Solution :
Given that in the right triangle , ∠A=30° and AB = 12√3 .
As the figure is missing and its not clearly mentioned that AB is the base or hypotenuse of the right triangle. So two cases arises-
Case 1: AB is the base for ∠A of the right triangle (as shown in figure 1).
As we know from the trigonometric ratio that, [tex]cos(\theta) = \frac{base}{hypotenuse}[/tex]
Here , AB is the base and AC is the hypotenuse , and ∠A=30°
[tex]\Rightarrow cos(30)=\frac{AB}{AC} \\\\\Rightarrow AC=\frac{AB}{cos(30)}[/tex]
The value of [tex]cos(30)=\frac{\sqrt{3} }{2}[/tex]
[tex]\Rightarrow AC=12\sqrt{3}\times\frac{2 }{\sqrt{3} }\\\\\Rightarrow AC=24[/tex]
Hence, AC is 24 unit long.
Case 2: AB is the hypotenuse for ∠A of the right triangle (as shown in figure 2).
As we know from the trigonometric ratio that, [tex]cos(\theta) = \frac{base}{hypotenuse}[/tex]
Here , AB is the hypotenuse and AC is the base, and ∠A=30°
[tex]\Rightarrow cos(30)=\frac{AC}{AB} \\\\\Rightarrow AC=AB\timescos(30)[/tex]
The value of [tex]cos(30)=\frac{\sqrt{3} }{2}[/tex]
[tex]\Rightarrow AC=12\sqrt{3}\times\frac{\sqrt{3}}{2}\\\\\Rightarrow AC=18[/tex]
Hence, AC is 18 unit long.
