Articles of trigonometry

Rotation of complex numbers in a complex plane. Check my work?

Say that $c_1 = -i$ and $c_2 = 3$. For this problem, let $z_0$ be an arbitrary complex number. We can rotate $z_0$ around $c_1$ by $\pi/4$ counterclockwise to get $z_1$. Next, we canrotate $z_1$ around $c_2$ by $\pi/4$ counter-clockwise to get $z_2$. There exists a complex number $c$ where we can get $z_2$ from […]

How do I find, by the definition of a derivative, the derivative of $\tan x$?

How do I find, by the definition of a derivative, the derivative of tanx? $$f'(x)=\lim_{\Delta x \to 0}{f(x+\Delta x)-f(x)\over \Delta x}=\lim_{\Delta x \to 0}{\tan(x+\Delta x)-\tan(x)\over \Delta x}=\lim_{\Delta x \to 0}{{\sin(x+\Delta x)\over \cos(x+\Delta x)}-{\sin(x)\over \cos(x)}\over \Delta x}$$ I tried using identities but I always reached something like ${0\over \cos(x+ \Delta x)\cos(x)}$ which is no good… Thanks […]

Why is the inverse tangent function not equivalent to the reciprocal of the tangent function?

I know that $$ {\tan}^2\theta = {\tan}\theta \cdot {\tan}\theta $$ So I guess the superscript on a trigonometric function is just like a normal superscript: $$ {\tan}^x\theta = {({\tan}\theta)}^{x} $$ Then why isn’t this true? $$ {\tan}^{-1}\theta = \dfrac{1}{{\tan}\theta} $$

Brouwer's fixed-point theorem and iterative convergence of a composition of circular functions

let $\psi:[0,1]\times \{0,1\} \rightarrow [0,1]$ be defined by: $$ \psi(x,\beta) = \beta \cos x + (1-\beta) \sin x $$ define $B_n$ as the set of $2^n$ binary strings $b=b_0b_1\dots b_{n-1}$ where each $b_j$ is a $0$ or a $1$. now, for a given $b \in B_n$ we define a set of $n$ functions: $\psi_{b,k}:[0,1] \rightarrow […]

Condition for $\tan A\tan B=\tan C\tan D$

Here, it is claimed that $$\tan A\tan B=\tan C\tan D$$ if one of the four following conditions holds $$\displaystyle A\pm B=C\pm D$$ If it is true, how to prove this? $\tan(x\pm y)$ did not help much. I am expecting some relation among $A,B,C,D$

$|\frac{\sin(nx)}{n\sin(x)}|\le1\forall x\in\mathbb{R}-\{\pi k: k\in\mathbb{Z}\}$

Find all real number $n$ such that $|\frac{\sin(nx)}{n\sin(x)}|\le1\forall x\in\mathbb{R}-\{\pi k: k\in\mathbb{Z}\}$.

Is it possible to solve this equation analytically?

The following equation holds: \begin{align} & \frac{9}{2}\pi \\[8pt] & = x \\[8pt] & {}+\cos (x) \\[8pt] & {}+\cos (x+\cos (x)) \\[8pt] & {}+\cos (x+\cos (x)+\cos (x+\cos (x))) \\[8pt] & {}+\cos (x+\cos (x)+\cos (x+\cos (x))+\cos (x+\cos (x)+\cos (x+\cos (x)))) \\[8pt] & {}+\cos (x+\cos (x)+\cos (x+\cos (x))+\cos (x+\cos (x)+\cos (x+\cos (x)))+\cos (x+\cos (x)+\cos (x+\cos (x))+\cos (x+\cos (x)+\cos […]

Does this inequality involving inverse tangent (arctan) hold?

I am wondering if the following statement is true for $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ and $x,y\in\mathbb{R}$: $$\tan^{-1}\left(\frac{\sin(\theta)+x}{\cos(\theta)+y}\right)\leq\theta+x\cos(\theta)-y\sin(\theta)+c(x^2+y^2),$$ where $c\geq2$ is a constant (though an answer showing that constant $c$ exists without quantifying what it is would be good enough). I’ve plotted the difference between RHS and LHS for multiple values of $\theta$ and the inequality seems to hold. […]

Integration by parts of $\cot x$

I attempted to integrate $\cot x$ by parts by taking $u$ as $\csc x$ and $\dfrac{dv}{dx}$ as $\sin x$. Then: $$\int \cot x\,dx = \int \csc x \cos x\,dx \\ = \sin x \csc x – \int- \sin x \csc x \cot x \, dx \\ = \frac{\sin x}{\sin x} + \int \frac{\sin x}{\sin x}\cot […]

How is $\sin 45^\circ=\frac{1}{\sqrt 2}$?

I’ve been reading about the proof of $\sin 45^\circ=\dfrac{1}{\sqrt 2}$ in my book. They did it as following, let $\triangle ABC$ be an isosceles triangle as shown, Since the triangle is isosceles with base and perpendicular equal the opposite angle $\angle C$ and $\angle B$ must be equal. Putting $\angle A=90^\circ$ and using $\angle A […]