Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2010  MayJun  (P19709/13)  Q#4
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Question
i. Show that the equation can be expressed as
.
ii. Solve the equation for .
Solution
i.
We have the equation;
We have the relation , therefore,
We have the trigonometric identity;
We can write it as;
Therefore;
ii.
To solve the equation for , as demonstrated in (i), we can write the given equation as;
To solve this equation for , we can substitute . Hence,
Now we have two options;









Since;
Using calculator we can find the values of .



NOT POSSIBLE

We utilize the symmetry property of to find other solutions (roots) of :


Symmetry 

Hence;
Therefore,
Therefore, we have two solutions (roots) of the equation;


To find all the solutions (roots) of we utilize the periodic property of .


Periodic 
or

Therefore;




For 
For 




For





























Hence all the solutions (roots) of the equation for are;


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