Articles of trigonometry

Are $\mathrm{arccot}(x)$ and $\arctan(1/x)$ the same function?

In my textbook it asks for me to: Prove that there is no constant $C$ such that $\text{arccot}(x) – \text{arctan}(\frac{1}{x}) = C $ for all $x \ne 0$. Explain why this does not violate the zero-derivative theorem. But I believe I have found such a $C$, i.e. $C =0$! I even asked WolframAlpha ( which […]

Explicitly finding the sum of $\arctan(1/(n^2+n+1))$

This is from a GRE prep book, so I know the solution and process but I thought it was an interesting question: Explicitly evaluate $$\sum_{n=1}^{m}\arctan\left({\frac{1}{{n^2+n+1}}}\right).$$

Solving $E=\frac{1}{\sin10^\circ}-\frac{\sqrt3}{\cos10^\circ}$

$$E=\frac{1}{\sin10^\circ}-\frac{\sqrt3}{\cos10^\circ}$$ I got no idea how to find the solution to this. Can someone put me on the right track? Thank you very much.

Is there a general formula for $\sin( {p \over q} \pi)$?

Virtually everyone knows the basic values of the unit circle, $\sin(\pi) = 0; \ \ \sin({\pi \over 2}) = 1; \ \ \sin({\pi \over 3}) = {\sqrt{3} \over 2} \\$ And other values can be calculated through various identities, like $\sin({\pi \over 8}) =\frac{1}{2} \sqrt{2 – \sqrt{2}}\\$ Does there exist a general formula for $\sin({p\over […]

A series expansion for $\cot (\pi z)$

How to show the following identity holds? $$ \displaystyle\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}=\pi\cot \pi z-\dfrac{1}{z}\qquad |z|<1 $$

What is $\arctan(x) + \arctan(y)$

I know $$g(x) = \arctan(x)+\arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right)$$ which follows from the formula for $\tan(x+y)$. But my question is that my book defines it to be domain specific, by which I mean, has different definitions for different domains: $$g(x) = \begin{cases}\arctan\left(\dfrac{x+y}{1-xy}\right), &xy < 1 \\[1.5ex] \pi + \arctan\left(\dfrac{x+y}{1-xy}\right), &x>0,\; y>0,\; xy>1 \\[1.5ex] -\pi + \arctan\left(\dfrac{x+y}{1-xy}\right), &x<0,\; […]

Evaluating limit (iterated sine function)

The limit is $$\lim_{x\rightarrow0} \frac{x-\sin_n(x)}{x^3},$$ where $\sin_n(x)$ is the $\sin(x)$ function composed with itself $n$ times: $$\sin_n(x) = \sin(\sin(\dots \sin(x)))$$ For $n=1$ the limit is $\frac{1}{6}$, $n=2$, the limit is $\frac{1}{3}$ and so on. Can we define a recurrent relation upon that given hypothesis? Also, how do I involve $n$ into calculation, because the final […]

How does e, or the exponential function, relate to rotation?

$e^{i \pi} = -1$. I get why this works from a sum-of-series perspective and from an integration perspective, as in I can evaluate the integrals and find this result. However, I don’t understand it intuitively. Intuitively, what this means to me is that if you rotate pi radians around a unit circle, you will end […]

Existence of continuous angle function $\theta:S^1\to\mathbb{R}$

Let $S^1\subseteq\mathbb{C}$ be the unit circle and let $U\subseteq S^1$ be open. How to show that there exist a continuous function $$\theta:U\to\mathbb{R}$$ such that $$e^{i\theta(z)}=z$$ for all $z\in U$ if and only if $U\neq S^1$? I have been able to show the $\Rightarrow$ direction but not the other one. $(\Rightarrow)$ If such a continuous $\theta:S^1\to\mathbb{R}$ […]

What is the value of $\frac{\sin x}x$ at $x=0$?

On plotting graph for $\frac{\sin x}{x}$ using Wolfram|Alpha and Google, got that : also, I can get the value of $\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$ using squeeze theorem and as illustrated on sources such as MIT. But I’m not able to understand how the function is defined at $x=0$ and its value came out […]