# 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;  