 2 ⋅ cos A ⋅ cos B ⋅ sin C − 2 ⋅ sin A ⋅ cos B ⋅ cos C − 2 ⋅ cos A ⋅ sin B ⋅ cos C 2 ⋅ cos A ⋅ sin B ⋅ cos C + 2 ⋅ cos A ⋅ cos B ⋅ sin C + 2 ⋅ sin A ⋅ cos B ⋅ cos C − 2 ⋅ sin A ⋅ cos B ⋅ cos C + 2 ⋅ cos A ⋅ cos B ⋅ sin C − 2 ⋅ cos A ⋅ sin B ⋅ cos C 4 ⋅ cos A ⋅ sin B ⋅ cos C + 0 + 0 4 ⋅ cos A ⋅ cos B ⋅ sin C ​ Dengan demikian, telah ditunjukan bahwa sin 2 A + sin 2 B − sin 2 C == 4 ⋅ cos A ⋅ cos B ⋅ sin C .