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Q.
Which of the following pieces of data does NOT uniquely determine an acute-angled triangle $A B C$ ( $R$ being the radius of the circumcircle)?
Trigonometric Functions
Solution:
Applying sine rule in $\triangle A B C$, we get
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin (\pi-A-B)}=2 R$
or $ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin (A+B)}=2 R$
(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.