Buktikan kesamaan trigonometri berikut. sin^(2)6°+sin^(2)42°+sin^(2)66°+sin^(2)78°=9/4 Petunjuk: cos 36°=(√5+1)/2

Jawaban : terbukti bahwa sin² 6° + sin² 42° + sin² 66° + sin² 78° = 9/4. Ingat! sin² a + cos² a = 1 sin (a-b) = sin a cos b - cos a sin b sin (a+b) = sin a cos b + cos a sin b sin (-a) = - sin a cos (-a) = cos a cos 2a = 1 - 2 sin² a cos (a+b) = cos a cos b - sin a sin b cos (a-b) = cos a cos b + sin a sin b Asumsi soal: cos 36° = (√5+1)/4 Dengan menggunakan identitas trigonometri diperoleh sin² 36° + cos² 36° = 1 sin² 36° + ((√5+1)/4)² = 1 sin² 36° + (5+2√5+1)/16 = 1 sin² 36° + (6+2√5)/16 = 16/16 sin² 36° = 16/16 - (6+2√5)/16 sin² 36° = (10-2√5)/16 sin 36° = ±√[(10-2√5)/16] sin 36° = ±[√(10-2√5)]/4 Karena 36° berada di kuadran I, maka sin 36° bernilai positif, diperoleh sin 36° = [√(10-2√5)]/4 >>> sin 6° = sin (-30°+36°) = sin (-30°) cos 36° + cos (-30°) sin 36° = -1/2 • (√5+1)/4 + 1/2 √3 • [√(10-2√5)]/4 = (-√5-1)/8 + [√3•√(10-2√5)]/8 = (-√5-1)/8 + [√(30-6√5)]/8 sin² 6° = ((-√5-1)/8 + [√(30-6√5)]/8)² = ((-√5-1)/8)² + 2•(-√5-1)/8 • [√(30-6√5)]/8 + ([√(30-6√5)]/8)² = (5+2√5+1)/64 - [(2√5+2)√(30-6√5)]/8 + (30-6√5)/64 = (5+2√5+1+30-6√5)/64 - [(2√5+2)√(30-6√5)]/8 = (36-4√5)/64 - [(2√5+2)√(30-6√5)]/8 >>> cos 84° = cos (120°-36°) = cos 120° cos 36° + sin 120° sin 36° = -1/2 • (√5+1)/4 + 1/2 √3 • [√(10-2√5)]/4 = (-√5-1)/8 + [√3•√(10-2√5)]/8 = (-√5-1)/8 + [√(30-6√5)]/8 cos 84° = 1 - 2 sin² 42° (-√5-1)/8 + [√(30-6√5)]/8 = 1 - 2 sin² 42° 2 sin² 42° = 1 - {(-√5-1)/8 + [√(30-6√5)]/8} 2 sin² 42° = 8/8 + (√5+1)/8 - [√(30-6√5)]/8 2 sin² 42° = (√5+9)/8 - [√(30-6√5)]/8 sin² 42° = (√5+9)/16 - [√(30-6√5)]/16 >>> sin 66° = sin (30°+36°) = sin 30° cos 36° + cos 30° sin 36° = 1/2 • (√5+1)/4 + 1/2 √3 • [√(10-2√5)]/4 = (√5+1)/8 + [√3•√(10-2√5)]/8 = (√5+1)/8 + [√(30-6√5)]/8 sin² 66° = ((√5+1)/8 + [√(30-6√5)]/8)² = ((√5+1)/8)² + 2•(√5+1)/8 • [√(30-6√5)]/8 + ([√(30-6√5)]/8)² = (5+2√5+1)/64 + [(2√5+2)√(30-6√5)]/8 + (30-6√5)/64 = (5+2√5+1+30-6√5)/64 + [(2√5+2)√(30-6√5)]/8 = (36-4√5)/64 + [(2√5+2)√(30-6√5)]/8 >>> cos 156° = cos (120°+36°) = cos 120° cos 36° - sin 120° sin 36° = -1/2 • (√5+1)/4 - 1/2 √3 • [√(10-2√5)]/4 = (-√5-1)/8 - [√3•√(10-2√5)]/8 = (-√5-1)/8 - [√(30-6√5)]/8 cos 156° = 1 - 2 sin² 78° (-√5-1)/8 - [√(30-6√5)]/8 = 1 - 2 sin² 78° 2 sin² 78° = 1 - {(-√5-1)/8 - [√(30-6√5)]/8} 2 sin² 78° = 8/8 + (√5+1)/8 + [√(30-6√5)]/8 2 sin² 78° = (√5+9)/8 + [√(30-6√5)]/8 sin² 78° = (√5+9)/16 + [√(30-6√5)]/16 Oleh karena itu, sin² 6° + sin² 42° + sin² 66° + sin² 78° = (36-4√5)/64 - [(2√5+2)√(30-6√5)]/8 + (√5+9)/16 - [√(30-6√5)]/16 + (36-4√5)/64 + [(2√5+2)√(30-6√5)]/8 + (√5+9)/16 + [√(30-6√5)]/16 = (36-4√5)/32 + (√5+9)/8 = (36-4√5)/32 + (4√5+36)/32 = 72/32 = 9/4 Jadi, terbukti bahwa sin² 6° + sin² 42° + sin² 66° + sin² 78° = 9/4.