Nilai dari sin 15°(cos 105° – sin 75°)adalah–¼(3 – √3).
 

Pembahasan

Trigonometri

Menghitung nilai sin 15°(cos 105° – sin 75°)

CARA PERTAMA

• sin 15° = sin(45°–30°)
• cos 105° = cos(90°+15°) = –sin 15°
• sin 75° = sin(45°+30°)

Ingat identitas trigonometri:
sin(α ± β) = sin α cos β ± cos α sin β

• sin 15° = sin(45°–30°)
  = sin 45° cos 30° – cos 45° sin 30°
  = ½√2 · ½√3 – ½√2 · ½
  = ¼√6 – ¼√2
  = ¼(√6 – √2)
• cos 105° = –sin 15° = –¼(√6 – √2)
• sin 75° = sin(45°+30°)
  = sin 45° cos 30° + cos 45° sin 30°
    (pakai hasil sin 15° di atas, ganti “–” menjadi “+”)
  = ¼(√6 + √2)

sin 15°(cos 105° – sin 75°)
= ¼(√6 – √2) · [ –¼(√6 – √2) – ¼(√6 + √2) ]
= ¼(√6 – √2) · ¼(–√6 + √2 – √6 – √2)
= ¼(√6 – √2) · ¼(–√6 – √6 + √2 – √2)
= ¼(√6 – √2) · ¼(–2√6)
= ¼(√6 – √2) · (–½√6)
= ¼ · (–½) · (√6 – √2)(√6)
= (–1/8)(√6√6 – √6√2)
= (–1/8)(6 – √3√2√2)
= (–1/8)(6 – 2√3)
= –¼(3 – √3)

CARA KEDUA
Dengan identitas trigonometri half-angle (sudut setengah)

sin 15°(cos 105° – sin 75°)

• cos 105° = –sin 15°

= sin 15°(–sin 15° – sin 75°)
= sin(½·30°) [cos(½·210°) – sin(½·150°)]

• sin(½α) = ±√[(1 – cos α) / 2] = ± (1/√2)√(1 – cos α)
  Dengan α = 30°, ½α = 15° (kuadran I, sinus positif):
  ⇒ sin(½·30°) = (1/√2)√(1 – cos 30°)
  ⇒ sin(½·30°) = (1/√2)√(1 – ½√3)
  ⇒ sin 15° = (1/√2)√(1 – ½√3)
  Dengan α = 150°, ½α = 75° (kuadran I, sinus positif):
  ⇒ sin(½·150°) = (1/√2)√(1 – cos 150°)
  ⇒ sin(½·150°) = (1/√2)√(1 – (–cos 30°))
  ⇒ sin(½·150°) = (1/√2)√(1 + cos 30°)
  ⇒ sin(½·150°) = (1/√2)√(1 +½√3)
  ⇒ sin 75° = (1/√2)√(1 + ½√3)

= (1/√2)√(1 – ½√3) · [ –(1/√2)√(1 – ½√3) – (1/√2)√(1 + ½√3) ]
= (1/√2)√(1 – ½√3) (–(1/√2)√(1 – ½√3))  – (1/√2)√(1 – ½√3)((1/√2)√(1 + ½√3))
= (1/√2)[–(1/√2)][√(1 – ½√3)]² – (1/√2)(1/√2)√[(1 – ½√3)(1 + ½√3)]
= –½(1 – ½√3) – ½√(1 – ¼·3)
= –½(1 – ½√3) – ½√(1 – 3/4)
= –½(1 – ½√3) – ½√(¼)
= –½(1 – ½√3) – ½·½
= –½(1 – ½√3) – ¼
= –¼·2(1 – ½√3) – ¼
= –¼(2 – √3) – ¼
= –¼(2 – √3 + 1)
= –¼(2 + 1– √3)
= –¼(3 – √3)