State whether the following are true or false. Justify your answer.
i) The value of $\tan A$ is always less than $1$.
ii) For some value of angle $A$, $\sec A=\dfrac{12}{5}$.
iii) $\cos A$ is the abbreviation used for the cosecant of angle $A$.
iv) $\cot A$ is the product of $\cot $ and $A$.
v) For some angle $\theta $, $\sin \theta =\dfrac{4}{3}$.
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Ans: The given statement is false. The value of $\tan A$ depends on the length of sides of a right triangle and sides of a triangle may have any measure.
Ans: We know that in the right triangle $\sec A=\dfrac{\text{hypotenuse}}{\text{adjacent side of }\angle \text{A}}$ .
We know that in the right triangle the hypotenuse is the largest side.
Therefore, the value of $\sec A$ must be greater than $1$.
In the given statement $\sec A=\dfrac{12}{5}$, which is greater than $1$.
Therefore, the given statement is true.
Ans: The given statement is false because $\cos A$ is the abbreviation used for the cosine of angle $A$. Abbreviation used for the cosecant of angle $A$ is $\operatorname{cosec}A$.
Ans: $\cot A$ is the abbreviation used for the cotangent of angle $A$. Hence the given statement is false.
Ans: We know that in the right triangle $\sin \theta =\dfrac{\text{opposite side}}{\text{hypotenuse}}$ .
We know that in the right triangle the hypotenuse is the largest side.
Therefore, the value of $\sin \theta $ must be less than $1$.
In the given statement $\sin \theta =\dfrac{4}{3}$, which is greater than $1$.
Therefore, the given statement is false.