Jika cos ( A − B ) = 5 3 ​ dan tan A tan B = 2 , hitunglah: c. cos ( A + B )

Pembahasan

Diketahui cos ( A − B ) = 5 3 ​ dan tan ( A ) tan ( B ) = 2 . Akan dicari cos ( A + B ) Ingat bahwa cos ( A − B ) = cos ( A ) cos ( B ) + sin ( A ) sin ( B ) cos ( A + B ) = cos ( A ) cos ( B ) − sin ( A ) sin ( B ) Diperhatikan tan ( A ) ⋅ tan ( B ) = 2 ⇒ c o s ( A ) s i n ( A ) ​ ⋅ c o s ( B ) s i n ( B ) ​ = 2 ⇒ sin ( A ) ⋅ sin ( B ) = 2 cos ( A ) ⋅ cos ( B ) ​ dan tan ( A ) ⋅ tan ( B ) = 2 ⇒ c o s ( A ) s i n ( A ) ​ ⋅ c o s ( B ) s i n ( B ) ​ = 2 ⇒ 2 s i n ( A ) ⋅ s i n ( B ) ​ = cos ( A ) ⋅ cos ( B ) ​ Diperoleh perhitungan cos ( A − B ) cos ( A ) cos ( B ) + sin ( A ) sin ( B ) cos ( A ) cos ( B ) + 2 cos ( A ) cos ( B ) 3 cos ( A ) cos ( B ) cos ( A ) cos ( B ) cos ( A ) cos ( B ) ​ = = = = = = ​ 5 3 ​ 5 3 ​ 5 3 ​ 5 3 ​ 5 3 ​ ⋅ 3 1 ​ 5 1 ​ ​ dan cos ( A − B ) cos ( A ) cos ( B ) + sin ( A ) sin ( B ) 2 s i n ( A ) s i n ( B ) ​ + sin ( A ) sin ( B ) 2 3 ​ sin ( A ) sin ( B ) sin ( A ) sin ( B ) sin ( A ) sin ( B ) ​ = = = = = = ​ 5 3 ​ 5 3 ​ 5 3 ​ 5 3 ​ 5 3 ​ ⋅ 3 2 ​ 5 2 ​ ​ Lebih lanjut, diperoleh cos ( A + B ) ​ = = = ​ cos ( A ) cos ( B ) − sin ( A ) sin ( B ) 5 1 ​ − 5 2 ​ − 5 1 ​ ​ Dengan demikian, diperoleh cos ( A + B ) = − 5 1 ​ .