Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  MayJun  (P19709/13)  Q#8
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Question
i. Show that can be written as a quadratic equation in and hence solve the equation for .
ii. Find the solutions to the equation for .
Solution
i.
We have the expression;
We know that ; therefore,
We have the trigonometric identity;
It can be rearranged to;
Therefore;
To solve this equation for , we can substitute . Hence,
Now we have two options;









Since;
Using calculator we can find the values of .




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


Symmetry 

Hence;
For 
For 




Therefore, we have only two solutions (roots) of the equation in the range of ;


ii.
We have the equation;
As demonstrated in (i) we can write the given equation as;
We can substitute . Hence,
Now we have two options;









Hence;


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 .
Using calculator we can find the values of .




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


Symmetry 

Hence;
For 
For 




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



To find all the solutions (roots) over the interval , we utilize the periodic property of for both these values of .


Periodic 
or

Therefore;




For 
For 
For 






For






















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




Since ;








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