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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 $\theta$.
Solution
$(iii)$ Abbreviation used for cosecant of angle $A$ is cosec $A$. And $\cos A$ is the abbreviation used for cosine of angle $A$
Hence, the given statement is false.
$(iv)$ cot $A$ is not the product of cot and $A$. It is the cotangent of $\angle A$.
Hence, the given statement is false.
$(v)$ $\sin \theta=\frac{4}{3}$
We know that in a right-angled triangle,
$\sin \theta=\frac{\text { Side opposite to } \angle \theta}{\text { Hypotenuse }}$
In a right-angled triangle, hypotenuse is always greater than the remaining two sides. Therefore, such value of $\sin \theta$ is not possible.
Hence, the given statement is false
