Propagation of uncertainties while calculating trigonometric ratios
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If one has an equation such as $$x=-(3.2±0.1)cos(30.3º±0.2º).$$
How does the error carry to be able to find the value of $x$? I have found that you have to -sine the error in the cosine, but then how do you deal with the value by which the scalar is multiplied, and its error?
homework-and-exercises error-analysis
edited Nov 12 at 16:24
Chair
3,59572034
asked Nov 10 at 21:48
John Arg
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up vote
4
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If one has an equation such as $$x=-(3.2±0.1)cos(30.3º±0.2º).$$
How does the error carry to be able to find the value of $x$? I have found that you have to -sine the error in the cosine, but then how do you deal with the value by which the scalar is multiplied, and its error?
homework-and-exercises error-analysis
edited Nov 12 at 16:24
Chair
3,59572034
asked Nov 10 at 21:48
John Arg
234
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
If one has an equation such as $$x=-(3.2±0.1)cos(30.3º±0.2º).$$
How does the error carry to be able to find the value of $x$? I have found that you have to -sine the error in the cosine, but then how do you deal with the value by which the scalar is multiplied, and its error?
homework-and-exercises error-analysis
edited Nov 12 at 16:24
Chair
3,59572034
asked Nov 10 at 21:48
John Arg
234
If one has an equation such as $$x=-(3.2±0.1)cos(30.3º±0.2º).$$
How does the error carry to be able to find the value of $x$? I have found that you have to -sine the error in the cosine, but then how do you deal with the value by which the scalar is multiplied, and its error?
homework-and-exercises error-analysis
homework-and-exercises error-analysis
edited Nov 12 at 16:24
Chair
3,59572034
asked Nov 10 at 21:48
John Arg
234
edited Nov 12 at 16:24
Chair
3,59572034
asked Nov 10 at 21:48
John Arg
234
edited Nov 12 at 16:24
Chair
3,59572034
edited Nov 12 at 16:24
Chair
3,59572034
edited Nov 12 at 16:24
Chair
3,59572034
3,59572034
asked Nov 10 at 21:48
John Arg
234
asked Nov 10 at 21:48
John Arg
234
asked Nov 10 at 21:48
John Arg
234
234
add a comment |
add a comment |
1 Answer
1
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5
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You can use the error propagation formula for the product of two numbers:
$$x=AB$$
$$Rightarrowleft(frac{Delta x}{x}right)^2=left(frac{Delta A}{A}right)^2+left(frac{Delta B}{B}right)^2$$
where $Delta x$ is the error in $x$ etc.
So in your case, you would identify $$A=3.2pm0.1$$ and $$B=cos((30.3pm0.2)^text{o})$$ (note that $Delta B$ is not simply $0.2^text{o}$, you have to work it out, but it seems you are fine with this).
The more general error propagation formula is (for any function $f(A,B,...)$):
$$sigma_f^2=sigma_A^2left(frac{partial f}{partial A}right)^2+sigma_B^2left(frac{partial f}{partial B}right)^2+...$$
which may be used to e.g. find the error in the cosine etc. Note though this assumes $A$ and $B$ are independent.
answered Nov 10 at 21:52
Garf
1,446317
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
accepted
You can use the error propagation formula for the product of two numbers:
$$x=AB$$
$$Rightarrowleft(frac{Delta x}{x}right)^2=left(frac{Delta A}{A}right)^2+left(frac{Delta B}{B}right)^2$$
where $Delta x$ is the error in $x$ etc.
So in your case, you would identify $$A=3.2pm0.1$$ and $$B=cos((30.3pm0.2)^text{o})$$ (note that $Delta B$ is not simply $0.2^text{o}$, you have to work it out, but it seems you are fine with this).
The more general error propagation formula is (for any function $f(A,B,...)$):
$$sigma_f^2=sigma_A^2left(frac{partial f}{partial A}right)^2+sigma_B^2left(frac{partial f}{partial B}right)^2+...$$
which may be used to e.g. find the error in the cosine etc. Note though this assumes $A$ and $B$ are independent.
answered Nov 10 at 21:52
Garf
1,446317
add a comment |
up vote
5
down vote
accepted
You can use the error propagation formula for the product of two numbers:
$$x=AB$$
$$Rightarrowleft(frac{Delta x}{x}right)^2=left(frac{Delta A}{A}right)^2+left(frac{Delta B}{B}right)^2$$
where $Delta x$ is the error in $x$ etc.
So in your case, you would identify $$A=3.2pm0.1$$ and $$B=cos((30.3pm0.2)^text{o})$$ (note that $Delta B$ is not simply $0.2^text{o}$, you have to work it out, but it seems you are fine with this).
The more general error propagation formula is (for any function $f(A,B,...)$):
$$sigma_f^2=sigma_A^2left(frac{partial f}{partial A}right)^2+sigma_B^2left(frac{partial f}{partial B}right)^2+...$$
which may be used to e.g. find the error in the cosine etc. Note though this assumes $A$ and $B$ are independent.
answered Nov 10 at 21:52
Garf
1,446317
add a comment |
up vote
5
down vote
accepted
up vote
5
down vote
accepted
You can use the error propagation formula for the product of two numbers:
$$x=AB$$
$$Rightarrowleft(frac{Delta x}{x}right)^2=left(frac{Delta A}{A}right)^2+left(frac{Delta B}{B}right)^2$$
where $Delta x$ is the error in $x$ etc.
So in your case, you would identify $$A=3.2pm0.1$$ and $$B=cos((30.3pm0.2)^text{o})$$ (note that $Delta B$ is not simply $0.2^text{o}$, you have to work it out, but it seems you are fine with this).
The more general error propagation formula is (for any function $f(A,B,...)$):
$$sigma_f^2=sigma_A^2left(frac{partial f}{partial A}right)^2+sigma_B^2left(frac{partial f}{partial B}right)^2+...$$
which may be used to e.g. find the error in the cosine etc. Note though this assumes $A$ and $B$ are independent.
answered Nov 10 at 21:52
Garf
1,446317
You can use the error propagation formula for the product of two numbers:
$$x=AB$$
$$Rightarrowleft(frac{Delta x}{x}right)^2=left(frac{Delta A}{A}right)^2+left(frac{Delta B}{B}right)^2$$
where $Delta x$ is the error in $x$ etc.
So in your case, you would identify $$A=3.2pm0.1$$ and $$B=cos((30.3pm0.2)^text{o})$$ (note that $Delta B$ is not simply $0.2^text{o}$, you have to work it out, but it seems you are fine with this).
The more general error propagation formula is (for any function $f(A,B,...)$):
$$sigma_f^2=sigma_A^2left(frac{partial f}{partial A}right)^2+sigma_B^2left(frac{partial f}{partial B}right)^2+...$$
which may be used to e.g. find the error in the cosine etc. Note though this assumes $A$ and $B$ are independent.
answered Nov 10 at 21:52
Garf
1,446317
edited Nov 10 at 21:58
answered Nov 10 at 21:52
Garf
1,446317
answered Nov 10 at 21:52
Garf
1,446317
answered Nov 10 at 21:52
Garf
1,446317
1,446317
add a comment |
add a comment |
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