Q4(iv):

State whether the following are true or false. Justify your answer. (iv) $\sin \theta = \cos \theta$ for all values of $\theta$.

Given: A statement $\sin \theta = \cos \theta$ for all values of $\theta$, where $\theta$ is an angle in a right-angled triangle.

To Find/Prove: Determine whether the given statement is True or False and provide a justification.

Step 1: Analyzing the definitions of Sine and Cosine

In a right-angled triangle $ABC$ (right-angled at $B$), for an acute angle $\theta$ at vertex $A$:

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

$\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}} = \frac{AB}{AC}$

Step 2: Testing the equality for specific values of $\theta$

The statement claims $\sin \theta = \cos \theta$ for all values of $\theta$. To disprove this, we only need to find one counter-example.

Let $\theta = 0^\circ$:

$\sin 0^\circ = 0$ [From trigonometric table values]

$\cos 0^\circ = 1$ [From trigonometric table values]

Since $0 \neq 1$, the statement $\sin \theta = \cos \theta$ is false for $\theta = 0^\circ$.

Let $\theta = 30^\circ$:

$\sin 30^\circ = \frac{1}{2} = 0.5$

$\cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.866$

Since $0.5 \neq 0.866$, the statement is false for $\theta = 30^\circ$.

Step 3: Identifying the condition where the statement holds

The equation $\sin \theta = \cos \theta$ is only true when $\frac{\sin \theta}{\cos \theta} = 1$, which implies $\tan \theta = 1$.

We know that $\tan 45^\circ = 1$. Therefore, $\sin \theta = \cos \theta$ only when $\theta = 45^\circ$ (within the range $0^\circ \le \theta \le 90^\circ$).

Conclusion:

Since the equality does not hold for all values of $\theta$ (e.g., it fails at $\theta = 0^\circ$ and $\theta = 30^\circ$), the statement is False.

Final Answer: False. The statement $\sin \theta = \cos \theta$ is only true when $\theta = 45^\circ$, not for all values of $\theta$.