0
votes
0answers
19 views

Strategies to work with system of trigonometric inequality

I'm trying solve this problem using matlab, anybody know good strategies to work with system of trigonometric inequalities such as $ ...
1
vote
1answer
36 views

Trigonometric inequality in a triangle

If $\alpha,\beta,\gamma$ are the interior angles in a triangle, the following inequality seems to hold: ...
3
votes
1answer
37 views

Range of $f(x)=\frac{\sin x -1}{\sqrt{3-2\cos x-2\sin x}}$ for a specified domain

We are asked to find the range of the function $$f(x)=\frac{\sin x -1}{\sqrt{3-2\cos x-2\sin x}}, \;\;\text{for}\;0\le x\le2\pi$$ I tried to find the range of each basic function of cos and sin then ...
4
votes
2answers
46 views

If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5>\ldots?$

Problem : If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5$ is always greater than : (a) $\ 7 $ (b) $\ 8 $ (c) $\ 9 $ (d) $\ $none of these Solution : We ...
1
vote
3answers
110 views

Is $-|x|\le\sin x\le|x|$ for all $x$ true?

I have seen in Thomas' Calculus that says to prove $\lim_{x\rightarrow0}\sin x=0$, use the Sandwich Theorem and the inequality $-|x|\le\sin x\le|x|$ for all $x$. My question is how could the ...
1
vote
3answers
66 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
44 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
35 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:
0
votes
1answer
27 views

Trig question, inequality

How can I find the following product using elementary trigonometry? Suppose $0 \lt x \lt \frac{\pi}{2}$ is an angle measured in radians. Use the trigonometric circle and show that $\cos(x) \le ...
5
votes
2answers
109 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 ...
3
votes
1answer
77 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? $$\frac{\sin x}{\sin y}\leq f(x,y)$$
6
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
47 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
95 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
40 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
206 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
93 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
93 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
36 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
66 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
396 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
68 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
321 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
66 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
140 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
69 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
81 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
91 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
137 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
53 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
51 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 :$$ \left|\prod_{k=0}^n \sin\left(2^k x\right)\right|\leq\left(\frac{\sqrt{3}}{2}\right)^n.$$