1
vote
2answers
34 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
18 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
66 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
45 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
147 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
81 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
70 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
56 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
19 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
34 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
61 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
376 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
28 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
66 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
224 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 ...
16
votes
4answers
260 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
63 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
112 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
62 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
66 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
56 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
3
votes
0answers
70 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
55 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
118 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
49 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 ...
2
votes
2answers
45 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
70 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
41 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
113 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
71 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
58 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 ...
0
votes
1answer
114 views

Prove using Jensen's Inequality

Let $\alpha_1, \alpha_2, . . . , \alpha_n$ be the interior angles of a convex (but not necessarily regular) n-gon. Prove, that for all integers $n\geq3$: $$\cos \alpha_1 + \cos \alpha_2 + \cdots + ...
3
votes
1answer
90 views

Solve an inequality $\sin t > \sin \left(t+\frac{\pi}{3}\right)$

I need to solve an inequality. $$\sin t > \sin \left(t+\frac{\pi}{3}\right)$$ I simplified the inequality to: $$\sin t > \sqrt3\ \cos t$$ I'm not sure what to do next. I know I can't square ...
13
votes
3answers
343 views

How prove this inequality $\sin{\left(\dfrac{\pi}{2}ab\right)}\le\sin{\left(\dfrac{\pi}{2}a\right )}\sin{\left(\dfrac{\pi}{2}b\right)}$

let $$0\le a\le 1,0\le b\le 1$$ prove or disprove $$\sin{\left(\dfrac{\pi}{2}ab\right)}\le\sin{\left(\dfrac{\pi}{2}a\right )}\sin{\left(\dfrac{\pi}{2}b\right)}$$ My try: since ...
5
votes
1answer
81 views

$\frac{\cos x}{1}+\frac{\cos(2x)}{2}+\cdots+\frac{\cos (nx)}{n}\gt -1$ is true for $n\in\mathbb N, 0\lt x\lt \pi$?

Let $n$ be a natural number and let $0\lt x\lt{\pi}$. Then, here are my questions. Question 1 : Is the following true? $$\sum_{k=1}^{n}\frac{\cos(kx)}{k}\gt -1$$ Question 2 : Is the ...
2
votes
1answer
32 views

Trigonometric inequality solving

How to solve this inequality $\left|\dfrac{\cos 2x + 3}{\cos x}\right|\geq 4$ ? I tried to consider 2 cases: 1) When $\cos 2x \geq 0$ and $0<\cos x<1$ 2) $\cos 2x\leq 0$ and $-1 < \cos x ...
8
votes
0answers
123 views

Verifying the inequality $\sum_{i=1}^n \frac{\cos y_i}{\sin x_i}≤ \sum_{i=1}^n\cot x_i $

Let $\sum_{i=1}^n x_i=\sum_{i=1}^ny_i= \pi$ , where $n>1$ and $x_i >0 , y_i>0 , \forall i=1,2,..,n$. How can we prove that $$\sum_{i=1}^n \frac{\cos y_i}{\sin x_i}≤ \sum_{i=1}^n\cot x_i $$ ? ...
2
votes
1answer
186 views

If $x+y+z=\pi/2$, $\sin{x}\sqrt{1-\sin x}+\sin y\sqrt{1-\sin y}+\sin z\sqrt{1-\sin z} \ge 4\sqrt{2} \sin{x}\sin{y}\sin{z}(\sin{x}+\sin{y}+\sin{z})$

Let: $x,y,z >0$ and $ x+y+z=\dfrac{\pi}{2}$ then prove: $$\sin{x}\sqrt{1-\sin{x}}+\sin{y}\sqrt{1-\sin{y}}+\sin{z}\sqrt{1-\sin{z}} \ge 4\sqrt{2} ...
4
votes
4answers
202 views

Prove that $|\cos(\sin(x_1)) - \cos(\sin(x_2))| \leq |x_1 - x_2|, \forall x_1, x_2 \in \mathbb R$.

I asked this question without any limitation on methods that might be used. I believe it's turned out to be interesting to see a variety of different approaches. It turns out that the aim of the ...
1
vote
3answers
202 views

Prove, that $\sin x- a^3\cos x\leq \frac 1 3 \sqrt{1+a^6}$

Let a and $x$ be natural numbers with the property that $\sin x\leq a\cos x$. Prove that $\sin x- a^3 \cos x\leq \frac 1 3 \sqrt{1+a^6}$. Again, I'm looking for a second solution. I don't know how to ...
2
votes
2answers
152 views

Proof of $X+\csc{\left(\frac{\pi}{X}\right)}>2\csc{\left(\frac{\pi}{2X}\right)}$

When $X$ is greater than 1, I want to prove that $X+\csc{\left(\frac{\pi}{X}\right)}>2\csc{\left(\frac{\pi}{2X}\right)}$ where $\csc{(\cdot)}=\frac{1}{\sin{(\cdot)}}$. Plotting the above ...
4
votes
0answers
68 views

Conditions to satisfy trigonometric inequality

I'm looking for sufficient (and necessary would be good too) conditions on $a,b,c$ such that \begin{align} a\cos\phi + b \cos 3\phi + c \cos 5\phi \geq -1 \hspace{20pt} (\forall \phi) \end{align} ...
12
votes
1answer
308 views

Prove $\left|\sum_{k=2001}^{m}a_{k}\sin{(kx)}\right|\le 1+\pi $ ,$m\ge 2001,x\in R$

let $\{a_{n}\}$ is non-increasing postive sequence;show that if for $n\ge 2001,na_{n}\le 1$, then for any positive integer numbers $m\ge 2001,x\in R$, we have ...
3
votes
2answers
210 views

Showing $|\sum_{k=1}^n\frac{\sin(kx)}k|<2\sqrt{\pi}$

For any real $x$ and positive integer $n$, is it true that: $$\left|\sum_{k=1}^n\frac{\sin(kx)}k\right|<2\sqrt{\pi}\quad ?$$ Please justify.
3
votes
2answers
131 views

Finding all $x$ such that $|\tan x | \leq 2\sin x$

I need to find all real numbers $x$ that satisfy: $$|\tan x | \leq 2\sin x \text{ and } x \in [ -\pi, \pi]$$ in terms of unions of intervals. I know it's equivalent to: $-2\sin x \leq \tan x \leq 2 ...
4
votes
1answer
109 views

An inequality involving arctan of complex argument

I have the following conjecture: \begin{equation} \text{Re}\left[(1+\text{i}y)\arctan\left(\frac{t}{1+\text{i}y}\right)\right] \ge \arctan(t), \qquad \forall y,t\ge0. \end{equation} Which seems to be ...