Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/23) | 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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