Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#5

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Question


i.               
Prove the identity .


ii.               
Solve the equation  for .

Solution


i.
 

We have the trigonometric identity;

We can rewrite it in two ways;


ii.
 

To solve the equation  for , we can express, as demonstrated in (i), the right hand side of given equation as;

Therefore the given equation can be written as;

We have the trigonometric relation;

To solve this equation for ,

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) are within the given interval ;

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