Cosine error

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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