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
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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
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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 |
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Periodicity |
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Odd/Even |
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Translation/ Symmetry |
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We utilize the periodicity/symmetry property of to find other solutions (roots) of
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Therefore;
For;
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Only following solutions (roots) are within the given interval ;
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iii.
We are required to find the exact value of , giving your answer as a fraction.
Therefore;
Hence, we need exact values of and
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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;
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Now we can substitute these values in the equation;