Page 54 - Trigonometry

Example 16
Prove that for any triangle whose angles are A, B and

C,

sin(2 ) + sin(2 ) + sin(2 )
= 2sin( ) sin( ) sin( )

Express as,

= [sin(2 ) + sin(2 )] + sin(2 )

Use the identity,

sin(2 ) + sin(2 ) = 2 sin( + ) cos( − )

Then

= 2 sin( + ) cos( − ) + cos(2 )

But + = 180 −

sin( + ) = sin(180 − )

= sin(180) cos( ) − cos(180) sin( )

= sin( )

= 2 sin( ) cos( − ) + cos(2 )

but

cos(2 ) = 2 sin( ) cos( )

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