Articles of trigonometry

Find a formula relating $\operatorname{arcsin}(x)$ and $\operatorname{arccos}⁡(x)$

From the formula $\sin\left(\frac{π}{2}−x\right)=\cos x$, find a formula relating $\operatorname{arcsin}(x)$ and $\operatorname{arccos}⁡(x)$. I have figured out that the domain of $x$ is $[-1;1]$, but I have no idea how to do this. I’ve tried letting $y=\cos x$ and the only result I’ve got is $$\operatorname{arccos}(y)+\operatorname{arcsin}(y)=\frac{π}{2}$$ I need a full answer.

Inequality involving inradius, exradii, sides, area, semiperimeter

In the $\Delta ABC$, let $r_a,r_b,r_c$ be the exradii, $S$ the area of the triangle, $a,b,c$ its sides, $p$ the semiperimeter and $r$ the inradius. Show that the following inequality holds: $$r_ar_b+r_br_c+r_cr_a \ge 2\sqrt3 S+\frac {abc}{p}+r^2.$$ I tried to express the exradii in terms of $p,a,b,c$, but I didn’t get to show that the inequality […]

Solve $\cos x+8\sin x-7=0$

Solve $\cos x+8\sin x-7=0$ My attempt: \begin{align} &8\sin x=7-\cos x\\ &\implies 8\cdot \left(2\sin \frac{x}{2}\cos \frac{x}{2}\right)=7-\cos x\\ &\implies 16\sin \frac{x}{2}\cos \frac{x}{2}=7-1+2\sin ^2\frac{x}{2}\\ &\implies 16\sin \frac{x}{2}\cos \frac{x}{2}=6+2\sin^2 \frac{x}{2}\\ &\implies 8\sin \frac{x}{2}\cos \frac{x}{2}=3+\sin^2 \frac{x}{2}\\ &\implies 0=\sin^2 \frac{x}{2}-8\sin \frac{x}{2}\cos \frac{x}{2}+3\\ &\implies 0=\sin \frac{x}{2}\left(\sin \frac{x}{2}-8\cos \frac{x}{2}\right)+3 \end{align} I’m not sure how to proceed from here (if this process is even […]

Simplifying this (perhaps) real expression containing roots of unity

Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ although I don’t think that is relevant. Let $\zeta:=\exp(2\pi i/k)$ and $\alpha_v:=\zeta^v+\zeta^{-v}+\zeta^{-1}$. As it comes from the trace of a positive matrix I know that the following is real: $$\sum_{v=1}^{\frac{k-1}{2}}\sec^2\left(\frac{2\pi v}{k}\right)\frac{(\overline{\alpha_v}^N+\alpha_v^N\zeta^{2N})(\zeta^{2v}-1)^2}{\zeta^N\zeta^{2v}}.$$ I am guessing, and numerical evidence suggests, that in fact $$\frac{(\overline{\alpha_v}^N+\alpha_v^N\zeta^{2N})(\zeta^{2v}-1)^2}{\zeta^N\zeta^{2v}}$$ is real […]

Is there any intuition behind why the derivative of $\cos(x)$ is equal to $-\sin(x)$ or just something to memorize?

why is $$\frac{d}{dx}\cos(x)=-\sin(x)$$ I am studying for a differential equation test and I seem to always forget \this, and i am just wondering if there is some intuition i’m missing, or is it just one of those things to memorize? and i know this is not very differential equation related, just one of those things […]

Did I write the right “expressions”?

$9$. Consider the parametric curve $K\subset R^3$ given by $$x = (2 + \cos(2s)) \cos(3s)$$ $$y = (2 + \cos(2s)) \sin(3s)$$ $$z = \sin(2s)$$ a) Express the equations of K as polynomial equations in $x,\ y,\ z,\ a = \cos(s),\ b = \sin(s)$. Hint: Trig identities. b) By computing a Groebner basis for the ideal […]

A trigonometric identity for special angles

Prove that for a natural number $n$, $$\prod_{k=1}^n \tan\left(\frac{k\pi}{2n+1}\right) = 2^n \prod_{k=1}^n \sin\left(\frac{k\pi}{2n+1}\right)=\sqrt{2n+1}.$$

How to prove $\left(1+\frac{1}{\sin a}\right)\left(1+\frac{1}{\cos a}\right)\ge 3+2\sqrt{2}$?

Prove the inequality: $$\left(1+\dfrac{1}{\sin a}\right)\left(1+\dfrac{1}{\cos a}\right)\ge 3+2\sqrt{2}; \text{ for } a\in\left]0,\frac{\pi}{2}\right[$$

Evaluate $\cot^2\left(\frac{1\pi}{13}\right)+\cot^2\left(\frac{2\pi}{13}\right)+\cdots+\cot^2\left(\frac{6\pi}{13}\right)$

Could anyone just evaluate this expression? I would be very grateful. $$\cot^2\left(\frac{1\pi}{13}\right)+\cot^2\left(\frac{2\pi}{13}\right)+\cot^2\left(\frac{3\pi}{13}\right)+\cot^2\left(\frac{4\pi}{13}\right)+\cot^2\left(\frac{5\pi}{13}\right)+\cot^2\left(\frac{6\pi}{13}\right)$$

Finding a diagonal in a trapezoid given the other diagonal and three sides

The figure below is a trapezoid, what is the length of the red line? Thank you very much in advance!