Errors with sine
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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 3 hours ago
Qmechanic♦
99.1k121781097
asked 4 hours ago
John Arg
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up vote
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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 3 hours ago
Qmechanic♦
99.1k121781097
asked 4 hours ago
John Arg
284
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 3 hours ago
Qmechanic♦
99.1k121781097
asked 4 hours ago
John Arg
284
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 3 hours ago
Qmechanic♦
99.1k121781097
asked 4 hours ago
John Arg
284
edited 3 hours ago
Qmechanic♦
99.1k121781097
asked 4 hours ago
John Arg
284
edited 3 hours ago
Qmechanic♦
99.1k121781097
edited 3 hours ago
Qmechanic♦
99.1k121781097
edited 3 hours ago
Qmechanic♦
99.1k121781097
99.1k121781097
asked 4 hours ago
John Arg
284
asked 4 hours ago
John Arg
284
asked 4 hours ago
John Arg
284
284
add a comment |
add a comment |
1 Answer
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4
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accepted
You can use the error propagation formula for the product of two numbers:
$$x=AB$$
$$Rightarrowleft(fracDelta xxright)^2=left(fracDelta AAright)^2+left(fracDelta BBright)^2$$
where $Delta x$ is the error in $x$ etc.
So in your case, you would identify $$A=3.2pm0.1$$ and $$B=cos((30.3pm0.2)^texto)$$ (note that $Delta B$ is not simply $0.2^texto$, you have to work it out, but it seems you are fine with this).
The more general error propagation formula is (for any function $f(A,B,...)$):
$$sigma_f^2=sigma_A^2left(fracpartial fpartial Aright)^2+sigma_B^2left(fracpartial fpartial Bright)^2+...$$
which may be used to e.g. find the error in the cosine etc. Note though this assumes $A$ and $B$ are independent.
answered 4 hours ago
Garf
1,279217
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
You can use the error propagation formula for the product of two numbers:
$$x=AB$$
$$Rightarrowleft(fracDelta xxright)^2=left(fracDelta AAright)^2+left(fracDelta BBright)^2$$
where $Delta x$ is the error in $x$ etc.
So in your case, you would identify $$A=3.2pm0.1$$ and $$B=cos((30.3pm0.2)^texto)$$ (note that $Delta B$ is not simply $0.2^texto$, you have to work it out, but it seems you are fine with this).
The more general error propagation formula is (for any function $f(A,B,...)$):
$$sigma_f^2=sigma_A^2left(fracpartial fpartial Aright)^2+sigma_B^2left(fracpartial fpartial Bright)^2+...$$
which may be used to e.g. find the error in the cosine etc. Note though this assumes $A$ and $B$ are independent.
answered 4 hours ago
Garf
1,279217
add a comment |
up vote
4
down vote
accepted
You can use the error propagation formula for the product of two numbers:
$$x=AB$$
$$Rightarrowleft(fracDelta xxright)^2=left(fracDelta AAright)^2+left(fracDelta BBright)^2$$
where $Delta x$ is the error in $x$ etc.
So in your case, you would identify $$A=3.2pm0.1$$ and $$B=cos((30.3pm0.2)^texto)$$ (note that $Delta B$ is not simply $0.2^texto$, you have to work it out, but it seems you are fine with this).
The more general error propagation formula is (for any function $f(A,B,...)$):
$$sigma_f^2=sigma_A^2left(fracpartial fpartial Aright)^2+sigma_B^2left(fracpartial fpartial Bright)^2+...$$
which may be used to e.g. find the error in the cosine etc. Note though this assumes $A$ and $B$ are independent.
answered 4 hours ago
Garf
1,279217
add a comment |
up vote
4
down vote
accepted
up vote
4
down vote
accepted
You can use the error propagation formula for the product of two numbers:
$$x=AB$$
$$Rightarrowleft(fracDelta xxright)^2=left(fracDelta AAright)^2+left(fracDelta BBright)^2$$
where $Delta x$ is the error in $x$ etc.
So in your case, you would identify $$A=3.2pm0.1$$ and $$B=cos((30.3pm0.2)^texto)$$ (note that $Delta B$ is not simply $0.2^texto$, you have to work it out, but it seems you are fine with this).
The more general error propagation formula is (for any function $f(A,B,...)$):
$$sigma_f^2=sigma_A^2left(fracpartial fpartial Aright)^2+sigma_B^2left(fracpartial fpartial Bright)^2+...$$
which may be used to e.g. find the error in the cosine etc. Note though this assumes $A$ and $B$ are independent.
answered 4 hours ago
Garf
1,279217
You can use the error propagation formula for the product of two numbers:
$$x=AB$$
$$Rightarrowleft(fracDelta xxright)^2=left(fracDelta AAright)^2+left(fracDelta BBright)^2$$
where $Delta x$ is the error in $x$ etc.
So in your case, you would identify $$A=3.2pm0.1$$ and $$B=cos((30.3pm0.2)^texto)$$ (note that $Delta B$ is not simply $0.2^texto$, you have to work it out, but it seems you are fine with this).
The more general error propagation formula is (for any function $f(A,B,...)$):
$$sigma_f^2=sigma_A^2left(fracpartial fpartial Aright)^2+sigma_B^2left(fracpartial fpartial Bright)^2+...$$
which may be used to e.g. find the error in the cosine etc. Note though this assumes $A$ and $B$ are independent.
answered 4 hours ago
Garf
1,279217
edited 4 hours ago
answered 4 hours ago
Garf
1,279217
answered 4 hours ago
Garf
1,279217
answered 4 hours ago
Garf
1,279217
1,279217
add a comment |
add a comment |
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