( c ) ( i ) By considering AOP ˆ
1
, express R ( 3 ) in terms of a sine ratio .
( ii ) Explain why L( 3 ) � 1. [ 4 ]
The points P
1
, P
2 and P
3 are constructed on the arc length of the semicircle such that arc AP1 � arc PP
1 2
� arc P2 P3 � arc P3
B , as shown in the following diagram :
P
2
P
1
P
3
A O
B
Let R ( 4 ) and L ( 4 ) be the area of the polygon AP1 P2 P3
B and the length AP
1 respectively .
( d ) Show that R( 4 ) � 2sin 45 .
( e ) Show that the exact value of L ( 4 ) satisfies the equation x
� 4x � 2 � 0 .
4 2
Similarly , the points P
1
, P
2
, P
3
, , P n� 1 are constructed on the arc of the semicircle such that arc AP1 � arc PP
1 2
� arc P2P 3
� � arc Pn�
1B
. Let Rn ( ) and
Ln ( ) be the area of the polygon APP
1 2P3 Pn� 1B and the length AP
1 respectively , where n � 4 .
( f ) ( i ) Find Rn ( ).
[ 2 ]
[ 5 ]
( ii ) By considering the properties of semicircle , write down an upper bound for Rn ( ).
( g ) ( i ) Show that
180 Ln ( ) � 2 � 2cos . n
[ 4 ]
( ii ) Hence , show that
Ln ( ) 2 90 � sec .
R( n) n n
[ 6 ]
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