Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2016  OctNov  (P19709/11)  Q#6
Question
i. Show that .
ii. Hence, or otherwise, solve the equation for .
Solution
i.
We have the trigonometric identity;
From this we can substitute ;
Hence, L.H.S=R.H.S.
ii.
We are required to solve the equation for .
From (i) we know that;
Substituting this in the given equation to solve;
We have the trigonometric identity;
From this we can substitute ;
Now we have two options.





NOT POSSIBLE 
Again we have two options.






Since we are required to solve the equation for , we find other solutions as well within the given range.
We utilize the periodic property of to find another solution (root) of :


Symmetry 

Hence;
For 
For 







Therefore, we have four solutions (roots) of the equation;
So we have four 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 
For 
For 




Now;































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




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