Ex.8.1 Q11 Introduction to Trigonometry Solution - NCERT Maths Class 10

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Question

State whether the following are true or false. Justify your answer.

(i) The value of \(\text{tan}\,A\) is always less than \(1. \)

(ii) \(\begin{align}\text{sec}A=\frac{\text{12}}{\text{5}}\end{align}\) for some value of \(\angle A\).

(iii) \(\text{cos}\,A\) is the abbreviation used for the cosecant of \({\angle A}\)

(iv) \(\text{cot}\, A\) is the product of \(\rm{cot} \) and \({A.}\)

(v) \(\begin{align}\sin \,\theta =\frac{4}{3},\end{align}\) for some \(\begin{align}\angle \theta\end{align}\)

Text Solution

Steps:

(i) False, because sides of a right-angled triangle may have any length. So \(\text{tan}\,A\) may have any value.

(ii) \(\begin{align}\text{sec}\,{A=}\,\frac{\text{hypotenuse}}{\text{side}\ \text{adjacent}\ \text{to}\ \angle {A}}\end{align}\)

As hypotenuse is largest side, the ratio on RHS will be greater than 1. Hence \(\begin{align}\text{sec }\text{A}>1.\end{align}\) Thus, the given statement is true.

(iii) Abbreviation used for cosecant of \( \angle A\) is \(\text{cosec}\,A\) and \(\text{cos}\,A\) is the abbreviation used for cosine of \(\rm \angle A\). Hence the given statement is false.

(iv) \(\text{cot}\,A\) is not the product of \(\rm{cot}\) and \({A.}\) It is the cotangent of \(\rm \angle A\) . Hence, the given statement is false.

(v)\(\begin{align}\text{Sin}\,\theta =\frac{4}{3}\end{align}\)

We know that in a right-angled triangle,

\[\begin{align}\text{Sin}\,\theta =\frac{\text{side}\ \text{adjacent}\ \text{to}\ \angle \theta }{\text{hypotenuse}}\end{align}\]

In a right-angled triangle, hypotenuse is always greater than the remaining two sides. Also, the value of Sine should be less than \(1.\) Therefore, such value of \(\rm{Sin\, \theta}\) is not possible. Hence the given statement is false.