Page 29 - Trigonometry

Example 9:

Find the solutions to equation between 0 ≤ ≤ 2 ,

sin(2 ) − cos( )
= 0
cos(2 ) + sin( ) − 1

Use the identities,

sin( + ) = sin( ) cos( ) + cos( ) sin( )
and

cos( + ) = cos( ) cos( ) − sin( ) sin( )
Then

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

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

Our equation becomes

2 sin( ) cos( ) − cos( )
= 0
2
2
cos ( ) − sin ( ) + sin( ) − 1

cos( ) [2 sin( ) − 1]
= 0
2
2
sin( ) − sin ( ) −[1 − cos ( )]

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