Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2010  MayJun  (P29709/23)  Q#8
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Question
i. Prove the identity
ii. Hence solve the equation
For .
Solution
i.
We are given that;
We utilize following two addition formulae;
ii.
We are required to solve;
As demonstrated in (i);
Therefore;
Since provided that ;
Since ;
Therefore, we solve for .
To solve this equation for , we can substitute . Hence,
Since given interval is , for interval can be found as follows;
Multiplying the entire inequality with 2;
Since ;
Hence the given interval for is .
To solve equation for interval ,
Using calculator we can find the value of .
Properties of 

Domain 

Range 

Odd/Even 

Periodicity 



Translation/ Symmetry 






To find other the solutions (roots) of , we utilize the symmetry property of .
Therefore;
Therefore, we have two solutions of .


To find all other solutions of in the interval we utilize periodicity property of .
For;















Only following solutions (roots) of the equation are within interval;




Since ;








Therefore, all solutions of within interval are;




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