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prove that sin 20*sin40+sin60*sin80=3/16 how does it comes

prove that sin 20*sin40+sin60*sin80=3/16 
how does it comes

Grade:9

3 Answers

Arun
25763 Points
3 years ago

since,sin60=√ 3/2

= √ 3/2( sin20sin40sin80)

=√ 3/2( sin20sin80sin40)

=√ 3/4 [(2sin20sin40)sin80]

on applying [cos(A-B)-cos(A+B) = 2sinAsinB]

we get,

= √ 3/4 (cos20-cos60)sin80 [since,cos(-a)=cosa]

= √ 3/4(cos20sin80-cos60sin80)

= √ 3/8(2sin80cos20-sin80)

= √ 3/8(sin100+sin60-sin80)

= √ 3/8( √ 3/2+sin100-sin80 )

= √ 3/8( √ 3/2+sin(180-80)-sin80 )

= √ 3/8( √ 3/2+sin80-sin80 ) [since,sin(180-a)=sina]

= √ 3/8( √ 3/2)

= 3/16

Chinmay sambhe
13 Points
2 years ago
Since sin 60= root3/2 
=> root3/2 [ sin 20 . Sin 40 . Sin 80]
=> root3/2 [ sin 20 . Sin (60-20) . Sin (60+ 20) ]
We know that sin x . Sin (60-x) . Sin (60+ x) = ¼ sin 3x
Therefore, root3/2 [ ¼ sin60]
Root 3/2 . ¼ . root 3/2 
=> 3/16.
Rishi Sharma
askIITians Faculty 646 Points
one year ago
Hello student,
The problem posted by you needs a small correction, it should be
sin20 * sin40 * sin60 * sin80 = 3/16
I hope the attached solution image will solve your doubts.
Thank You
All the best for your exams.

645-1163_WhatsApp Image 2020-06-04 at 8.11.48 PM(1).jpeg

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