simplify sin20sin40sin80
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Answer:
Step-by-step explanation:
___________
SinASin( 60 - A ) sin ( 60 + A )
=SinA [(Sin60cosA- cos60sinA)
(Sin60cosA + cos60sinA)]
=SinA [ (Sin60cosA)² - (cos60sinA)² ]
=SinA [ (√3/2×cosA )² - ( 1/2 × sinA )² ]
=SinA(3 Cos² A )/4 - ( sin² A) / 4
= 1/4×SinA [ 3cos² A - sin² A]
= 1/4 × SinA[3( 1- sin²A) - sin² A ]
= (SinA)/4[3 - 3sin² A - sin² A ]
= ( SinA)/4 [ 3- 4sin² A]
= [ 3sinA - 4sin³ A ]/4
=( Sin3A)/4 ------( 1 )
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Here A = 20,
Sin 20sin40sin80
= Sin20sin(60-20)sin(60+20)
=[ Sin (3×20)]/4 from ( 1 )
= (Sin60)/4
= ( √3/2 ) / 4
= √3 / 8
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