tag:blogger.com,1999:blog-38193052.post363197089132074190..comments2024-03-20T08:57:17.447-03:00Comments on Jornalheiros: Matemática - Senos multiplicadosPC Filhohttp://www.blogger.com/profile/16547063456626761789noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-38193052.post-62452243932250587542021-09-01T23:34:49.452-03:002021-09-01T23:34:49.452-03:00Here’s a shorter proof:
sin 6° sin 42° sin 66° sin...Here’s a shorter proof:<br />sin 6° sin 42° sin 66° sin 78°<br />= sin 6° sin 66° sin 42° sin 78°<br />= (1/4)(cos 60° – cos 72°)(cos 36° – cos 120°)<br />= (1/4)[1/2 – (√5 – 1)/4][(√5 + 1)/4 – (–1/2)]<br />= (1/64)[2 – (√5 – 1)][(√5 + 1) + 2]<br />= (1/64)(3 – √5)(3 + √5)<br />= (1/64)(9 – 5)<br />= (1/64)(4)<br />= 1/16<br />QEDjrh150482https://www.blogger.com/profile/10502831081969372299noreply@blogger.comtag:blogger.com,1999:blog-38193052.post-90484068549382283712021-08-31T08:12:56.979-03:002021-08-31T08:12:56.979-03:00Well done, Jake. :)Well done, Jake. :)PC Filhohttps://www.blogger.com/profile/16547063456626761789noreply@blogger.comtag:blogger.com,1999:blog-38193052.post-50660901151290224652021-08-31T07:52:45.289-03:002021-08-31T07:52:45.289-03:00Prove that sin 6° sin 42° sin 66° sin 78° = 1/16.
...Prove that sin 6° sin 42° sin 66° sin 78° = 1/16.<br /><br />sin 6° sin 42° sin 66° sin 78° = sin 6° sin 42° sin [(180 – 66)°] sin 78°<br />= sin 6° sin 42° sin 78° sin 114°<br />= sin (60 – 54)° sin (60 – 18)° sin (60 + 18)° sin (60 + 54)°<br />= sin (60 – 54)° sin (60 + 54)° sin (60 – 18)° sin (60 + 18)°<br />= (sin² 60° cos² 54° – cos² 60° sin² 54°)(sin² 60° cos² 18° – cos² 60° sin² 18°)<br />= [(3/4)(1 – sin² 54°) – (1/4) sin² 54°][(3/4)(1 – sin² 18°) – (1/4) sin² 18°]<br />= (3/4 – sin² 54°)(3/4 – sin² 18°)<br />= (3/4 – cos² 36°)[3/4 – (1/2)(1 – cos 36°)]<br />= (3/4 – cos² 36°)[1/4 + (1/2) cos 36°]<br />= [3/4 – ((√5 + 1)/4)²][1/4 + (√5 + 1)/8]<br />= [3/4 – (3 + √5)/8][1/4 + (√5 + 1)/8]<br />= [(3 – √5)/8][(3 + √5)/8]<br />= (9 – 5)/64<br />= 4/64<br />= 1/16<br />QEDjrh150482https://www.blogger.com/profile/10502831081969372299noreply@blogger.com