Calculating the uncertainty with this measurement

[LCSphysicist]

Homework Statement

Be θ = (59,3 ± 1,2)° an angle, and n=tan(θ) a function. Obtain the uncertainty and the n with the right significant figures

Relevant Equations

\n

It is supposed to find it with the uncertainty propagation, so i tried by it.
$$\delta _{n} = \delta_{\theta}\frac{dn}{d\theta}=\frac{\delta _{\theta}\pi sec(\theta )^{2}}{180} = sec(59,3)^2*\pi*1.2/180$$
$$n = 1.68 \pm 0.08$$
But this uncertainty is quite low, so i am trying to understand if i made any mistake of if it is right.

 

[Gaussian97]

Well, in case you want to double-check your computation you can compute the following quantity:
$$f(x+\delta x)-f(x-\delta x)$$
this quantity should be, approximately, near to $2\delta f$, in your case give $0.16$ so the uncertainty seems plausible.

 

[haruspex]

this uncertainty is quite low

Not really. The uncertainty in the angle was 1 in 50; in n it is higher at 1 in 21.

 

[Steve4Physics]

tan(59.3º - 1.2º) = 1.61
tan(59.3º) = 1.68
tan(59.3º + 1.2º) = 1.77
In good agreement with n = 1.68 ± 0.08