# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/11) | Q#5

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Question

The function  is such that    for .

i.       Express  in the form , stating the values of  and .

ii.       State the greatest and least values of .

iii.       Solve the equation .

Solution

i.

We have the function;

We have the trigonometric identity;

We can rewrite it as;

Therefore the function becomes;

Hence;

ii.

From (i), we can rewrite the given function as;

We know that;

Substituting extreme values of ;

Hence;

iii.

We have

We are given that;

Therefore we can write;

Using calculator we can find the values of .

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

 Symmetry Property

Hence;

 For For

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

To find all the solutions (roots) over the interval , we utilize the periodic property of   for both these values of .

 Periodic Property or

Therefore;

 For For For For

For

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