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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.^[1]^[2]

Approximate error sizes for a few example angles are:^[3]

Angle	Error
10°	1.5%	= 1 part in 65 or 66^[4]
1°	0.015%	= 1 part in 6,600
0.1°	0.00015%	= 1 part in 660,000
0.01°	0.0000015%	= 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 ratio^[5] 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]} 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).^[2]^[6]^{[verification needed]}^{[better source needed]} 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:

Mitigation

The longer the length of the instrument, the easier it is to control cosine error.^[2] If the instrument is very small, then optical alignment techniques can be used to reduce cosine error.^[2]

References

^ Bosch, John A. (1995-04-10). Coordinate Measuring Machines and Systems. CRC Press. ISBN 978-0-8247-9581-8.
^ ^a ^b ^c ^d "Cosine Error". Dover Motion. Retrieved 2021-09-25.
^ Calculated directly from the values of the cosines of these angles, which are approximately:
$\cos 10^{\circ }=0.9848,$

$\cos 1^{\circ }=0.999848,$

$\cos 0.1^{\circ }=0.99999848,$ and

$\cos 0.01^{\circ }=0.9999999848.$
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 1.519%, while dividing by it adds 1.543%.
^ 65 when dividing by the cosine; 66 when multiplying.
^ Strictly, the smaller ratio: the shorter length divided by the longer one.
^ ^a ^b Carosell, Philip J.; Coombs, William C. (1955). "Radar Evidence in the Courts". Dicta. 32: 323.
^ Pieczynski, Joe (17 January 2018). Cosine Error Demonstrated and Challenged !. Retrieved 25 September 2021.
^ Mekid, Samir (2008-12-23). Introduction to Precision Machine Design and Error Assessment. CRC Press. ISBN 978-0-8493-7887-4.
^ "ProLaser 4 OPERATOR'S MANUAL" (PDF). www.whatdotheyknow.com. Retrieved 25 September 2021.

[1] Bosch, John A. (1995-04-10). Coordinate Measuring Machines and Systems. CRC Press. ISBN 978-0-8247-9581-8.

[:0-2] "Cosine Error". Dover Motion. Retrieved 2021-09-25.

[3] Calculated directly from the values of the cosines of these angles, which are approximately:
$\cos 10^{\circ }=0.9848,$

$\cos 1^{\circ }=0.999848,$

$\cos 0.1^{\circ }=0.99999848,$ and

$\cos 0.01^{\circ }=0.9999999848.$
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 1.519%, while dividing by it adds 1.543%.

[4] 65 when dividing by the cosine; 66 when multiplying.

[5] Strictly, the smaller ratio: the shorter length divided by the longer one.

[:1-6] Carosell, Philip J.; Coombs, William C. (1955). "Radar Evidence in the Courts". Dicta. 32: 323.

[7] Pieczynski, Joe (17 January 2018). Cosine Error Demonstrated and Challenged !. Retrieved 25 September 2021.

[8] Mekid, Samir (2008-12-23). Introduction to Precision Machine Design and Error Assessment. CRC Press. ISBN 978-0-8493-7887-4.

[6-9] "ProLaser 4 OPERATOR'S MANUAL" (PDF). www.whatdotheyknow.com. Retrieved 25 September 2021.

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+{{short description|Type of measurement error}}{{See also|Solar tracker}}
-{{See also|solar tracker}}{{More citations needed|date=September 2021}}
+'''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.
-'''Cosine error''' occurs in [[measuring instrument]] readings when the user of an instrument does not realize that the [[euclidean vector|vector]] that an instrument is measuring does not coincide with the vector that the user wishes to measure.<ref>{{Cite book|last=Bosch|first=John A.|url=https://books.google.com/books?id=YUz5XpLUH9gC&newbks=0&printsec=frontcover&pg=PA182&hl=en|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> Often the lack of coincidence is subtle (with vectors ''almost'' coinciding), which is why the user does not notice it (or notices but fails to appreciate its importance). A simple example 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|last=Carosell|first=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]]. Although many workers might not use the term "cosine error" to name this mistake (instead calling it "failing to measure squarely"), the underlying concept is the same. For example, a novice at [[carpentry]] might make this kind of mistake with a [[tape measure]] that is slightly [[Wiktionary:askew#Adjective|askew]], whereas a master carpenter would know by ingrained experience to measure squarely.{{Citation needed|date=September 2021}}
+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>
-A context in which potential cosine error must often be considered is [[indicator (distance amplifying instrument)#Cosine error|in the use of an indicator (distance amplifying instrument)]].<ref>{{Cite Youtube|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}}
+Approximate error sizes for a few example angles are:<ref>Calculated directly from the values of the cosines of these angles, which are approximately:
-Cosine error can also affect [[Interferometry|laser interferometry]].<ref>{{Cite book|last=Mekid|first=Samir|url=https://books.google.com/books?id=ClbLBQAAQBAJ&newbks=0&printsec=frontcover&pg=PA42&hl=en|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>
+:<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 1.519%, while dividing by it adds 1.543%.</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.
-Another context in which potential cosine error draws attention is in [[LIDAR traffic enforcement|lidar traffic enforcement]] and [[radar gun|radar traffic enforcement]], where drivers assert that the speed measurement was in error because the lidar or radar signal was emitted in a direction not directly along the line of travel.<ref name=":2">{{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|url-status=live|access-date=25 September 2021|website=www.whatdotheyknow.com}}</ref> (Cosine error always reduces the measured speed, thus favoring the motorist.<ref name=":2" />) The extent to which it is true that the equipment is prone to this error, {{Citation needed span|text=as opposed to successfully compensating for angles automatically|date=September 2021}}, has been argued in traffic courts.<ref name=":1" /> It is demonstrably true that [[missile guidance|missile-guiding radars]] are capable of accurately measuring the oblique movements of enemy aircraft under a variety of conditions, but to what degree traffic enforcement radar or lidar succeeds at this problem has been challenged by defendants, who speak of the '''cosine effect''' or '''cosine error effect'''.{{Citation needed|date=September 2021}}
-== Mitigation ==
+==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 name=>{{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}}</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" />