Question

Calculate the length of AB using Sine rule

Answer 1

Answer:

Approximately $22.2\; \rm m$.

Step-by-step explanation:

By sine rule, the length of each side of a triangle is proportional to the sine value of the angle opposite to that side. For example, in this triangle $\triangle ABC$, angle $\angle A$ is opposite to side $BC$, while $\angle C$ is opposite to side $AB$. By sine rule, $\displaystyle \frac{BC}{\sin{\angle A}} = \frac{AB}{\sin \angle C}$.

It is already given that $BC = 22.4\; \rm m$ and $\angle A = 58^\circ$. The catch is that the value of $\angle C$ needs to be calculated from $\angle A$ and $\angle B$.

The sum of the three internal angles of a triangle is $180^\circ$. In $\triangle ABC$, that means $\angle A + \angle B + \angle C = 180^\circ$. Hence,

$\begin{aligned}\angle C &= 180^\circ - \angle A - \angle B \\ &= 180^\circ - 58^\circ - 65^\circ \\ &= 57^\circ\end{aligned}$.

Apply the sine rule:

$\begin{aligned} & \frac{BC}{\sin{\angle A}} = \frac{AB}{\sin \angle C} \\ \implies & AB = \frac{BC}{\sin{\angle A}} \cdot \sin \angle C \end{aligned}$.

$\begin{aligned}AB &= \frac{BC}{\sin{\angle A}} \cdot \sin \angle C \\ &= \frac{22.4\; \rm m}{\sin 58^\circ} \times \sin 57^\circ \\ &\approx 22.2\; \rm m\end{aligned}$.