State whether the following statements 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) $sin\ θ = \frac{4}{3}$ for some angle.

To do:

We have to state whether the given statements are true or false.

Solution:

(i) $\tan\ A =\frac{side\ opposite\ to\ A}{side\ adjacent\ to\ A}$

$=\frac{BC}{AB}$

In a triangle ABC, $BC$ can be greater than $AB$.

Therefore,

$\frac{BC}{AB}$ can be greater than 1.

This implies the value of $tan\ A$ can be greater than 1.

The given statement is false.  

(ii) $\sec\ A =\frac{Hypotenuse}{side\ adjacent\ to\ A}$

$=\frac{AC}{AB}$

In a triangle, ABC, $AC$(Hypotenuse) is greater than $AB$.

Therefore,

$\frac{AC}{AB}$ is greater than 1.

This implies \( \sec A=\frac{12}{5} \) for some value of angle \( A \).

The given statement is true.

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

\( \cos A \) is the abbreviation used for the cosine of angle \( A . \)

The given statement is false.  

(iv) $\cot\ A =\frac{side\ adjacent\ to\ A}{side\ opposite\ to\ A}$

$=\frac{AB}{BC}$

$cot\ A$ is a single term. It is not a product of cot and A.

The given statement is false.  

(v) Let $\sin \theta=\sin\ A =\frac{side\ opposite\ to\ A}{Hypotenuse}$

$=\frac{BC}{AC}$

In a triangle, ABC, $AC$(Hypotenuse) is greater than $BC$.

Therefore,

$\frac{BC}{AC}$ is less than 1. It is equal to 1 when angle A is equal to zero but never greater than 1.

This implies \( \sin \theta≠\frac{4}{3} \) for any value of angle \( \theta \).

The given statement is false. 

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Updated on: 10-Oct-2022

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