Question

Calculate the length of AB using Sine rule

Answer 1

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 (BC)/(sin(\angle A)) = (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} & (BC)/(sin(\angle A)) = (AB)/(\sin \angle C) \\ \implies & AB = (BC)/(sin(\angle A)) \cdot \sin \angle C \end{aligned}`.

`\begin{aligned}AB &= (BC)/(sin(\angle A)) \cdot \sin \angle C \\ &= (22.4\; \rm m)/(\sin 58^\circ) * \sin 57^\circ \\ &\approx 22.2\; \rm m\end{aligned}`.