1
vote
3answers
65 views

Proof of $\sin2x+x\sin^2x \lt\dfrac{1}{4}x^2+2$

How can be proven the following inequality? $$\forall{x\in\mathbb{R}},\left[\sin(2x)+x\sin(x)^2\right]\lt\dfrac{1}{4}x^2+2$$ Thanks
2
votes
2answers
111 views

Proving $\displaystyle \frac{\sin^3x}{x}\lt 0.69$ for any $x\gt 0$

Question : How can we prove strictly that the following inequality holds for any $x\gt0$?$$\frac{\sin^3x}{x}\lt 0.69$$ This seems difficult though it doesn't look so. Can anyone help?
0
votes
1answer
34 views

$f(x)=sec(x)$ inequality inconsistency\trouble

I'm currently attempting to find the range of $f(x)=\sec(x)$ by considering $\cos(x)$ in the intervals of $0<\cos(x)\leqslant 1$ and $-1\leqslant \cos(x)<0$ (as $\sec(x)$ is undefined for ...
1
vote
0answers
24 views

$\frac {1 } {10 }(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2)^2+(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1)^2) \le (y_1-x_1)^2+(y_2-x_2)^2$?

Is it true that: $$\frac {1 } {10 }\left(\left(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2\right)^2+\left(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1\right)^2\right) \le (y_1-x_1)^2+(y_2-x_2)^2$$ I think I should ...
2
votes
1answer
42 views

(Elementary) Trigonometric inequality

Any idea for proving the following inequality: $5+8\cos x+4 \cos 2x+ \cos3x\geq 0$ for all real x? I've tried trigonometric identities to make squares appear, and other tricks; but nothing has worked ...
1
vote
1answer
34 views

Inequality involving $|\cos(x)|$

If we have: $$\prod_{k=1}^{N}|\cos(\omega x_k)|=1$$ then, how can the following inequality: $$\sum_{k=1}^N\frac{1}{N-1+|\cos(\omega x_k)|}\le1$$ be proven? Thanks:
5
votes
2answers
106 views

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 ...
2
votes
0answers
66 views

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 ...
0
votes
1answer
34 views

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? $$\frac{\sin x}{\sin y}\leq f(x,y)$$
7
votes
2answers
196 views

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 $y$? $$ |\sin^2x - \sin^2y | \le K|x - y|$$
0
votes
3answers
46 views

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 ...
3
votes
1answer
106 views

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 ...
1
vote
1answer
39 views

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: $\frac{\pi}{2}-x<\tan^{-1}(x)$ ...
3
votes
3answers
94 views

Proving $\cos 36° > \tan 36° $

How do we prove that $\cos 36° > \tan 36° $ ? Please help . Thank you.
0
votes
2answers
30 views

Inequality: $\tan(x) > 1$

So far, I've not come very... far. It ends up with me trying to solve it more intuitively than mathematically. I figured, first I'll find the place of equality, which is at $x = \arctan 1 = ...
1
vote
2answers
39 views

Inequality with trigonometric functions

Find all values for $a$ such that the following inequality holds: $$\sin^6x + \cos^6x + a\sin x \cos x \ge 0$$ To be fair, I didn't manage to get anything helpful wiht my calculations. I tried to ...
0
votes
1answer
23 views

Arccos and inequalities?

There is something I don't understand with arccos and inequalities. Suppose I have this inequality $cos(x) ≤ \frac{1}{2}$ Having $x = 90$, satisfies this since $cos(90) = 0$. Then since arccos is ...
2
votes
1answer
78 views

Putnam inspired problem

The following is a beautiful problem from Putnam 2003 minimize $|\sin x + \cos x + \tan x + \csc x + \sec x + \cot x|$ I was thinking about a small variation of the above problem minimize $|\sin ...
0
votes
1answer
48 views

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 ...
1
vote
6answers
200 views

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 ...
2
votes
2answers
91 views

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 ...
1
vote
2answers
92 views

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 ...
1
vote
1answer
35 views

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 $m,n\in \mathbb N\setminus \{ 0\}$.
2
votes
4answers
58 views

How can I prove the trigonometric Problem?

How can I show the following trigonometric problem : $$\frac{1}{3}\leq \frac{\sec^2\theta-\tan^2\theta}{\sec^2\theta+\tan^2 \theta}\leq 3$$ I have tried in the following way : $$ ...
0
votes
1answer
21 views

When is $\cos\frac{\pi}{x}<0$?

This seems really simple, but I'm trying to find a way to solve $\cos\frac{\pi}{x}<0$. I get $$x\geq \frac{\pi}{\arccos0}=\frac{\pi}{(2k+1)\frac{\pi}{2}} = \frac{2}{2k+1}$$ for $k\in\mathbb{Z}$. ...
1
vote
1answer
35 views

Which one is valid: $ 2\cos((a+b)/2)<2$ or $ 2\cos((a+b)/2)\leq2$?

