Trigonometry> prove that sin 20sin40+sin60sin80=3/16 ...

prove that sin 20sin40+sin60sin80=3/16
how does it comes

sai ramya , 8 Years ago

Grade 9

3 Answers

Arun

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

Last Activity: 8 Years ago

Chinmay sambhe

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.

Last Activity: 7 Years ago

Rishi Sharma

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