Resultant of two equal coplanar forces: Two equal forces, each of magnitude P, act at a point with an included angle θ between them. Determine the magnitude of their resultant vector.

Introduction / Context:
Vector addition of forces is central to statics and dynamics. For the special case of two equal forces, a compact formula relates the resultant to the half-angle between them, simplifying quick hand calculations in design problems.

Given Data / Assumptions:

• Two coplanar forces of equal magnitude P.
• Included angle between the forces is θ.
• Standard Euclidean vector addition applies.

Concept / Approach:

Using the parallelogram (or triangle) law, the magnitude of the resultant R of two forces P and P at angle θ is R = √(P^2 + P^2 + 2P·P cos θ) = √(2P^2 (1 + cos θ)). Apply the half-angle identity 1 + cos θ = 2 cos^2(θ/2) to obtain a simplified expression.

Step-by-Step Solution:

Verification / Alternative check:

Limit cases: θ = 0 → R = 2P (forces in the same direction). θ = 180° → R = 0 (equal and opposite). Both confirm the formula.

Why Other Options Are Wrong:

Other trigonometric forms do not satisfy the limiting cases or dimensional reasoning; only option (d) matches vector addition.

Common Pitfalls:

Using sin instead of cos; forgetting to halve the angle; misinterpreting θ as the supplement.

Final Answer:

2P cos(θ/2)