NCERT Solutions
Class 10 - Mathematics - Chapter 8: INTRODUCTION TO TRIGONOMETRY - Exercise 8.1

Question 11

We will check each statement using basic facts of trigonometry and write a clear justification.

(i) “The value of \(\tan A\) is always less than 1.”

False. \(\tan A\) can be less than 1, equal to 1, or greater than 1 depending on the angle.

Example: if \(A = 45^\circ\), then \(\tan 45^\circ = 1\) (not less than 1).

Also, if \(A = 60^\circ\), then \(\tan 60^\circ = \sqrt{3} \approx 1.732\), which is greater than 1. So it is not always less than 1.

(ii) “\(\sec A = \dfrac{12}{5}\) for some value of angle \(A\).”

True. We know \(\sec A = \dfrac{1}{\cos A}\). So \(\sec A = \dfrac{12}{5}\) means:

\(\cos A = \dfrac{5}{12}\).

This is possible because for an acute angle, \(\cos A\) can take any value between 0 and 1, and \(\dfrac{5}{12}\) lies between 0 and 1.

Student Note: A quick check: in a right triangle, \(\cos A = \dfrac{\text{adjacent}}{\text{hypotenuse}}\). Taking adjacent = 5 and hypotenuse = 12 is valid (hypotenuse is bigger), so such an angle \(A\) can exist.

(iii) “\(\cos A\) is the abbreviation used for the cosecant of angle \(A\).”

False. \(\cos A\) means cosine of angle \(A\).

The abbreviation for cosecant is \(\text{cosec } A\), not \(\cos A\).

(iv) “\(\cot A\) is the product of cot and \(A\).”

False. \(\cot A\) is a single trigonometric ratio called cotangent of angle \(A\).

It does not mean (cot) × (A). It is like \(\sin A\) or \(\tan A\): a function value, not a multiplication.

(v) “\(\sin \theta = \dfrac{4}{3}\) for some angle \(\theta\).”

False. The value of \(\sin\theta\) for any real angle always lies between \(-1\) and \(+1\), i.e.,

\(-1 \le \sin\theta \le 1\).

But \(\dfrac{4}{3} \approx 1.333\), which is greater than 1, so it is not possible for any angle.

Student Note: The same idea is true for \(\cos\theta\) as well (it also lies between \(-1\) and \(+1\)).