6. Jika cos x = 2 1 ​ , maka cos 3 x sama dengan ....

Pembahasan

Ingat kembali rumus penjumlahan dua sudut: cos ( α + b ) = cos α ⋅ cos β − sin α ⋅ sin β cos 2 α = 2 ⋅ cos 2 α − 1 sin 2 α = 2 ⋅ sin α ⋅ cos α dan identitas trigonometri sin 2 α = 1 − cos 2 α sehingga dapat diperoleh rumus berikut: cos 3 α cos 3 α ​ = = = = = = = = = = ​ cos ( 2 α + α ) cos 2 α ⋅ cos α − sin 2 α ⋅ sin α ( 2 ⋅ cos 2 α − 1 ) ⋅ co s α − ( 2 ⋅ sin α ⋅ cos α ) ⋅ sin α 2 ⋅ cos 3 α − co s α − 2 ⋅ sin 2 α ⋅ cos α 2 ⋅ cos 3 α − co s α − 2 ⋅ ( 1 − cos 2 α ) ⋅ cos α 2 ⋅ cos 3 α − co s α − 2 ⋅ ( 1 − cos 2 α ) ⋅ cos α 2 ⋅ cos 3 α − co s α − 2 ⋅ 1 ⋅ cos α + 2 ⋅ cos 2 α ⋅ cos α 2 ⋅ cos 3 α − co s α − 2 ⋅ cos α + 2 ⋅ cos 3 α 2 ⋅ cos 3 α + 2 ⋅ cos 3 α − co s α − 2 ⋅ cos α 4 ⋅ cos 3 α − 3 ⋅ cos α ​ Oleh karena cos x = 2 1 ​ , maka dengan menggunakan rumus di atas, diperoleh: cos 3 x ​ = = = = = = = = = = = ​ 4 ⋅ cos 3 x − 3 ⋅ cos x 4 × ( 2 1 ​ ) 3 − 3 × ( 2 1 ​ ) 4 × 2 3 1 3 ​ − 3 × ( 2 1 ​ ) 4 × 8 1 ​ − 3 × 2 1 ​ 8 4 × 1 ​ − 2 3 × 1 ​ 8 4 ​ − 2 3 ​ 8 ÷ 4 4 ÷ 4 ​ − 2 3 ​ 2 1 ​ − 2 3 ​ 2 1 − 3 ​ 2 − 2 ​ − 1 ​ Dengan demikian, diperoleh cos 3 x sama dengan − 1 . Oleh karena itu, jawaban yang benar adalah E.