Q11(i):

State whether the following are true or false. Justify your answer. (i) The value of $\tan A$ is always less than 1.

Given: A statement regarding the trigonometric ratio $\tan A$, where $A$ is an acute angle in a right-angled triangle.

To Find/Prove: Determine whether the statement "The value of $\tan A$ is always less than 1" is True or False, and provide a mathematical justification.

Visual Representation:

Step 1: Definition of Tangent Ratio

In a right-angled triangle $ABC$ right-angled at $B$, the tangent of angle $A$ is defined as the ratio of the length of the side opposite to angle $A$ to the length of the side adjacent to angle $A$.

Mathematically: $\tan A = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{BC}{AB}$

Step 2: Analyzing the Ratio

In a right-angled triangle, the lengths of the sides $BC$ (opposite) and $AB$ (adjacent) are independent of each other, provided they are both positive real numbers. There is no geometric constraint that forces the opposite side to be smaller than the adjacent side.

Step 3: Counter-Example Verification

Let us consider a specific case where the opposite side is greater than the adjacent side. Suppose $BC = 4$ units and $AB = 3$ units.

Then, $\tan A = \frac{4}{3}$

Performing the division: $\frac{4}{3} = 1.333...$

Since $1.333... > 1$, we have found a case where $\tan A$ is greater than 1.

Step 4: Theoretical Justification

The tangent function $\tan \theta$ is defined for $0^\circ \le \theta < 90^\circ$. As $\theta$ approaches $90^\circ$, the value of $\tan \theta$ increases without bound (tends to infinity). For example, $\tan 60^\circ = \sqrt{3} \approx 1.732$, which is clearly greater than 1.

Conclusion:

Since there exist values of $A$ for which $\tan A > 1$, the statement "The value of $\tan A$ is always less than 1" is incorrect.

Final Answer: False. The value of $\tan A$ can take any real value. For instance, if $\angle A = 60^\circ$, then $\tan 60^\circ = \sqrt{3} \approx 1.732$, which is greater than 1.