# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2002 | Oct-Nov | (P2-9709/02) | Q#5

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Question

The angle x, measured in degrees, satisfies the equation i.       By expanding each side, show that the equation may be simplified to ii.       Find the two possible values of x lying between and .

iii.       Find the exact value of , giving your answer as a fraction.

Solution

i.

We are given; We apply following two addition formulae on both sides of given equation.  Therefore; Since;        ii.

We are required to find the two possible values of x lying between and We have found an equation in (i) as; We can rearrange the equation as follows to solve it.  We know that ,therefore;  Using calculator we can find the value of . Properties of Domain Range Periodicity  Odd/Even Translation/ Symmetry  We utilize the periodicity/symmetry property of to find other solutions (roots) of  Therefore;  For;        Only following solutions (roots) are within the given interval ;  iii.

We are required to find the exact value of , giving your answer as a fraction. Therefore;  Hence, we need exact values of and .

Expression for trigonometric ratio in right-triangle is; We have already found in (ii) that; Therefore, for this right-triangle;  Pythagorean Theorem      Now we have a right-angled triangle with;             Now we can substitute these values in the equation;   