Cofunction Series

The Cofunction Taylor Series

Taylor Series Constructed

The Taylor Series around zero is the Maclarin Series, and for Sine and Cosine, they are as follows:

Sine Taylor Series

From The Cofunction Derivatives , we have for Sine:

$$ \sin(x) = \sin(0) + \cos(0) x - \frac{1}{2!}\sin(0) x^2 - \frac{1}{3!}\cos(0) x^3 + \cdots $$

Here we see that the sine function is to first order linear in its argument—applicable when its argument is much less than one, $x\ll 1$, which we knew from geometric argument, above, and is now reflected in the arbitrarily accurate, Taylor series, having more terms of precision and extent.

Cosine Taylor Series

$$ \cos(x) = \cos(0) - \sin(0) x - \frac{1}{2!}\cos(0) x^2 + \frac{1}{3!}\sin(0) x^3 + \cdots $$

Figure 4. The Sine and Sine Taylor Series functions, plotted over the domain in radians from $-\pi$ to $\pi$. The sine function has a range of $[-1,1]$, while the Taylor series is unbounded, as the highlights of the Taylor Series at $-\pi/2$ (a little less than negative one), and the Sine at $\pi/2$ (exactly one) demonstrate.

Tangent Taylor Series

As a rational construction of the cofunctions, the tangent has no dependence on the hypotenuse as the Sine in the numerator and Cosine in the denominator have the same factor of it which forms a factor of unity upon reduction. The derivatives of tangent rely on the derivatives of the cofunctions (via the Product Rule here) starting procedurally with the definition of the derivative.

Derivative of Tangent

$$ \frac{d}{dx}\tan(x) = \lim_{dx\to 0}\frac{tan(x+dx)-tan(x)}{dx} $$

Using the angle addition rule on the tangent is a matter of using the angle addition formulas for the cofunctions:

$$ \tan(x+dx)=\frac{\sin(x)\cos(dx)+\sin(dx)\cos(x)}{\cos(x)\cos(dx) -\sin(x)\sin(dx)} $$

Applying our small angle approximations for the cofunctions in order to get first-order approximations for terms with argument of $dx$, and plugging into the numerator:

$$ \tan(x+dx)-\tan(x)=\frac{\sin(x)+dx\cos(x)}{\cos(x) -\sin(x)dx} - \frac{\sin(x)}{\cos(x)} $$

Finding equivalent fractions with a common denominator, and reducing:

$$ \frac{d}{dx}\tan(x) = \lim_{dx\to 0}\frac{1}{\cos^2(x)-\sin(x)\cos(x)dx}=\cos^{-2}(x) = \sec^2(x) $$

Second Derivative of Tangent

Applying the Chain Rule to the secant-squared function gives us:

$$ \frac{d^2}{dx^2}\tan(x)=\frac{d}{dx}\cos^{-2}(x)=\frac{2\sin(x)}{\cos^3(x)}=2\sec^2(x)\tan(x) $$

Constructing the Tangent Taylor Series

We know what the zeroth derivative of tangent is at zero, $\tan(0)=0$, and its first derivative at zero, $\sec^2(0)=1$, and calcululating the derivatives through the fifth order, and evaluating them at zero, we have for the Taylor series of tangent:

$$ \tan(x) = x + \frac{2}{3!}x^3 + \frac{16}{5!}x^5 \cdots $$

Figure 5. The Tangent and Tangent Taylor Series functions, plotted over the domain containing the origin, in radians from $-\pi/2 + 0.4$ to $\pi/2 - 0.4$. The Taylor Series is the lesser at the edges of its domain to order-five, since the source function is hyperbolic in Cosine and the polynomial is finite-ordered.

The domain of tangent is $(-\pi/2, \pi/2)$, which repeats every $\pi$ radians in both directions.

Inversion

The description of the inverse of Sine is with the understanding that an interval of $2\pi$ radians defines a domain of every period of the cyclical function.

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