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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?

Dando18
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JavaNoob
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  • This is really more subtle than elementary books give it credit for, but most commonly you model something like $\sin(30 \pm 1)$ as $\sin(30) \pm \sin'(30) \cdot 1 = \sin(30) \pm \cos(30) \cdot \frac{\pi}{180} \cdot 1 = \frac{1}{2} \pm \frac{\sqrt{3} \pi}{360}$. (I assume things are in degrees, otherwise this makes little sense.) – Ian Jul 20 '17 at 01:13

4 Answers4

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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.

kimchi lover
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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.

Kaster
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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

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

Stu
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The error can be approximated by the differential of the funcion

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

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

dromastyx
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    In elementary physics books this is not usually assumed to be how errors propagate in multiplication because the errors are (implicitly) assumed to be independent so that there is some sort of cancellation effect when multiplying. – Ian Jul 20 '17 at 01:29
  • Er, to be more correct, I should say that there is cancellation when adding, i.e. errors in added variables combine in a concave manner. But if we additionally assume (as they usually do) that measurement errors are small relative to overall measured values, then multiplication reflects a similar structure (specifically, relative errors combine in a concave manner). – Ian Jul 20 '17 at 22:27
  • @Ian Yes, it appears that when multiplication is involved, a better approximation of the relative error would be the square root of the sum of squares of the fractional errors. – dromastyx Jul 20 '17 at 23:50