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.
Approximate error sizes for a few example angles are:Calculated directly from the values of the cosines of these angles, which are approximately:
and
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 subtracts 1.52%, while dividing by it adds 1.54%. Angle Error 10° 1.5% = 1 part in 65 or 6665 when dividing by the cosine; 66 when multiplying. 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 ratioStrictly, the smaller ratio: the shorter length divided by the longer one. 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. 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). 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: The use of an indicator (distance amplifying instrument) Laser interferometry Speed limit enforcement Lidar traffic enforcement Radar traffic enforcement
Mitigation The longer the length of the instrument, the easier it is to control cosine error. If the instrument is very small, then optical alignment techniques can be used to reduce cosine error.
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