Question
Which of the following pieces of data does NOT uniquely determine an acute-angled triangle $$ABC$$ ($$R$$ being the radius of the circumcircle) ?
A.
$$a,\sin A,\sin B$$
B.
$$a, b, c$$
C.
$$a,\sin B,R$$
D.
$$a,\sin A,R$$
Answer :
$$a,\sin A,R$$
Solution :
We know by Sine law in $$\Delta ABC\,\,{\text{as}}$$
$$\eqalign{
& \frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin \left( {\pi - A - B} \right)}} = 2R \cr
& \Rightarrow \,\,\frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin \left( {A + B} \right)}} = 2R \cr} $$
(a) If we know $$a, \sin A, \sin B$$ we can find $$b, c ;$$ values of $$\angle 's\,A,B$$ and $$C$$ all.
(b) Using $$a, b, c$$ we can find $$\angle A,\angle B,\angle C$$ using cosine law.
(c) $$a, \sin B, R$$ are given then $$\sin A, b$$ and hence sin$$(A + B)$$ and then $$C$$ can be found.
(d) If we know $$a, \sin A , R$$ then we know only the ratio $$\frac{b}{{\sin B}} = \frac{c}{{\sin \left( {A + B} \right)}};$$ we can not determine the values of $$b, c, \sin B, \sin C$$ separately.
$$∴ \Delta $$ can not be determined in this case.