# 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

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;