**Question:**

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

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

(ii) $\sec A=\frac{12}{5}$ for some value of angle $\mathrm{A}$.

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

(iv) cot A is the product of cot and A.

(v) $\sin \theta=\frac{4}{3}$ for some angle $\theta$.

**Solution:**

(i) In $\tan A, \angle A$ is acute an angle

Therefore,

Minimum value of $\angle A$ is $0^{\circ}$ and

Maximum value of $\angle A$ is $90^{\circ}$

We know that $\tan 0^{\circ}=0$ and

$\tan 90^{\circ}=\infty$

Therefore the statement that;

"The value of $\tan A$ is always less than 1 " is false

(ii) $\sec A=\frac{1}{\cos A}$

In $\sec A$ and $\cos A, \angle A$ is acute angle

Therefore,

Minimum value of $\angle A$ is $0^{\circ}$ and

Maximum value of $\angle A$ is $90^{\circ}$

$\cos 90^{\circ}=0$

Now,

$\sec 0^{\circ}=\frac{1}{\cos 0^{\circ}}$

$=\frac{1}{1}$

$=1$

Therefore minimum value of $\sec A$ is $\sec 0^{\circ}=1$.....(1)

Now,

$\sec 90^{\circ}=\frac{1}{\cos 90^{\circ}}$

$=\frac{1}{0}$

$=\infty$

Therefore maximum value of $\sec A$ is $\sec 90^{\circ}=\infty$.....(2)

Now consider the given value

$\sec A=\frac{12}{5}$

Here, $\frac{12}{5}=2.4$

This value $2.4$ lies in between 1 and $\infty$

Now from equation (1) and $(2)$, we can say that the value $\frac{12}{5}=2.4$ lies in between minimum value of $\sec A$ (that is 1 ) and maximum value of sec $A$ (that is $\infty$ )

Hence, $\sec A=\frac{12}{5}$, for some value of angle $\mathrm{A}$ is true

(iii) Cosecant of angle $\mathrm{A}$ is defined as $\operatorname{cosec} A=\frac{1}{\sin A}$

Also, $\sin A$ is defined as $\sin A=\frac{\text { Perpendicular side opposite to } \angle A}{\text { Hypotenuse }}$

Therefore,

$\operatorname{cosec} A=\frac{\text { Hypotenuse }}{\text { Perpendicular side opposite to } \angle A}$.....(1)

And

$\cos A$ is defined as $\cos A=\frac{\text { Base side adjacent to } \angle A}{\text { Hypotenuse }}$......(2)

Therefore from equation (1) and (2), it is clear that $\cos A$ and $\operatorname{cosec} A$ (that is $\operatorname{cosecant}$ of angle A) are two different trigonometric angles

Hence, $\cos A$ is the abbreviation used for cosecant of angle $\mathrm{A}$ is False

(iv) cot A is a trigonometric ratio which means cotangent of angle A

Hence, $\cot A$ is the product of $\cot$ and $\mathrm{A}$ is False

(v) $\sin \theta=\frac{4}{3}$

The value $\frac{4}{3}=1.333$

In $\sin \theta, \angle \theta$ is acute an angle

Therefore,

Minimum value of $\angle \theta$ is $0^{\circ}$ and

Maximum value of $\angle \theta$ is $90^{\circ}$

We know that $\sin 0^{\circ}=0$ and

$\sin 90^{\circ}=1$

Therefore the value of $\sin \theta$ should lie between 0 and 1 and must not exceed 1

Hence the given value for $\sin \theta$ (that is $\frac{4}{3}=1.333$ ) is not possible

Therefore, $\sin \theta=\frac{4}{3}$, for some angle $\theta=$ False

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