Resultant of Two Equal Forces — Magnitude Formula Two equal forces each of magnitude P act at an angle θ to each other. What is the magnitude of their resultant?

Introduction / Context: Vector addition of two forces is a staple of statics and dynamics. For equal magnitudes with angle θ between them, the resultant has a compact expression that is widely used in engineering and physics.

Given Data / Assumptions:

• Force magnitudes: P and P.
• Angle between forces: θ.

Concept / Approach: Use the law of cosines for vectors: R^2 = P^2 + P^2 + 2 P P cos θ. Then take the square root to obtain R. Trigonometric half-angle identities lead to a simple form involving cos(θ/2).

Step-by-Step Solution:

Verification / Alternative check: Special cases: θ = 0 ⇒ R = 2P (collinear same direction). θ = 180° ⇒ R = 0 (equal and opposite), consistent with formula since cos 90° = 0.

Why Other Options Are Wrong: Forms with sin, tan, or cot of θ/2 do not satisfy the cosine-law derivation and fail at boundary checks (e.g., θ = 0 or 180°).

Common Pitfalls: Forgetting the half-angle identity or misapplying parallelogram/triangle construction with wrong included angle.

Final Answer: 2 P cos(θ/2).