Which one is valid: $ 2\cos((a+b)/2)<2$ or $ 2\cos((a+b)/2)\leq2$? I need this to be true for my proof.
1
vote
1answer
63 views

Prove this trigonometry inequality

I'm having difficulty proving that tan(26°) < 0.5 < tan(27°) . Any idea ? Thanks. p.s. 26 and 27 are in degrees.
1
vote
3answers
64 views

Proving that $\sin x \gt \dfrac x2$

I was working on this question and I got a contradiction. $\sin x \gt \dfrac x2$ for $0 \lt x \lt \dfrac {\pi}{2}$ $\arccos ( \sin x)) \gt \arccos (\dfrac x2)$ $\dfrac {\pi}{2} -x \gt \arccos ...
2
votes
4answers
391 views

Proof that $\sin(x) > x/2$

I need to prove that $\sin(x) > \frac{x}{2}$ if $0<x<\pi/2$ I've started working with the derivative, but if it's possible, I'd rather something simpler than that.
0
votes
1answer
31 views

How to show that $ 0< a\leq\cos^2(\theta)\leq b<1$ in this problem?

The inequality $2\cos^4(\theta/2)-2\cos^2(\theta/2)+1/4\leq 0$ means that $\cos^2(\theta/2)$ lies between the roots of $2x^2-2x+1/4$ i. e., we can conclude that $$ ...
3
votes
2answers
67 views

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: $$\sin(x)+\cos(x)>1-\sin(2x)$$ ...
6
votes
4answers
228 views

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 ...
17
votes
4answers
319 views

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 ...
3
votes
2answers
65 views

How to solve this inequality.

Let $\alpha$, $\beta$, $\gamma$ be the angles of a triangle. Show that $\sin\frac{\alpha}{2}.\sin\frac{\beta}{2}.\sin\frac{\gamma}{2}<{\frac{1}{4}}$
5
votes
3answers
138 views

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 $-3/2$
1
vote
3answers
68 views

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 ...
5
votes
2answers
77 views

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 $\cos{2x}\cdot \cos{2y}\cdot \cos{2z}\le 0$.
0
votes
2answers
59 views

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
4
votes
0answers
89 views

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 ...
2
votes
1answer
68 views

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 ...
2
votes
2answers
56 views

Proof of $\frac{1}{\sin{(\frac{\pi}{2x})}}<\frac{2x}{\pi}+1$

I want to show that \begin{equation} \frac{1}{\sin{(\frac{\pi}{2x})}}<\frac{2x}{\pi}+1 \end{equation} for any positive integer $x$. Seems that it is related to the well-known inequality ...
5
votes
5answers
135 views

Prove $\frac{2 \sin x}{3}+\frac{\tan x}{3} > x$ for $x \in (0, \frac{\pi}{2})$

Please, help Prove $\frac{2 \sin x}{3}+\frac{\tan x}{3} > x$ $x \in (0, \frac{\pi}{2})$
1
vote
2answers
52 views

Inequality on trigonometry

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 ...
3
votes
2answers
50 views

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 :$$ \left|\prod_{k=0}^n \sin\left(2^k x\right)\right|\leq\left(\frac{\sqrt{3}}{2}\right)^n.$$
2
votes
0answers
73 views

Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
1
vote
2answers
34 views

Trigonometric inequality to prove

Prove the following inequality for each $x$: $$|\sin x+7\cos x| \leq \sqrt{50}$$ My remark: It seems to me that the solution uses the fact that $1^2+7^2=50$. Thanks in advance!
3
votes
2answers
44 views

$\sin(\pi x)\geq\frac{x}{2}$ for $0\leq x\leq \frac34$

How can I prove that $\sin(\pi x)\geq\dfrac{x}{2}$ for $0\leq x\leq \dfrac34$? It's very simple-looking, but the $\sin x\leq x$ doesn't seem to help.
0
votes
1answer
137 views

how to solve trigonometric inequalities?

how does one solve trigonometric inequalities? Is there a method to this or is every solution done ad hoc? simple equations of the type: $cos3x \leq 0$ when: $0\leq x \leq 2π$ The attempt at a ...
4
votes
0answers
82 views

How prove this inequality $\sin^2{\frac{A}{2}}+\sin^3{\frac{B}{2}}+\sin^4{\frac{C}{2}}\ge\frac{7}{16}$

in $\Delta ABC$,such $$5\cos{A}+6\cos{B}+7\cos{C}=9$$ show that $$\sin^2{\dfrac{A}{2}}+\sin^3{\dfrac{B}{2}}+\sin^4{\dfrac{C}{2}}\ge\dfrac{7}{16}$$ By the way: This inequality is my favourite ...
4
votes
2answers
63 views

Third-degree cosine inequality for obtuse triangle

Suppose $\triangle ABC$ is an obtuse triangle with side lengths $a=BC, b=CA, c=AB$. I want to show that $$a^3\cos A+b^3\cos B+c^3\cos C<abc.$$ My idea is to use the cosine rule. I have $\cos ...