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