For a quantity m = πtanθ measured with a small fixed absolute angular uncertainty δθ (radians), the percentage error in m is minimum at θ = (15n)°. Find n.
Physics · Mixed Chapter
JEE Mains

Question

For a quantity m = πtanθ measured with a small fixed absolute angular uncertainty δθ (radians), the percentage error in m is minimum at θ = (15n)°. Find n.

G1 Physics · Accepted Answer

Step 1 — Fractional error: For m = πtanθ, differentiation gives δm/m ≈ [sec²θ/tanθ]δθ. Step 2 — Simplify angular factor: sec²θ/tanθ = 1/(sinθcosθ) = 2/sin2θ. Step 3 — Minimize error for 0°<θ<90°: The factor is minimum when sin2θ is maximum, i.e. 2θ = 90° ⇒ θ = 45°. Step 4 — Match θ = 15n°: 15n = 45 ⇒ n = 3. Answer: 3.