9. Jika A + B + C = 18 0 ∘ , tunjukkan bahwa: b. cos 2 A + cos 2 B + cos 2 C = − 1 − 4 cos A ⋅ cos B ⋅ cos C

Pembahasan

Ingat kembali: sudut berelasi: cos ( 180 − α ) = − cos α rumus cosinus untuk penjumlahan dua sudut: cos ( α + β ) = cos α ⋅ cos β − sin α ⋅ sin β r umus cosinus untuk sudut ganda: cos 2 α = 2 ⋅ cos 2 α − 1 identitas trigonometri: sin 2 α = 1 − cos 2 α Olehkarena itu, dengan menggunakan sudut berelasi dapatdiperoleh: cos C ​ = = = = = ​ cos ( 18 0 ∘ − ( A + B ) ) − cos ( A + B ) − ( cos A ⋅ cos B − sin A ⋅ sin B ) − cos A ⋅ cos B + sin A ⋅ sin B sin A ⋅ sin B − cos A ⋅ cos B ​ sehingga diperoleh: cos 2 C ​ = = = = = = = = = = = = ​ ( cos C ) 2 ( sin A ⋅ sin B − cos A ⋅ cos B ) 2 ( sin A ⋅ sin B ) 2 + ( cos A ⋅ cos B ) 2 − 2 ⋅ sin A ⋅ sin B ⋅ cos A ⋅ cos B sin 2 A ⋅ sin 2 B + cos 2 A ⋅ cos 2 B − 2 ⋅ sin A ⋅ sin B ⋅ cos A ⋅ cos B ( 1 − cos 2 A ) ⋅ ( 1 − cos 2 B ) + cos 2 A ⋅ cos 2 B − 2 ⋅ sin A ⋅ sin B ⋅ cos A ⋅ cos B 1 − cos 2 A − cos 2 B + cos 2 A ⋅ cos 2 B + cos 2 A ⋅ cos 2 B − 2 ⋅ sin A ⋅ sin B ⋅ cos A ⋅ cos B 1 − cos 2 A − cos 2 B + 2 ⋅ cos 2 A ⋅ cos 2 B − 2 ⋅ sin A ⋅ sin B ⋅ cos A ⋅ cos B 1 − cos 2 A − cos 2 B + 2 ⋅ ( cos A ⋅ cos B ) ( cos A ⋅ cos B − sin A ⋅ sin B ) 1 − cos 2 A − cos 2 B + 2 ⋅ ( cos A ⋅ cos B ) ( − sin A ⋅ sin B + cos A ⋅ cos B ) 1 − cos 2 A − cos 2 B − 2 ⋅ ( cos A ⋅ cos B ) ( sin A ⋅ sin B − cos A ⋅ cos B ) 1 − cos 2 A − cos 2 B − 2 ⋅ ( cos A ⋅ cos B ) ( cos C ) 1 − cos 2 A − cos 2 B − 2 ⋅ cos A ⋅ cos B ⋅ cos C ​ Selanjutnya, dengan menggunakan rumus sudut ganda dan kemudian menyubtitusi bentuk cos 2 C yang diperoleh di atas, maka diperoleh: ​ = = = = = = = = ​ cos 2 A + cos 2 B + cos 2 C 2 ⋅ cos 2 A − 1 + 2 ⋅ cos 2 B − 1 + 2 ⋅ cos 2 C − 1 2 ⋅ cos 2 A + 2 ⋅ cos 2 B + 2 ⋅ cos 2 C − 1 − 1 − 1 2 ⋅ cos 2 A + 2 ⋅ cos 2 B + 2 ⋅ cos 2 C − 3 2 ⋅ cos 2 A + 2 ⋅ cos 2 B + 2 ⋅ ( 1 − cos 2 A − cos 2 B − 2 ⋅ cos A ⋅ cos B ⋅ cos C ) − 3 2 ⋅ cos 2 A + 2 ⋅ cos 2 B + 2 − 2 ⋅ cos 2 A − 2 ⋅ cos 2 B − 4 ⋅ cos A ⋅ cos B ⋅ cos C − 3 2 − 3 + 2 ⋅ cos 2 A − 2 ⋅ cos 2 A + 2 ⋅ cos 2 B − 2 ⋅ cos 2 B − 4 ⋅ cos A ⋅ cos B ⋅ cos C − 1 + 0 + 0 − 4 ⋅ cos A ⋅ cos B ⋅ cos C − 1 − 4 ⋅ cos A ⋅ cos B ⋅ cos C ​ Dengan demikian, telah ditunjukan bahwa cos 2 A + cos 2 B + cos 2 C = − 1 − 4 ⋅ cos A ⋅ cos B ⋅ cos C .