State whether the following are true or false. Justify your answer.
$(i)$ $\cos A$ is the abbreviation used for the cosecant of angle $A$.
$(ii)$ $\cot A$ is the product of $\cot$ and $A$.
$(iii)$ $\sin \theta = \frac{4}{3}$ for some angle $\theta$.

(NONE) $(i)$ The abbreviation used for the cosecant of angle $A$ is $\text{cosec } A$. The abbreviation $\cos A$ is used for the cosine of angle $A$. Hence, the statement is false.
$(ii)$ $\cot A$ is not the product of $\cot$ and $A$. It represents the cotangent of angle $A$. Hence, the statement is false.
$(iii)$ We know that in a right-angled triangle, $\sin \theta = \frac{\text{Side opposite to } \theta}{\text{Hypotenuse}}$. Since the hypotenuse is the longest side in a right-angled triangle, the value of $\sin \theta$ must always be $\le 1$. Since $\frac{4}{3} > 1$, this value is not possible. Hence, the statement is false.