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cosine error

{{short description|Type of measurement error}}

{{See also|Solar tracker}}

Cosine error is a type of measurement error caused by the difference between the intended and actual directions in which a measurement is taken. Depending on the type of measurement, it either multiplies or divides the true value by the cosine of the angle between the two directions.

For small angles the resulting error is typically very small, since an angle needs to be relatively large for its cosine to depart significantly from 1.{{Cite book|last=Bosch|first=John A.|url=https://books.google.com/books?id=YUz5XpLUH9gC&pg=PA182|title=Coordinate Measuring Machines and Systems|date=1995-04-10|publisher=CRC Press|isbn=978-0-8247-9581-8|language=en}}{{Cite web|title=Cosine Error|url=https://dovermotion.com/resources/motion-control-handbook/cosine-error/|access-date=2021-09-25|website=Dover Motion|language=en-US}}

Approximate error sizes for a few example angles are:Calculated directly from the values of the cosines of these angles, which are approximately:

:\cos 10^\circ=0.9848,

:\cos 1^\circ=0.999 848,

:\cos 0.1^\circ=0.999 998 48, and

:\cos 0.01^\circ=0.999 999 984 8.

Although multiplying and dividing by the cosine give slightly different error sizes, the difference is too small to affect the rounded percentages in the table. For example, multiplying by \cos 10^\circ subtracts 1.52%, while dividing by it adds 1.54%.

style="padding-bottom:0.5em;" | Anglestyle="padding-left: 1.2em;padding-bottom:0.5em;" | Error
10°style="padding-left: 1.2em;" | 1.5%style="padding-left: 0.5em;" | = 1 part in 65 or 6665 when dividing by the cosine; 66 when multiplying.
1°style="padding-left: 1.2em;" | 0.015%style="padding-left: 0.5em;" | = 1 part in 6,600
0.1°style="padding-left: 1.2em;" | 0.00015%style="padding-left: 0.5em;" | = 1 part in 660,000
0.01°style="padding-left: 1.2em;" | 0.0000015%style="padding-left: 0.5em;" | = 1 part in 66,000,000

The error is equivalent to treating the hypotenuse and one of the other sides of a right-angled triangle as if they were equal; the cosine of the angle between them is the ratioStrictly, the smaller ratio: the shorter length divided by the longer one. of their lengths.

Concept

A simple example of cosine error is taking a measurement across a rectangle but failing to realize that the line of measurement is not quite parallel with the edges, being slightly diagonal.{{Citation needed|date=September 2021}} Rather than measuring the desired vector (in this case, orthogonal width), the instrument is measuring the hypotenuse of a triangle in which the desired vector is in fact one of the legs. The cosine of this triangle correlates to how much error exists in the measurement (hence the name cosine error).{{Cite journal|last1=Carosell|first1=Philip J.|last2=Coombs|first2=William C.|date=1955|title=Radar Evidence in the Courts|url=https://digitalcommons.du.edu/cgi/viewcontent.cgi?article=4451&context=dlr|journal=Dicta|volume=32|pages=323}}{{Verify source|date=September 2021}}{{Better source needed|date=September 2021}} Thus the user might measure a block of metal and come away with a width of 208.92 mm when the true width is 208.91 mm, a difference that matters to the subsequent machining.

==Examples==

Some practical examples in which the potential for cosine error must be considered include:

  • The use of an indicator (distance amplifying instrument){{Cite AV media|url=https://www.youtube.com/watch?v=dsWSxpwCPUg#t=10m18s|title=Cosine Error Demonstrated and Challenged !|date=17 January 2018|last=Pieczynski|first=Joe|language=en|access-date=25 September 2021}}{{Better source needed|date=September 2021}}
  • Laser interferometry{{Cite book|last=Mekid|first=Samir|url=https://books.google.com/books?id=ClbLBQAAQBAJ&pg=PA42|title=Introduction to Precision Machine Design and Error Assessment|date=2008-12-23|publisher=CRC Press|isbn=978-0-8493-7887-4|language=en}}
  • Speed limit enforcement
  • Lidar traffic enforcement{{Cite web|title=ProLaser 4 OPERATOR'S MANUAL|url=https://www.whatdotheyknow.com/request/342357/response/840504/attach/7/PL%204%20UK%20Operator%20s%20Manual%20V%201.3%20Feb%2016.pdf|access-date=25 September 2021|website=www.whatdotheyknow.com|date=27 June 2016 }}
  • Radar traffic enforcement

Mitigation

The longer the length of the instrument, the easier it is to control cosine error. If the instrument is very small, then optical alignment techniques can be used to reduce cosine error.

References

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Category:Error detection and correction