{{short description|Type of measurement error}} '''Cosine error''' is a type of [[Observational error|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 [[Small-angle approximation|very small]], since an angle needs to be relatively large for its cosine to depart significantly from 1.<ref>{{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}}</ref><ref name=":0">{{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}}</ref>
Approximate error sizes for a few example angles are:<ref>Calculated directly from the values of the cosines of these angles, which are approximately: :<math>\cos 10^\circ=0.9848, </math> :<math>\cos 1^\circ=0.999 848, </math> :<math>\cos 0.1^\circ=0.999 998 48, </math> and :<math>\cos 0.01^\circ=0.999 999 984 8.</math> 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 <math>\cos 10^\circ</math> subtracts 1.52%, while dividing by it adds 1.54%.</ref> {| |- | style="padding-bottom:0.5em;" | '''Angle''' || style="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 66<ref>65 when dividing by the cosine; 66 when multiplying.</ref> |- | 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 triangle|right-angled triangle]] as if they were equal; the cosine of the angle between them is the ratio<ref>Strictly, the smaller ratio: the shorter length divided by the longer one.</ref> 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, [[orthogonality|orthogonal]] width), the instrument is measuring the [[hypotenuse]] of a triangle in which the desired vector is in fact one of the legs. The [[trigonometric functions#cosine|cosine]] of this triangle correlates to how much error exists in the measurement (hence the name ''cosine error'').<ref name=":0" /><ref name=":1">{{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}}</ref>{{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: *[[indicator (distance amplifying instrument)#Cosine error|The use of an indicator (distance amplifying instrument)]]<ref>{{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}}</ref>{{Better source needed|date=September 2021}} *[[Interferometry|Laser interferometry]]<ref>{{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}}</ref> *[[Speed limit enforcement#Instantaneous speed measurement|Speed limit enforcement]] **[[Lidar traffic enforcement#Limitations|Lidar traffic enforcement]]<ref>{{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 }}</ref> **[[Radar speed gun#Limitations|Radar traffic enforcement]]<ref name=":1" />
==Mitigation== The longer the length of the instrument, the easier it is to control cosine error.<ref name=":0" /> If the instrument is very small, then optical alignment techniques can be used to reduce cosine error.<ref name=":0" />
== References == {{Reflist}}
[[Category:Error detection and correction]]