Given that $\arctan \frac{1}{2} - \arctan \frac{1}{3} = \arctan a,{\text{ }}a \in {\mathbb{Q}^ + }$, find the value of a.

Hence, or otherwise, solve the equation $\arcsin x = \arctan a$.

$\tan \left( {\arctan \frac{1}{2} - \arctan \frac{1}{3}} \right) = \tan (\arctan a)$     (M1)

$a = 0.14285 \ldots  = \frac{1}{7}$     (A1)A1

[3 marks]

$\arctan \left( {\frac{1}{7}} \right) = \arcsin (x) \Rightarrow x = \sin \left( {\arctan \frac{1}{7}} \right) \approx 0.141$     (M1)A1

Note: Accept exact value of $\left( {\frac{1}{{\sqrt {50} }}} \right)$.

[2 marks]

Many candidates failed to give the answer for (a) in rational form. The GDC can render the answer in this form as well as the decimal approximation, but this was obviously missed by many candidates.

(b) was generally answered successfully.