Uncertainty/error calculation (product of number and trig)

Question

I understand how to multiply two numbers with errors together. However, I am unsure how to do a problem when there is trig involved:

$$ (100 \pm 10) \cdot \sin(30 \pm 1) $$

What are the steps I should take to solve it?

Answer 1

Is it right that? If it's not, I'm curious to know why?

$\sin(30+1)= \sin(30)\cos(1)+\sin(1)\cos(30)=\sin(30)\left(\cos(1)+\dfrac{\sin(1)\cos(30)}{\sin(30)}\right)$

$\sin(30-1)= \sin(30)\cos(1)-\sin(1)\cos(30)=\sin(30)\left(\cos(1)-\dfrac{\sin(1)\cos(30)}{\sin(30)}\right)$

and

$\boxed{\scriptstyle 90\sin(30)\left(\cos(1)-\frac{\sin(1)\cos(30)}{\sin(30)}\right)\leq(100 \pm 10) \cdot \sin(30 \pm 1)\leq110\sin(30)\left(\cos(1)+\frac{\sin(1)\cos(30)}{\sin(30)}\right)}$

Answer 2

There is something called propagation of errors invented for your sort of problem: read about it here. (That web page has a chart with a formula for the $sin$ function.) It is more-or-less the cousin of the delta method, but more aimed at practical measurement problems instead of proving theorems.

Answer 3

Hint

Use Taylor series expansing of the function of multiple variables. For example if $$ z(x, y) = x \sin y $$ then linear approximation would be \begin{align} z(x_0 + \Delta x, y_0 + \Delta y) &\approx z(x_0, y_0) + z_x(x_0, y_0)\Delta x + z_y(x_0, y_0) \Delta y = \\ &= x_0 \sin y_0 + \sin y_0 \Delta x + x_0 \cos y_0 \Delta y \end{align} If you want say second order approximation, just use this more compact form \begin{align} z(\mathbf x_0 + \Delta \mathbf x) \approx z(\mathbf x_0) + \Delta \mathbf x \cdot \nabla z(\mathbf x_0) + \Delta \mathbf x \cdot \left[ H(\mathbf x_0) \cdot \Delta \mathbf x\right] \end{align} where $H(\mathbf x)$ is a Hessian of your function $z(\mathbf x)$, and $\mathbf x = [x, y]$.

PS

Angles in trig functions here are in radians.

Answer 4

The error can be approximated by the differential of the funcion

$$f(x,y)=x\sin(y).$$

$$\Delta f\approx \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy=\sin(y)dx+x\cos(y)dy\approx\sin(y)|\Delta x|+x\cos(y)|\Delta y|$$