Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#6

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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
Property

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
Property

or

For the given case,

For

For

For

For

Now;

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

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