= Solution
The central value is $S=150\cos45^\circ=106.07$. Treating the stated errors as <independent> <standard uncertainties>, convert the angular uncertainty to radians and propagate derivatives:
$$
\sigma_S^2=(\cos2\theta\,\sigma_{S_0})^2+
(2S_0\sin2\theta\,\sigma_\theta)^2,
\qquad
\boxed{S=106.1\pm6.5.}
$$
The two measured values differ by $10.8/\sqrt{6.0^2+5.6^2}=1.32$ combined <standard deviations>. Our prediction differs from them by $0.19$ and $1.07$ combined <standard deviations> respectively. \b[All three values are statistically compatible.] If the input errors are strict bounds instead, the first-order bound on the calculated value is about $\pm9.0$; consistency remains possible, but a Gaussian significance test then has no stated justification.
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