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 1

Answer: (D)

Applying sine rule in $△ABC$, we get
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin (\pi -A-B)}=2R$
or $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin (A+B)}=2R$
(a) If we know $a,\sin A,\sin B$, we can find $b,c$, and the value of angles $A,B$, and $C$.
(b) Using $a,b,c$, we can find $\angle A,\angle B,\angle C$ using the cosine law.
(c) $a,\sin B,R$ are given, so $\sin A,b$ 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 $\frac{b}{\sin B}$ or $\frac{c}{\sin (A+B)}$; we cannot determine the values of $b,c,\sin B,\sin C$ separately. Therefore, the triangle cannot be determined in this case.