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.

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

[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.

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