1. Geometric Derivation: Consider a right-angled triangle where the other two angles are $45^\circ$ each. In such a triangle, the two legs must be equal in length. Let the length of each leg be $x$.
2. Using Pythagoras Theorem: $$\text{Hypotenuse}^2 = \text{side}_1^2 + \text{side}_2^2$$
$$h^2 = x^2 + x^2 = 2x^2$$
$$h = \sqrt{2x^2} = x\sqrt{2}$$
3. Calculating Sine: By definition, $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$.
For $\theta = 45^\circ$:
$$\sin 45^\circ = \frac{x}{x\sqrt{2}}$$
$$\sin 45^\circ = \frac{1}{\sqrt{2}}$$
Numerically, this value is approximately $0.7071$.