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;

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