# Tagged Questions

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### Show that $\sin(x)\sin(y)\sin(z) \leq \frac{1}{8}$

Let $(x,y,z) \in (\mathbb{R}^+)^3$ such that $x + y + z \leq \frac{\pi}{2}$. Show that $$\sin(x)\sin(y)\sin(z) \leq \frac{1}{8}$$ I have a solution using convexity of $\sin$ but I am looking ...
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### Proving a tough geometrical inequality, with equality in equilateral triangles.

For any triangle with sides $a ,b, c$ prove or disprove (1) and (2) : $$\sum_\mathrm{cyc} \frac{1}{\frac{(a+b)^2-c^2}{a^2}+1}\ge \frac34$$ Equality in (1) holds if and only if the triangle is ...
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### Inequality for $\frac{\sin x}{\sin y}$

We know that $$\sqrt{\sin x \sin y}\leq \sin\left(\frac{x+y}{2}\right)$$ Is there an useful inequality for $\dfrac{\sin x}{\sin y}$ like this? $$\frac{\sin x}{\sin y}\leq f(x,y)$$
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### Smallest K for which $|\sin^2x - \sin^2y | \le K|x - y|$ holds [closed]

What is the smallest positive number K for which the following inequality holds $\forall$ $x$ and $y$? $$|\sin^2x - \sin^2y | \le K|x - y|$$
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### Solve $6\sin x \cos 2x\ge 0$

How can I solve the following inequality: $${6\sin x\cos 2x\ge 0}$$ Can you give me an explicit explanation of how this exercise can be understood. I have no problems with trigonometric equations, but ...
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### Inequality problem about sides of a triangle and the semiperimeter

Let $a,b,c$ the sides of a triangle and $s$ be the semi perimeter. Then show that $$a^2+b^2+c^2 > \frac{36}{35}(a^2+\frac{abc}{s})$$ I tried it doing in many ways using some ...
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### Prove that $\frac{\pi}{2}-x<\tan^{-1}(x)<\frac{\pi}{2}-x+\frac{x^3}{3}$

Prove that for every $x>0$, it is true: $$\frac{\pi}{2}-x<\tan^{-1}(x)<\frac{\pi}{2}-x+\frac{x^3}{3}$$ We can split it into two statements: $\frac{\pi}{2}-x<\tan^{-1}(x)$ ...
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### Proving $\cos 36° > \tan 36°$

How do we prove that $\cos 36° > \tan 36°$ ? Please help . Thank you.
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### Prove that $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$

Prove the following: $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$ Hello everyone, I am trying to create some sort of ...
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### How would you solve the inequality $\sin x \gt \cos x$?

$$\sin x \gt \cos x, \qquad (-2\pi <x <2\pi)$$ I tried an approach saying that $\tan x\gt1$ but apparently the solution, which is $\frac{\pi}{4}<x<\frac{5\pi}{4}$ is not good. It's a ...
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### Trigonometric Inequality $\cos 1 +\cos2+\ldots +\cos n < 0.55$ can be solved with the help of Integrals?

How can I prove for every $n \in \mathbb{N}$ $$\cos 1 +\cos2+\ldots +\cos n < 0.55$$ Any idea, any solution? Thanks! EDIT Can be solved this inequality with the help of integrals, because I met ...
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### Positivness of the sum of $\frac{\sin(2k-1)x}{2k-1}$.

For $n\in \mathbb{N}$, $x\in (0,\pi)$. Prove that : $$f_n(x)=\sum_{k=1}^n \frac{\sin [(2k-1)x]}{2k-1} \geq 0.$$ I've tried to do it by differentiation : I Calculate $f_n'(x)$ (sum of ...
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### Solutions to an inequality involving trigonometric terms

I would like to determine the angles $\theta\in [0,2\pi[$ such that $$\big(1-\cos(n\theta)\big)\big(1-\cos(m\theta)\big)\geq 1/4$$ for every positive integers $m,n\in \mathbb N\setminus \{ 0\}$.
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### Trigonometric inequality

