# Cosine error

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Type of measurement error

**Cosine error** is a type of [measurement error](/source/Observational_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](/source/Cosine) of the angle between the two directions.

For small angles the resulting error is typically [very small](/source/Small-angle_approximation), 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](/source/Right_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](/source/Rectangle) but failing to realize that the line of measurement is not quite parallel with the edges, being slightly [diagonal](/source/Diagonal).[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*] Rather than measuring the desired vector (in this case, [orthogonal](/source/Orthogonality) width), the instrument is measuring the [hypotenuse](/source/Hypotenuse) of a triangle in which the desired vector is in fact one of the legs. The [cosine](/source/Trigonometric_functions#cosine) of this triangle correlates to how much error exists in the measurement (hence the name *cosine error*).[2][6][*[verification needed](https://en.wikipedia.org/wiki/Wikipedia:Verifiability)*][*[better source needed](https://en.wikipedia.org/wiki/Wikipedia:Verifiability#Questionable_sources)*] 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](/source/Machining).

## Examples

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

- [The use of an indicator (distance amplifying instrument)](/source/Indicator_(distance_amplifying_instrument)#Cosine_error)[7][*[better source needed](https://en.wikipedia.org/wiki/Wikipedia:Verifiability#Questionable_sources)*]

- [Laser interferometry](/source/Interferometry)[8]

- [Speed limit enforcement](/source/Speed_limit_enforcement#Instantaneous_speed_measurement) - [Lidar traffic enforcement](/source/Lidar_traffic_enforcement#Limitations)[9] - [Radar traffic enforcement](/source/Radar_speed_gun#Limitations)[6]

## 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

1. **[^](#cite_ref-1)** Bosch, John A. (1995-04-10). [*Coordinate Measuring Machines and Systems*](https://books.google.com/books?id=YUz5XpLUH9gC&pg=PA182). CRC Press. [ISBN](/source/ISBN_(identifier)) [978-0-8247-9581-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8247-9581-8).

1. ^ [***a***](#cite_ref-:0_2-0) [***b***](#cite_ref-:0_2-1) [***c***](#cite_ref-:0_2-2) [***d***](#cite_ref-:0_2-3) ["Cosine Error"](https://dovermotion.com/resources/motion-control-handbook/cosine-error/). *Dover Motion*. Retrieved 2021-09-25.

1. **[^](#cite_ref-3)** Calculated directly from the values of the cosines of these angles, which are approximately: 1. cos ⁡ 10 ∘ = 0.9848 , {\displaystyle \cos 10^{\circ }=0.9848,} 1. cos ⁡ 1 ∘ = 0.999848 , {\displaystyle \cos 1^{\circ }=0.999848,} 1. cos ⁡ 0.1 ∘ = 0.99999848 , {\displaystyle \cos 0.1^{\circ }=0.99999848,} and 1. cos ⁡ 0.01 ∘ = 0.9999999848. {\displaystyle \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 ∘ {\displaystyle \cos 10^{\circ }} subtracts 1.52%, while dividing by it adds 1.54%.

1. **[^](#cite_ref-4)** 65 when dividing by the cosine; 66 when multiplying.

1. **[^](#cite_ref-5)** Strictly, the smaller ratio: the shorter length divided by the longer one.

1. ^ [***a***](#cite_ref-:1_6-0) [***b***](#cite_ref-:1_6-1) Carosell, Philip J.; Coombs, William C. (1955). ["Radar Evidence in the Courts"](https://digitalcommons.du.edu/cgi/viewcontent.cgi?article=4451&context=dlr). *Dicta*. **32**: 323.

1. **[^](#cite_ref-7)** Pieczynski, Joe (17 January 2018). [*Cosine Error Demonstrated and Challenged !*](https://www.youtube.com/watch?v=dsWSxpwCPUg#t=10m18s). Retrieved 25 September 2021.

1. **[^](#cite_ref-8)** Mekid, Samir (2008-12-23). [*Introduction to Precision Machine Design and Error Assessment*](https://books.google.com/books?id=ClbLBQAAQBAJ&pg=PA42). CRC Press. [ISBN](/source/ISBN_(identifier)) [978-0-8493-7887-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8493-7887-4).

1. **[^](#cite_ref-9)** ["ProLaser 4 OPERATOR'S MANUAL"](https://www.whatdotheyknow.com/request/342357/response/840504/attach/7/PL%204%20UK%20Operator%20s%20Manual%20V%201.3%20Feb%2016.pdf) (PDF). *www.whatdotheyknow.com*. 27 June 2016. Retrieved 25 September 2021.

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Adapted from the Wikipedia article [Cosine error](https://en.wikipedia.org/wiki/Cosine_error) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Cosine_error?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.