Past exam of the physics A course of the University of Cambridge / 2011 / Paper 2 / Section A / 4

Solution

The central estimate is

z = 3.2743

. propagation of uncertainty must include the occurrence of

x

in both numerator and denominator. With angular derivatives measured per radian,

z_{x} = \frac{s e c ^{2} ( x + y )}{t a n x} - \frac{t a n ( x + y )}{s i n ^{2} x} = - 3.5382, z_{y} = \frac{s e c ^{2} ( x + y )}{t a n x} = 6.6497.

Interpreting the two quoted errors as independent standard deviations, convert them to radians before calculating

σ_{z}^{2} = z_{x}^{2} (4 π /180)^{2} + z_{y}^{2} (3 π /180)^{2}, z ≃ 3.27 \pm 0.43 .

If the quoted errors are instead absolute bounds, the first-order worst-case estimate is

∣ δz ∣ ≲ ∣ z_{x} ∣ δ x + ∣ z_{y} ∣ δy = 0.60

. Correlated errors add

2 z_{x} z_{y} Cov (x, y)

to the variance. Treating

x + y

and

x

as independent would lose their shared measurement uncertainty.

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