Question

Which of the following pieces of data does not uniquely determine acute angled triangle $ABC$($R=$circum-radius )

• A. $a,\sin A,\sin B$
• B. $a,b,c$
• C. $a,\sin B,R$
• D. $a,\sin A,R$

Answer 1

The correct option is D $a,\sin A,R$

$\because$In a $△ABC,\angle A+\angle B+\angle C=\pi$
$\Rightarrow C=\pi -(A+B)$
Using Sine rule we have
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$
$\Rightarrow \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin [\pi -(A+B)]}=2R$
$\Rightarrow \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin (A+B)}=2R$
Option$\left(a\right)$: If we know $a,\sin A,\sin B$ then we can find $b,c,\angle A,\angle B$ and $\angle C$.
Option$\left(b\right):$ We can find $\angle A,\angle B$ and $\angle C$ using cosine rule.
Option$\left(c\right):a,\sin B,R$ are given then we can find $\sin A,b$ and hence
$\sin C=\sin \left[\pi -(A+B)\right]=\sin C$
Option$\left(d\right):a,\sin A,R$ are given 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.
$\therefore △ABC$ cannot be determined in this case.