Linearizing Trigonometric Functions Near Standard Angles

Why Linearize Trigonometric Functions?

A trigonometric value at a nearby angle can often be estimated without evaluating the function directly. The idea is to use the tangent line at a familiar angle, such as \(\pi/6\), \(\pi/4\), or \(\pi/3\), where sine, cosine, and tangent have known values.

This method is useful only when the angle change is expressed in the right units. The standard derivative formulas for trigonometric functions assume that the input is measured in radians. A change of \(1\) degree is not a change of \(1\) radian, so mixing those units changes the calculation.

The Linearization Formulas

The linearization of a differentiable function \(f\) at a point \(a\) is the linear function that has the same value and the same derivative as \(f\) at \(a\). For trigonometric functions with radian inputs, the standard derivative formulas are \(\frac{d}{dx}\sin x=\cos x\), \(\frac{d}{dx}\cos x=-\sin x\), and \(\frac{d}{dx}\tan x=\sec^2 x\), with tangent defined only where \(\cos x\ne0\).

The tangent formula requires \(\cos a\ne0\), so its base point cannot be an angle where tangent is undefined. The target angle must also be in the same interval of definition as the base point: do not use a tangent linearization across an asymptote. Sine and cosine are defined for all real inputs, so they have no corresponding restriction.

For any of these formulas, \(x-a\) is the signed change in angle. A positive change moves to a larger angle; a negative change moves to a smaller angle. Because the derivative gives change in function value per radian, \(x-a\) must be in radians when the displayed formulas are used.

Why the Formulas Work

At the base point \(a\), the linearization must match the original function’s value. Its slope must match the original function’s derivative there. These two requirements determine the line: a line with value \(f(a)\) at \(a\) and slope \(f'(a)\) has equation \(f(a)+f'(a)(x-a)\).

Also, differentiating the linear function gives \(L_a'(x)=f'(a)\), so \(L_a'(a)=f'(a)\). Thus it agrees with \(f\) in both value and slope at \(a\). When \(x\) is near \(a\), this matching tangent line provides a local estimate.

Applying this reasoning to sine gives \(f(a)=\sin a\) and \(f'(a)=\cos a\), producing \(L_{\sin,a}(x)\). For cosine, \(f(a)=\cos a\) and \(f'(a)=-\sin a\), producing \(L_{\cos,a}(x)\). For tangent, \(f(a)=\tan a\) and \(f'(a)=\sec^2 a\), producing \(L_{\tan,a}(x)\), provided tangent is defined at \(a\).

Worked Examples

When Angles Are Given in Degrees

The formulas above use radians. If the angle is given in degrees, first convert its change to radians, or explicitly include the conversion factor in the derivative with respect to degrees. Since \(x^\circ=\pi x/180\) radians, a change of \(h\) degrees corresponds to a radian change of \(h\pi/180\).

Common Mistake: Using Degrees as if They Were Radians

A frequent error is to insert a degree change directly into a formula whose derivative assumes radians. For instance, using the sine linearization at \(30^\circ\) but treating the change to \(31^\circ\) as the number \(1\) would give

This is not the correct linearization calculation: the input change is one degree, not one radian. Convert the change first:

The wrong calculation does not merely round poorly; it uses a slope with the wrong angle unit. Always identify the units of both the base point and the change before substituting.

Choosing a Base Point and Respecting the Domain

Use a standard angle close to the target and with known trigonometric values. A closer base point generally makes the tangent-line estimate more local, but a familiar value is also useful because it keeps the arithmetic manageable. The estimate is a local model, not an identity, so it should not be treated as an exact value or extended arbitrarily far from its base point.

For tangent, domain checks are essential. Tangent is undefined when \(\cos x=0\), at angles \(\pi/2+k\pi\) for integers \(k\). Its graph has a vertical asymptote at each such angle. A tangent line at a valid base point cannot provide a model across one of these breaks. For sine and cosine, there are no excluded real angles, though their linearizations are still intended for nearby inputs.