Question

Prove that sin 13pi/3.sin 8pi/3+cos 2pi/3.sin 5pi /6=1/2.

Answer 1

Hi, do you have any thoughts on how to get started with this?

Answer 2

If your expression mean:

sin ( 13 π / 3 ) ∙ sin ( 8 π / 3 ) + cos ( 2 π / 3 ) ∙ sin ( 5 π / 6 ) = 1 / 2

then:

13 π / 3 = 12 π / 3 + π / 3 = 4 π + π / 3 = 2 ∙ 2 π + π / 3

sin ( 2 π ) = 0

sin ( 2 π + x ) = sin ( x )

sin ( 2 ∙ 2 π + x ) = sin ( x )

sin ( 13 π / 3 ) = sin ( 2 ∙ 2 π + π / 3 ) = sin ( π / 3 ) = √ 3 / 2


8 π / 3 = 6 π / 3 + 2 π / 3 = 2 π + 2 π / 3

sin ( 2 π + x ) = sin ( x )

sin ( 8 π / 3 ) = sin ( 2 π + 2 π / 3 ) = sin ( 2 π / 3 ) = √ 3 / 2


2 π / 3 = 2 ∙ 2 π / 2 ∙ 3 = 4 π / 6 = 3 π / 6 + π / 6 = π / 2 + π / 6

cos ( π / 2 + x ) = - sin ( x )

cos ( 2 π / 3 ) = cos ( π / 2 + π / 6 ) = - sin ( π / 6 ) = - 1 / 2


5 π / 6 = 3 π / 6 + 2 π / 6 = π / 2 + 2 π / 6

sin ( π / 2 + x ) = cos ( x )

sin ( 5 π / 6 ) = sin ( π / 2 + 2 π / 6 ) = cos ( 2 π / 6 ) = 1 / 2


sin ( 13 π / 3 ) ∙ sin ( 8 π / 3 ) + cos ( 2 π / 3 ) ∙ sin ( 5 π / 6 ) =

( √ 3 / 2 ) ∙ ( √ 3 / 2 ) + ( - 1 / 2 ) ∙ ( 1 / 2 ) =

( √ 3 ∙ √ 3 ) / ( 2 ∙ 2 ) - ( 1 ∙ 1 ) / ( 2 ∙ 2 ) =

3 / 4 - 1 / 4 =

2 / 4 = 1 / 2