# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#2

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Question

Find all the values of  in the interval  which satisfy the equation;

Solution

First we need to manipulate the given equation to write it in a single trigonometric
ratio;

Dividing the entire equation by ;

Using the relation  we can rewrite the equation as,

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

Since given interval is  , for  interval can be found as follows;

Multiplying the entire inequality with 3;

Since ;

Hence the given interval for  is .

To solve  equation for interval ,

Using calculator we can find the value of .

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

 Periodic/Symmetry Property or

Therefore;

For;

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

Since  ;

Therefore;

Hence, all the solutions of the equation  within  interval are;