Q4(iii):

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

Given: A trigonometric function $f(\theta) = \cos \theta$, where $\theta$ is an angle in a right-angled triangle, typically considered in the interval $0^\circ \le \theta \le 90^\circ$.

To Determine: Whether the statement "The value of $\cos \theta$ increases as $\theta$ increases" is true or false, with justification.

Step 1: Definition of Cosine
In a right-angled triangle, the cosine of an angle $\theta$ is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse:
$\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}}$

Step 2: Evaluating specific values of $\cos \theta$
To test the statement, we evaluate $\cos \theta$ at standard angles within the range $0^\circ$ to $90^\circ$:

Step 3: Analyzing the trend
Comparing the values calculated in Step 2:
As $\theta$ increases from $0^\circ$ to $90^\circ$:
$1 > 0.866 > 0.707 > 0.5 > 0$
We observe that as the angle $\theta$ increases, the value of $\cos \theta$ decreases.

Step 4: Justification
In a right-angled triangle, as the angle $\theta$ increases, the side adjacent to $\theta$ decreases in length while the hypotenuse remains constant. Since $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$, a decreasing numerator with a constant denominator results in a decreasing value for the fraction.

Conclusion:
Since the value of $\cos \theta$ decreases as $\theta$ increases from $0^\circ$ to $90^\circ$, the given statement is false.

Final Answer: False. The value of $\cos \theta$ decreases as $\theta$ increases in the interval $0^\circ \le \theta \le 90^\circ$.