Dimensional Collateral: Do Two Sines Equal A Cosine?

It is not just irony to say that comparative gages have their greatest accuracy at zero. Even though such a gage could provide a direct reading measurement, it is always best to use it as a comparator. One of the most common sources of error when using a comparative gage over long range is cosine error. For reliable measurement, one must understand cosine error.

Cosine error is most typically seen with test-style indicators and lever type electronic probes doing run-out and concentricity checks on shafts and bores; or, in engineering and tool making, doing checks of parallelism and alignment of flat faces. With a test-style indicator, accuracy is greatest when the axis of the contact point is perpendicular to the measuring direction. This is seldom the case, however, and as the angle of the contact to the surface increases, the amount of vertical distance encompassed (change in height) also increases. The result is cosine error. The following table can be used to correct for this error. “A” is the angle between the probe and the surface of the part

Angle A Correction Factor
5° .996
10° .985
15° .965
20° .940
30° .866

In circumstances where a larger cosine error exists—where the angle of the probe is greater than 30 degrees—it may be better to zero the comparator closer to the actual part size. This will minimize cosine error. To do this, select a zero master that is closer to the calculated reading (based on the angle on the probe) than the actual standard size.

The rule is to try to maintain the probe angle to within ±15 degrees in either direction. Special contacts with a special involute shape manufactured into them are available to help minimize this type of error.

Another place where cosine error can have a negative effect is in a standard benchtop comparator. If the axis of the indicator is out of alignment with the line of measurement on the part, then a cosine error will result. A 1-degree out of alignment condition starts to become noticeable. If the indicator is set with a 2-inch master and a part is placed in the gage, a 0.050-inch deviation will result in a 0.00001-inch error, as follows:

Change in height = X

X = (deviation) × (cos 1)

= (0.050") × (.99985)

= 0.04999”

(deviation) - (change in height) = error

0.050" – 0.04999 = 0.00001"

This may not be serious for most measurement applications, but it becomes important in gage calibration.

These types of cosine errors are the least serious errors caused by gage misalignment. In the example, if a flat contact were used (rather than the normal radius version), then the error becomes a function of the radius of the contact surface. Stated mathematically, if the misalignment is 1 degree, the angle of the contact face to the surface is 1 degree and the diameter of the contact is 0.250 inch (radius 0.125 inch), then:

Error = (radius) × (sin 1°)

= (0.125") × (0.01745)

= 0.0022"

This is more serious than the previous example of cosine error. Such a misalignment in a micrometer or snap gage would repeat this error at each contact surface. This also shows why flat contact points should only be used when absolutely necessary; using a spherical point eliminates part of this error. Because both the contact point and the part being measured are compressible, there is area contact. With area contact comes the potential for sine error.