Solve inequality per x: $$\sin(x)+\cos(x)+\sin(2x)>1$$ I need some start, I tried to factor but i can't get something easier to solve, for example: $$\sin(x)+\cos(x)>1-\sin(2x)$$ ...
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### Prove the given inequality

$$\sin^{2}A(\tan(B-C))>\sin^{2}B(\tan(A-C))$$ $$\implies \frac{\sin^2 A}{\sin^2 B} > \frac{\tan(A-C)}{\tan(B-C)}$$ Given if $A>B>C$ and $A+B+C=180^\circ$ Is that implication correct if ...
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### Prove: $\sin (\tan x) \geq {x}$

I bumped into this question: Question: Prove that for $x\in \Bigl[0,\dfrac {\pi}{4}\Bigr]$, $$\sin (\tan x) \geq {x}$$ This seems to be an innocent inequality but I am already exhausted trying ...
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### How to solve this inequality.

Let $\alpha$, $\beta$, $\gamma$ be the angles of a triangle. Show that $\sin\frac{\alpha}{2}.\sin\frac{\beta}{2}.\sin\frac{\gamma}{2}<{\frac{1}{4}}$
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### minimum value of $\cos(A-B)+\cos(B-C) +\cos(C-A)$ is $-3/2$

How to prove that the minimum value of $\cos(A-B)+\cos(B-C) +\cos(C-A)$ is $-3/2$
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### Trigonometric inequality bounded by lines

How can it be shown that $$16x\cos(8x)+4x\sin(8x)-2\sin(8x)<|17x|?$$ This problem arises from work with damped motion in spring-mass systems in Differential Equations. I have gotten to this ...
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### Weird inequality

Let $x,y,z$ be real numbers such that $\cos x+\cos y+\cos z=0$ and $\cos{3x}+\cos{3y}+\cos{3z}=0$ prove that $\cos{2x}\cdot \cos{2y}\cdot \cos{2z}\le 0$.
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### proving a specific trig inequality

I can't figure out how to prove the following inequality: $$1/2 < 4\sin^2\left(\frac{\pi}{14}\right) + \frac{1}{4\cos^2\left(\frac{\pi}{7}\right)} < 2 - \sqrt{2}$$ Thanks
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### Trigonometry or inequality problem

Today, I saw this question: If $x,y,z \in [0,\frac\pi 2]$, $x+y+z=\frac{3\pi}{4}$ and $\sec^2(x)\sec^2(y)\sec^2(z)=8$, calculate $E=\tan x\tan y+\tan y\tan z+\tan z\tan x$ My first thought was ...
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### Prove the following trig inequality

Show that $$\frac{\pi}{4} +\frac{3}{25}< \arctan \frac{4}{3} < \frac{\pi}{4} +\frac{1}{6}$$ I tried using some inequalities I know like $\ln(1+x) < \arctan(x) < \arcsin(x)$ on ...
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### Proof of $\frac{1}{\sin{(\frac{\pi}{2x})}}<\frac{2x}{\pi}+1$

I want to show that $$\frac{1}{\sin{(\frac{\pi}{2x})}}<\frac{2x}{\pi}+1$$ for any positive integer $x$. Seems that it is related to the well-known inequality ...
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### Prove $\frac{2 \sin x}{3}+\frac{\tan x}{3} > x$ for $x \in (0, \frac{\pi}{2})$

Please, help Prove $\frac{2 \sin x}{3}+\frac{\tan x}{3} > x$ $x \in (0, \frac{\pi}{2})$
Let $a,b,c$ be in $\left(0;\dfrac{\pi}2\right)$ such that $\cos^2a+\cos^2b+\cos^2c=1$. I am trying to prove the following inequality: $\tan a+\tan b+\tan c\geq 2\left(\cot a+\cot b+\cot c\right)$, but ...
### upper bound for the product of $\sin (2^k x)$.
Let $n\geq 1$ be an integer and $x$ is a real number. Prove or disprove that :$$\left|\prod_{k=0}^n \sin\left(2^k x\right)\right|\leq\left(\frac{\sqrt{3}}{2}\right)^n.$$