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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