# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#4

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Question

i.       Show that the equation  can be expressed as

.

ii.       Solve the equation   for .

Solution

i.

We have the equation;

We have the relation , therefore,

We have the trigonometric identity;

We can write it as;

Therefore;

ii.

To solve the equation  for , as demonstrated in (i), we can write the given equation as;

To solve this equation for , we can substitute . Hence,

Now we have two options;

Since;

Using calculator we can find the values of .

 NOT POSSIBLE

We utilize the symmetry property of   to find other solutions (roots) of :

 Symmetry Property

Hence;

Therefore,

Therefore, we have two solutions (roots) of the equation;

To find all the solutions (roots) of  we utilize the periodic property of  .

 Periodic Property or

Therefore;

 For For

For

Hence all the solutions (roots) of the equation   for  are;