Answer on Question #53373 Trigonometry
Answer on Question #53373 ? Math ? Trigonometry Prove that
3 sin + sin(2) 1 + 3 cos + cos(2) = tan where is a constant.
Solution
We'll use next trigonometric identities
sin(2) = 2 sin() cos() ;
cos(2) = cos2 - sin2 ;
sin2 = 1 - cos2 .
Thus we have
3 sin + sin(2)
3 sin + 2 sin() cos()
1 + 3 cos + cos(2) = 1 + 3 cos + cos2 - sin2 =
sin() (3 + 2 cos())
sin() (3 + 2 cos())
= 1 + 3 cos + cos2 - (1 - cos2 ) = 1 + 3 cos + cos2 - 1 + cos2 =
sin() (3 + 2 cos()) sin() (3 + 2 cos()) sin() = 3 cos + 2 cos2 = cos() (3 + 2 cos()) = cos() = tan .
So we proved
3 sin + sin(2) 1 + 3 cos + cos(2) = tan .
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