Mixing
Absolute error and inverse trig functions
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I am doing a physics project, and am using the sine law to find the angle of my final momentum vector. Using absolute error arithmetic (i.e. add absolute error when adding/subtracting, add relative error when multiplying/dividing), I have arrived at the expression
$$sin alpha = 0.503 pm 0.08.$$
How do I find the absolute error $x$ for my final value of $alpha = 30^circ pm x$? If there is a rule for this (which I cannot seem to find on the internet), does it hold for logarithms, normal trig functions, etc.?
trigonometry mathematical-physics error-propagation
asked Feb 2 at 21:58
A. LavieA. Lavie
274
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add a comment |
$begingroup$
I am doing a physics project, and am using the sine law to find the angle of my final momentum vector. Using absolute error arithmetic (i.e. add absolute error when adding/subtracting, add relative error when multiplying/dividing), I have arrived at the expression
$$sin alpha = 0.503 pm 0.08.$$
How do I find the absolute error $x$ for my final value of $alpha = 30^circ pm x$? If there is a rule for this (which I cannot seem to find on the internet), does it hold for logarithms, normal trig functions, etc.?
trigonometry mathematical-physics error-propagation
asked Feb 2 at 21:58
A. LavieA. Lavie
274
$endgroup$
add a comment |
$begingroup$
I am doing a physics project, and am using the sine law to find the angle of my final momentum vector. Using absolute error arithmetic (i.e. add absolute error when adding/subtracting, add relative error when multiplying/dividing), I have arrived at the expression
$$sin alpha = 0.503 pm 0.08.$$
How do I find the absolute error $x$ for my final value of $alpha = 30^circ pm x$? If there is a rule for this (which I cannot seem to find on the internet), does it hold for logarithms, normal trig functions, etc.?
trigonometry mathematical-physics error-propagation
asked Feb 2 at 21:58
A. LavieA. Lavie
274
$endgroup$
I am doing a physics project, and am using the sine law to find the angle of my final momentum vector. Using absolute error arithmetic (i.e. add absolute error when adding/subtracting, add relative error when multiplying/dividing), I have arrived at the expression
$$sin alpha = 0.503 pm 0.08.$$
How do I find the absolute error $x$ for my final value of $alpha = 30^circ pm x$? If there is a rule for this (which I cannot seem to find on the internet), does it hold for logarithms, normal trig functions, etc.?
trigonometry mathematical-physics error-propagation
trigonometry mathematical-physics error-propagation
asked Feb 2 at 21:58
A. LavieA. Lavie
274
asked Feb 2 at 21:58
A. LavieA. Lavie
274
asked Feb 2 at 21:58
A. LavieA. Lavie
274
asked Feb 2 at 21:58
A. LavieA. Lavie
274
asked Feb 2 at 21:58
A. LavieA. Lavie
274
274
add a comment |
add a comment |
1 Answer
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We have $alpha=arcsin{(0.503 pm 0.08)}$ from your original statement. As $arcsin(x)$ is an increasing function we can then write $arcsin{(0.503 - 0.08)} le alpha le arcsin{(0.503 + 0.08)}$ or approximately, $25.0^circ le alpha le 35.7^circ$. We can calculate the uncertainty of $alpha$ = $frac{1}{2} times$ range = $frac{1}{2} times (35.7-25.0)=5.32^circ$. So approximately we have:
$$alpha=30.3^circ pm 5.32^circ$$
In general for an increasing function $f(x)$, if we have $f^{-1}(x)=apm b$, then
$$x=frac{f(a+b)+f(a-b)}{2} pm frac{|f(a+b)-f(a-b)|}{2}$$
answered Feb 2 at 22:12
Peter ForemanPeter Foreman
7,3831319
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1 Answer
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1 Answer
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$begingroup$
We have $alpha=arcsin{(0.503 pm 0.08)}$ from your original statement. As $arcsin(x)$ is an increasing function we can then write $arcsin{(0.503 - 0.08)} le alpha le arcsin{(0.503 + 0.08)}$ or approximately, $25.0^circ le alpha le 35.7^circ$. We can calculate the uncertainty of $alpha$ = $frac{1}{2} times$ range = $frac{1}{2} times (35.7-25.0)=5.32^circ$. So approximately we have:
$$alpha=30.3^circ pm 5.32^circ$$
In general for an increasing function $f(x)$, if we have $f^{-1}(x)=apm b$, then
$$x=frac{f(a+b)+f(a-b)}{2} pm frac{|f(a+b)-f(a-b)|}{2}$$
answered Feb 2 at 22:12
Peter ForemanPeter Foreman
7,3831319
$endgroup$
add a comment |
$begingroup$
We have $alpha=arcsin{(0.503 pm 0.08)}$ from your original statement. As $arcsin(x)$ is an increasing function we can then write $arcsin{(0.503 - 0.08)} le alpha le arcsin{(0.503 + 0.08)}$ or approximately, $25.0^circ le alpha le 35.7^circ$. We can calculate the uncertainty of $alpha$ = $frac{1}{2} times$ range = $frac{1}{2} times (35.7-25.0)=5.32^circ$. So approximately we have:
$$alpha=30.3^circ pm 5.32^circ$$
In general for an increasing function $f(x)$, if we have $f^{-1}(x)=apm b$, then
$$x=frac{f(a+b)+f(a-b)}{2} pm frac{|f(a+b)-f(a-b)|}{2}$$
answered Feb 2 at 22:12
Peter ForemanPeter Foreman
7,3831319
$endgroup$
add a comment |
$begingroup$
We have $alpha=arcsin{(0.503 pm 0.08)}$ from your original statement. As $arcsin(x)$ is an increasing function we can then write $arcsin{(0.503 - 0.08)} le alpha le arcsin{(0.503 + 0.08)}$ or approximately, $25.0^circ le alpha le 35.7^circ$. We can calculate the uncertainty of $alpha$ = $frac{1}{2} times$ range = $frac{1}{2} times (35.7-25.0)=5.32^circ$. So approximately we have:
$$alpha=30.3^circ pm 5.32^circ$$
In general for an increasing function $f(x)$, if we have $f^{-1}(x)=apm b$, then
$$x=frac{f(a+b)+f(a-b)}{2} pm frac{|f(a+b)-f(a-b)|}{2}$$
answered Feb 2 at 22:12
Peter ForemanPeter Foreman
7,3831319
$endgroup$
We have $alpha=arcsin{(0.503 pm 0.08)}$ from your original statement. As $arcsin(x)$ is an increasing function we can then write $arcsin{(0.503 - 0.08)} le alpha le arcsin{(0.503 + 0.08)}$ or approximately, $25.0^circ le alpha le 35.7^circ$. We can calculate the uncertainty of $alpha$ = $frac{1}{2} times$ range = $frac{1}{2} times (35.7-25.0)=5.32^circ$. So approximately we have:
$$alpha=30.3^circ pm 5.32^circ$$
In general for an increasing function $f(x)$, if we have $f^{-1}(x)=apm b$, then
$$x=frac{f(a+b)+f(a-b)}{2} pm frac{|f(a+b)-f(a-b)|}{2}$$
answered Feb 2 at 22:12
Peter ForemanPeter Foreman
7,3831319
answered Feb 2 at 22:12
Peter ForemanPeter Foreman
7,3831319
answered Feb 2 at 22:12
Peter ForemanPeter Foreman
7,3831319
answered Feb 2 at 22:12
Peter ForemanPeter Foreman
7,3831319
7,3831319
add a comment |
add a comment |
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