Question 11

Let's check each statement about trigonometric ratios.

Step 1 — Evaluate tan A

$\tan A$ has no upper or lower bound — it equals Opposite/Adjacent, and either leg can be longer than the other.

We define $\tan A$ using a right triangle.

$\tan A = \frac{\text{Opposite side}}{\text{Adjacent side}}$

If the opposite side is longer than the adjacent side, then $\tan A$ will be greater than $\mathbf{1}$.

For example, if Opposite = $\mathbf{3}$ and Adjacent = $\mathbf{2}$:

$\tan A = \frac{3}{2}$

$= 1.5$

$\boxed{\text{False}}$

Step 2 — Evaluate sec A

$\sec A = \frac{\text{Hypotenuse}}{\text{Adjacent}}$. Since the hypotenuse is always the longest side in a right triangle, $\sec A$ is always $> 1$.

We define $\sec A$ using a right triangle.

$\sec A = \frac{\text{Hypotenuse}}{\text{Adjacent side}}$

In any right triangle, the hypotenuse is always the longest side.

So, Hypotenuse is always greater than the Adjacent side.

This means $\frac{\text{Hypotenuse}}{\text{Adjacent side}}$ must always be greater than $\mathbf{1}$.

The given value is $\frac{12}{5}$.

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

Since $\mathbf{2.4}$ is greater than $\mathbf{1}$, this value is possible.

$\boxed{\text{True}}$

Step 3 — Abbreviation for cos A

Naming: $\cos A$ stands for cosine of A. Cosecant of A is a different ratio — written as $\csc A$ or $\text{cosec } A$, defined as $\frac{\text{Hyp}}{\text{Opp}}$.

We know that $\cos A$ is the abbreviation for cosine of angle A.

Cosecant of angle A is abbreviated as $\csc A$ or $\text{cosec } A$.

$\boxed{\text{False}}$

Step 4 — Meaning of cot A

Notation: Trigonometric names like $\cot$, $\sin$, $\cos$ are functions — they have no meaning without an angle. $\cot A$ means cotangent of angle A, not "cot" multiplied by A.

We know that $\cot A$ represents the cotangent of angle A.

It is a single trigonometric ratio.

It is not a product of 'cot' and 'A'.

'cot' by itself has no mathematical meaning.

$\boxed{\text{False}}$

Step 5 — Evaluate sin $\theta$

$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$. Since opposite < hypotenuse always, $\sin\theta$ is always strictly between 0 and 1 for acute angles.

We define $\sin \theta$ using a right triangle.

$\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}$

In any right triangle, the hypotenuse is always the longest side.

So, the Opposite side is always shorter than the Hypotenuse.

This means $\frac{\text{Opposite side}}{\text{Hypotenuse}}$ must always be less than $\mathbf{1}$.

The given value is $\frac{4}{3}$.

$\frac{4}{3} \approx 1.33$

Since $\mathbf{1.33}$ is greater than $\mathbf{1}$, this value is not possible for $\sin \theta$.

$\boxed{\text{False}}$

Answer

(i) False (ii) True (iii) False (iv) False (v) False