Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2015  OctNov  (P19709/11)  Q#4
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Question
i. Show that the equation can be expressed as
ii. Hence solve the equation for .
Solution
i.
We are given that;
Since ;
We have the trigonometric identity;
From this we can write;
Therefore;
ii.
We are required to solve the equation for .
From (i), we know that;
Therefore;
Let ;









Since;
We have two options;


We know that;
Therefore is not possible. Hence;
Using calculator we can find the value of .
We utilize the periodic property of to find another solution (root) of :


Symmetry 

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


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


Periodic 
or

For the given case,




For 
For 


Now;
For 















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


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