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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. It causes the measured value to differ from the true one by a factor equal to the cosine of the angle between the two directions.^[1]^[2]

The resulting error may be very small, since a relatively large angle is needed for its cosine to depart significantly from 1.

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]^[3]^{[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.
^ ^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.

[:1-3] Carosell, Philip J.; Coombs, William C. (1955). "Radar Evidence in the Courts". Dicta. 32: 323.

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

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

[3-6] "ProLaser 4 OPERATOR'S MANUAL" (PDF). www.whatdotheyknow.com. Retrieved 25 September 2021.

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 '''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. It causes the measured value to differ from the true one by a factor equal to the [[cosine]] of the angle between the two directions.<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>
+The resulting error may be [[Small-angle approximation|very small]], since a relatively large angle is needed for its cosine to depart significantly from 1.
-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).
 ==Concept==