Question

find the value of sin 20°sin40°sin80°​

Answer 1

sin20 sin40 sin80

sin40 = 2 sin20 cos20

2 sin20 sin20 cos20 sin80

2 cos20 sin80 sin^2 20

sin( 100) + sin60 ) sin^2 20

sin(90+10) + sin60) sin^2 20

cos10 + sin60) sin^2 20

cos10 sin20 + sin60 sin20) sin20

sin30 + sin10 + cos40 - cos80)( sin20)/2

sin30 + sin10 + cos40 - cos(90-10) )( sin20)/2

sin30 + sin10 + cos40 - sin10) sin20))/2

sin30 + cos40)) sin20 )/2

1/2. + cos40))sin20))/2

(sin(20))/4 + sin20 cos40))/2

(sin(20))/4. +(( sin(60) - sin20))/4

sin20 + sin60 - sin20))/4

sin60))/4

√3/8

Answer 2

Answer:

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Hi ,

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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 )

____________________

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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