# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/13) | Q#8

Question

i.
Show that  can be written as a quadratic equation in  and hence solve the equation  for .

ii.       Find the solutions to the equation  for .

Solution

i.

We have the expression;

We know that ; therefore,

We have the trigonometric identity;

It can be rearranged to;

Therefore;

To solve this equation for , we can substitute . Hence,

Now we have two options;

Since;

Using calculator we can find the values of .

We utilize the periodic property of   to find other solutions (roots) of :

 Symmetry Property

Hence;

 For For

Therefore, we have only two solutions (roots) of the equation in the range of ;

ii.

We have the equation;

As demonstrated in (i) we can write the given equation as;

We can substitute . Hence,

Now we have two options;

Hence;

To solve this equation for , we can substitute . Hence,

Since given interval is  , for  interval can be found as follows;

Multiplying the entire inequality with 2;

Since ;

Hence the given interval for  is .

Using calculator we can find the values of .

We utilize the periodic property of   to find other solutions (roots) of :

 Symmetry Property

Hence;

 For For

Therefore, we have three solutions (roots) of the equation;

To find all the solutions (roots) over the interval , we utilize the periodic property of    for both these values of .

 Periodic Property or

Therefore;

 For For For

For

Hence all the solutions (roots) of the equation  for  are;

Since ;