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 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 θ.

Answer 1

(i) In, is acute an angle

Therefore,

Minimum value of is 0° and

Maximum value of is 90°

We know that and

tan90° = ∞

Therefore the statement that;

“The value of is always less than1” is false

(ii)

In and, is acute angle

Therefore,

Minimum value of is 0°and

Maximum value of is

We know that cos0° = 1 and

cos90° = 0

Now,

Therefore minimum value of is …… (1)

Now,

Therefore maximum value of is …… (2)

Now consider the given value

Here,

This value 2.4 lies in between 1 and

Now from equation (1) and (2) , we can say that the value lies in between minimum value of (that is 1) and maximum value of (that is)

Hence, , for some value of angle A is true

(iii) Cosecant of angle A is defined as

Also, is defined as

Therefore,

…… (1)

And

is defined as …… (2)

Therefore from equation (1) and (2) , it is clear that and (that is cosecant of angle A) are two different trigonometric angles

Hence, is the abbreviation used for cosecant of angle A is False

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

Hence, is the product of cot and A is False

(v)

The value

In, is acute an angle

Therefore,

Minimum value of is 0° and

Maximum value of is 90°

We know that and

sin90° = 1

Therefore the value of should lie between 0 and 1 and must not exceed 1

Hence the given value for (that is) is not possible

Therefore, , for some angle = False