In the triangle ABD, AC is perpendicular to BD. $\angle ADC = 48^\circ$, $AC = 8.4\text{ cm}$ and $BC = 5\text{ cm}$. Calculate

1. Identify the correct trigonometric ratio for angle $ABC$.

In right-angled triangle $ABC$, relative to angle $ABC$:

• opposite side $= AC = 8.4$
• adjacent side $= BC = 5$

So
$\tan \angle ABC = \frac{AC}{BC} = \frac{8.4}{5}.$

2. Calculate the angle using inverse tangent.

$\angle ABC = \tan^{-1}\left(\frac{8.4}{5}\right)$

$\angle ABC = 59.237\ldots^\circ$

$\angle ABC \approx 59.2^\circ$

1. Identify and set up the correct trigonometric ratio in right triangle $ACD$.

In right-angled triangle $ACD$,
$\sin 48^\circ=\frac{AC}{AD}=\frac{8.4}{AD}.$

2. Rearrange and calculate $AD$.

$AD=\frac{8.4}{\sin 48^\circ}$
$AD=11.303\ldots$
So, to 3 significant figures,
$AD=11.3\text{ cm}.$