Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2002  OctNov  (P29709/02)  Q#5
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 righttriangle is;
We have already found in (ii) that;
Therefore, for this righttriangle;
Pythagorean Theorem
Now we have a rightangled triangle with;










Now we can substitute these values in the equation;