Kuadran IV : 270° < α < 360° ( < α < 2π) sin (360° - α) = sin (-α) = -sin α cos (360° - α) = cos (-α) = cos α tan (360° - α) = tan (-α) = -tan α cotan (360° - α) = cotan (-α) = -cotan α secan (360° - α) = secan (-α) = -secan α sec (360° - α) = sec (-α) = sec α
Jika α dan β merupakan sudut-sudut sebarang, maka 1. sin α + sin β = 2 sin (α + β) cos (α - β) 2. sin α - sin β = 2 cos (α + β) sin (α - β) 3. cos α + cos β = 2 cos (α + β) cos (α - β) 4. cos α - cos β = -2sin (α + β) sin (α - β)
Nilai perbandingan trigonometri sudut-sudut istimewa, antara lain : sin 30° = , cos 30° = √3, tan 30° = √3 sin 45° = √2, cos 45° = √2, tan 45° = 1 sin 60° = √3, cos 60° = , tan 60° = √3 sin 90° = 1, cos 90° = 0, tan 90° = ∞
Mari kita lihat soal tersebut. sin 51° + cos 81° = sin 51° + cos (90 - 9)° = sin 51° + sin 9° = 2 sin (51 + 9)° cos (51 - 9)° = 2 sin 60° cos 42° = 2 sin 30° cos 21° = 2 × × cos 21° = cos 21° Jadi, sin 51° + cos 81° sama dengan cos 21°. Semangat!
Materi : Trigonometri
Kata Kunci : trigonomteri, perbandingan, penjumlahan, sudut, sebarang, istimewa
Pembahasan :
Dalam trigonometri, besar sudut α dibagi menjadi 4 kelompok, yaitu :
Kuadran I : 0° < α < 90° (0 < α < )
sin (90° - α) = cos α
cos (90° - α) = sin α
tan (90° - α) = cotan α
cotan (90° - α) = tan α
secan (90° - α) = sec α
sec (90° - α) = cosec α
Kuadran II : 90° < α < 180° ( < α < π)
sin (180° - α) = sin α
cos (180° - α) = -cos α
tan (180° - α) = -tan α
cotan (180° - α) = -cotan α
secan (180° - α) = secan α
sec (180° - α) = -sec α
Kuadran III : 180° < α < 270° (π < α < )
sin (270° - α) = -cos α
cos (270° - α) = -sin α
tan (270° - α) = cotan α
cotan (270° - α) = tan α
secan (270° - α) = -sec α
sec (270° - α) = -cosec α
Kuadran IV : 270° < α < 360° ( < α < 2π)
sin (360° - α) = sin (-α) = -sin α
cos (360° - α) = cos (-α) = cos α
tan (360° - α) = tan (-α) = -tan α
cotan (360° - α) = cotan (-α) = -cotan α
secan (360° - α) = secan (-α) = -secan α
sec (360° - α) = sec (-α) = sec α
Jika α dan β merupakan sudut-sudut sebarang, maka
1. sin α + sin β = 2 sin (α + β) cos (α - β)
2. sin α - sin β = 2 cos (α + β) sin (α - β)
3. cos α + cos β = 2 cos (α + β) cos (α - β)
4. cos α - cos β = -2sin (α + β) sin (α - β)
Nilai perbandingan trigonometri sudut-sudut istimewa, antara lain :
sin 30° = , cos 30° = √3, tan 30° = √3
sin 45° = √2, cos 45° = √2, tan 45° = 1
sin 60° = √3, cos 60° = , tan 60° = √3
sin 90° = 1, cos 90° = 0, tan 90° = ∞
Mari kita lihat soal tersebut.
sin 51° + cos 81°
= sin 51° + cos (90 - 9)°
= sin 51° + sin 9°
= 2 sin (51 + 9)° cos (51 - 9)°
= 2 sin 60° cos 42°
= 2 sin 30° cos 21°
= 2 × × cos 21°
= cos 21°
Jadi, sin 51° + cos 81° sama dengan cos 21°.
Semangat!
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