$\begingroup$

In the $\sin(x+h)$ Taylor’s series, can $h$ be in degrees or it has to be in radians?

$\endgroup$ 2

3 Answers

$\begingroup$

Radians. Generally, whenever in Mathematics done above high school level, angles are always in radians. If degrees are to be used, they will be labelled by $^o$

$\endgroup$ $\begingroup$

Let $f(x) = \sin( \frac{\pi x}{180})$. You can think of $f$ as computing the sine of an angle $x$ which is in degrees rather than radians. The Taylor series for $f$ (centered at $0$) is

\begin{align*} f(x) &= f(0) + f'(0)x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3!} x^3 + \cdots \\ &= (\pi/180) x - \frac{(\pi/180)^3}{3!} x^3 + \frac{(\pi/180)^5}{5!} x^5 - \cdots. \end{align*}

This is of course the same result you would obtain by plugging $u = \pi x/ 180$ into the series$$ \sin(u) = u - \frac{u^3}{3!} + \frac{u^5}{5!} - \cdots. $$

Note that you can't compute the sine of a 30 degree angle by just plugging the number 30 into the standard Taylor series for sine. You must first convert to radians.

$\endgroup$ $\begingroup$

It can be either, but it must be the same as $x$.

As Ishan Deo said, it is generally going to be radians if you're in a college course.

$\endgroup$