Rail diamond geometry: The side length of a rail diamond (formed by two tracks crossing) can be obtained by dividing the gauge of the track by which trigonometric function of the acute crossing angle?

Introduction / Context:
A rail diamond is a rhombus-like assembly where two tracks intersect. Knowing how to compute its side from gauge and crossing angle is fundamental to detailing and setting out.

Given Data / Assumptions:

• Gauge G is the perpendicular spacing between track centerlines (or running faces) as appropriate.
• Acute crossing angle θ is the small angle between the two track centerlines.
• Side of the diamond is measured along one of its four equal edges.

Concept / Approach:
Projecting the gauge onto the direction of the diamond side uses right-triangle relations. The component relationship leads to: side = G / sin θ for the typical geometric construction where the gauge spans the short diagonal direction and the side is inclined at angle θ.

Step-by-Step Solution:
Let diamond side length be S and acute crossing angle be θ.By geometry, G corresponds to S * sin θ.Therefore, S = G / sin θ.

Verification / Alternative check:
Check limiting behavior: as θ increases, sin θ increases and S decreases, which is consistent with a steeper crossing yielding a shorter side for the same gauge.

Why Other Options Are Wrong:

• Cosine / tangent / cotangent / secant do not match the standard projection relation used in diamond layout.

Common Pitfalls:
Mixing up the roles of diagonal vs side; using cos instead of sin due to a misdrawn triangle.

Final Answer:
Sine of the acute crossing angle