Q4(ii):

State whether the following are true or false. Justify your answer. (ii) The value of $\sin \theta$ increases as $\theta$ increases.

Given: A trigonometric function $f(\theta) = \sin \theta$, where $\theta$ represents an angle in a right-angled triangle.

To Determine: Whether the statement "The value of $\sin \theta$ increases as $\theta$ increases" is True or False, and provide a justification.

Step 1: Definition of Sine Function
In a right-angled triangle, the sine of an angle $\theta$ is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse:
$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$

Step 2: Evaluating $\sin \theta$ at specific angles in the interval $0^\circ \le \theta \le 90^\circ$
We use the standard trigonometric values for common angles:

Step 3: Analyzing the Trend
By observing the values in the table above:
- At $\theta = 0^\circ$, $\sin \theta = 0$
- At $\theta = 30^\circ$, $\sin \theta = 0.5$
- At $\theta = 45^\circ$, $\sin \theta \approx 0.707$
- At $\theta = 60^\circ$, $\sin \theta \approx 0.866$
- At $\theta = 90^\circ$, $\sin \theta = 1$
As $\theta$ increases from $0^\circ$ to $90^\circ$, the value of $\sin \theta$ consistently increases from $0$ to $1$.

Step 4: Geometric Justification
As the angle $\theta$ increases in a right-angled triangle (while keeping the hypotenuse constant), the length of the side opposite to $\theta$ increases. Since $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$, an increase in the numerator (Opposite side) while the denominator (Hypotenuse) remains fixed results in an increase in the overall value of the fraction.

Conclusion:
Since the value of $\sin \theta$ rises as $\theta$ increases from $0^\circ$ to $90^\circ$, the statement is mathematically correct.

Final Answer: True. The value of $\sin \theta$ increases as $\theta$ increases from $0^\circ$ to $90^\circ$.