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 .

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;