Angular Measurement Error In Trigonometric Models

Why Angular Uncertainty Needs Special Care

A small uncertainty in an angle can produce a small or substantial change in a trigonometric model, depending on the measured angle and the function being used. For instance, the sensitivity of sine changes with the angle, while the sensitivity of tangent becomes very large near its vertical asymptotes.

The central calculation is the same local error-propagation principle used for other functions: multiply the input uncertainty by the magnitude of the derivative evaluated at the measured input. For trigonometric models, however, one condition is essential: the angle used in the derivative must be measured in radians.

Convert Angular Uncertainty to Radians

The standard derivative formulas for sine, cosine, and tangent assume that their angle inputs are measured in radians. Degrees can still be used to report a measured angle, but convert both the angle and its uncertainty before applying those formulas:

For example, an uncertainty of \(0.2^\circ\) is not \(0.2\) in the derivative calculation. In radians it is \(0.2\pi/180=\pi/900\approx 0.003491\).

An angle and its uncertainty are different quantities. The angle \(\theta_0\) is the measured input at which sensitivity is evaluated; \(u_\theta\) describes how far that input may be from its true value. If the measurement is stated as \(\theta_0\pm u_\theta\), the first-order estimate is made around \(\theta_0\).

First-Order Propagation for Trigonometric Models

Let \(f\) be differentiable at the measured angle \(\theta_0\). For a small signed angular error \(\Delta\theta\), linearization gives \(f(\theta_0+\Delta\theta)\approx f(\theta_0)+f'(\theta_0)\Delta\theta\). Thus the signed predicted output change is \(f'(\theta_0)\Delta\theta\). If only the size of the input uncertainty is known, use its magnitude to estimate the output uncertainty.

Here \(u_y\) is the estimated absolute uncertainty in the output. Since \(\sec^2\theta_0\) is positive wherever tangent is defined, no absolute-value signs are needed around that factor. The trigonometric outputs are dimensionless; the derivatives describe output change per radian.

Why the Derivative Gives the Estimate

The estimate follows directly from the definition of the derivative. For an increment \(\Delta\theta\), differentiability at \(\theta_0\) means

where the remainder satisfies \(\lim_{\Delta\theta\to 0}r(\Delta\theta)/\Delta\theta=0\). As the angular error becomes small, the remainder becomes small relative to \(\Delta\theta\). The leading change is therefore \(f'(\theta_0)\Delta\theta\). Taking magnitudes gives the first-order estimate \(\lvert f'(\theta_0)\rvert\lvert\Delta\theta\rvert\), and replacing the unknown error size by its uncertainty \(u_\theta\) gives the rule above.

Worked Examples

Signed Change, Sensitivity, and Limits

The signed derivative indicates the direction of the predicted change when the angle increases. Cosine has a negative derivative where \(\sin\theta>0\), so an increase in angle there predicts a decrease in cosine. For an uncertainty reported only as a nonnegative size, use the absolute value of the derivative to give an output-uncertainty magnitude.

A zero derivative means that the first-order estimate is zero; it does not guarantee that the output is unchanged. For example, at \(\theta_0=\pi/2\), the derivative of sine is \(\cos(\pi/2)=0\). Yet for a small nonzero \(h\), \(\sin(\pi/2+h)=\cos h\), which generally differs from \(1\). In this situation, the first-order estimate misses the leading nonzero change.

The tangent model needs additional care. Tangent is undefined at \(\theta=\pi/2+k\pi\), where \(k\) is an integer, and its derivative grows without bound as an angle approaches one of these values. The measured angle and the plausible range of angles must stay away from these asymptotes for a local estimate to be meaningful. Even away from an asymptote, a large derivative signals that a small angular uncertainty may create a relatively large output uncertainty.

Common Mistake: Using Degrees in the Derivative

Suppose the angle is \(45^\circ\) with uncertainty \(0.2^\circ\), and the model is \(y=\tan\theta\). A common incorrect calculation is \(u_y\approx \sec^2(45^\circ)(0.2)=2(0.2)=0.4\). This treats the numerical degree uncertainty as though it were in radians, so the result is too large by the conversion factor \(180/\pi\).

The correction is to convert the uncertainty first: \(0.2^\circ=\pi/900\) radians. Then \(u_y\approx 2(\pi/900)=\pi/450\approx0.006981\). The derivative formula \(\sec^2\theta\) is being used with a radian input and a radian increment. The same conversion requirement applies to sine and cosine.