# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | 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 Property

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 Property or

For the given case,

 For For

Now;

 For

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