Note: All working is performed with angles in radian measure.

(a)    Consider the statement:  If \(\tan \left( A \right) = \tan \left( B \right)\), then \(A = B\).

(i)     Disprove this statement by finding a counterexample.

(ii)    If \(A\) and \(B\) are angles in the same triangle, then explain why the statement in (a) is true.

(b)    The figure below shows a rectangle that measures 3 units by 1 unit with one vertex at \(\left( {0,0} \right)\). 

         Three line segments with lengths \(a\), \(b\) and \(c\) are drawn from the rectangle’s vertex at \(\left( {0,1} \right)\) to the       

         rectangle’s side on the \(x\)-axis forming three acute angles \(\alpha \) (alpha), \(\beta\) (beta) and \(\gamma\) (gamma). 

(i)     Calculate the radian measure of \(\alpha\), \(\beta\) and \(\gamma\) to an accuracy of four significant figures.

(ii)    Hence, suggest a mathematical statement relating the measures of \(\alpha\), \(\beta\) and \(\gamma\).

(iii)   Given your explanation in (a) (ii), prove your statement.