1
vote
2answers
42 views

$m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $

Prove that $m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $ for every $m, n, l >0$.
0
votes
1answer
64 views

Prove $\sin(x)< x$ when $x>0$ using LMVT

According to Lagrange's Mean Value Theorem (LMVT), if a function $f(x)$ is continuous on $\left[a,b\right]$ and differentiable on $\left(a,b\right)$, then there exists some constant $c$ such that ...
3
votes
5answers
69 views

Proving a trigonometric inequality $(1-\sin a)x^2 -2x\cos a + 1+ \sin a \ge 0$

$(1-\sin (a))x^2 -2x\cos(a) + 1+ \sin( a) \ge 0$, where $a,x$ are any two real constants. Any suggestions on how to prove this ? I tried playing with it, but nothing useful came out.
11
votes
5answers
240 views

Proof: $\cos^p (\theta) \le \cos(p\theta)$

I came across this problem when I was at a book store inside of a book made to prepare Berkeley graduates to pass a mandatory exam. I wanted to buy the book, but, alas, I didn't have the money (forty ...
3
votes
2answers
123 views

How to prove this $\frac{\sin{(nx)}}{\sin{x}}\ge\frac{\sqrt{3}}{3}(2n-1)^{\frac{3}{4}}$

let $n<\dfrac{\pi}{2\arccos{\dfrac{c}{2}}},c\in (0,2),c=2\cos{x}$, show that $$\dfrac{\sin{(nx)}}{\sin{x}}\ge\dfrac{\sqrt{3}}{3}(2n-1)^{\frac{3}{4}}$$ where $0<x<\dfrac{\pi}{2}$ My idea: ...
1
vote
1answer
35 views

Trigonometric inequality question [closed]

Let $0 < A < \frac {\pi}{2}$ and $0 < B < \frac {\pi}{2}$. (a) prove that $\sec^2 A + \csc^2 A \cdot \csc^2 B \cdot \sec^2 B \geq 9.$ (b) determine values of $\sec A$ and $\sec B$ when ...
1
vote
0answers
79 views

Proving $\frac\pi{22}\cos\frac\pi{22}+\frac{2\pi}{11}\cos\frac{5\pi }{22}+\frac{2\pi}{ 11}\cos\frac{9\pi}{22}+\frac\pi{22}\cos\frac{5\pi}{11}<\cdots$

$$(\frac{\pi}{22}) \cos (\frac{\pi}{22}) +(\frac{2\pi}{11}) \cos (\frac{5\pi }{22}) + (\frac{2\pi}{ 11}) \cos (\frac{9\pi}{22}) + (\frac{\pi}{22}) \cos(\frac{5\pi}{11}) < (\frac{\pi}{26}) ...
5
votes
1answer
62 views

Prove $\sin^{2m}\alpha\cdot\cos^{2n}\alpha\leq\frac{m^m n^n}{(m+n)^{(m+n)}}$

If $n$ and $m$ are natural numbers, Prove: $$\sin^{2m}\alpha\cos^{2n}\alpha\leq\frac{m^mn^n}{(m+n)^{(m+n)}}$$ Additional info:We should only use AM-GM inequality.We can use Trigonometry ...
2
votes
2answers
57 views

Trigonometric inequation $\sin x \ne \sin y$

How can I solve the following trigonometric inequation? $$\sin\left(x\right)\ne \sin\left(y\right)\>,\>x,y\in \mathbb{R}$$ Why I'm asking this question... I was doing my calculus homework, ...
1
vote
1answer
34 views

How prove $ \frac{\cos x\cos y-4}{\cos x+\cos y-4}\le1+\frac{1}{2}\cos(\frac{x+y}{\cos x+\cos y-4}) $?

For any $x,y\in[0,\frac{\pi}{2}]$ , how prove the inequality $\frac{\cos x\cos y-4}{\cos x+\cos y-4}\le1+\frac{1}{2}\cos(\frac{x+y}{\cos x+\cos y-4})$?
1
vote
1answer
23 views

$S$, $I$, $O$ are circumcenter, incenter and orthocenter then $SO\ge IO \sqrt2$

Let $S$, $I$ and $O$ be the circumcenter, incenter and orthocenter of $\triangle ABC$ then prove that $SO\ge IO \sqrt2$, or equivalently $SO^2\ge 2IO^2$. I was able to derive an expression for $SO^2$ ...
2
votes
1answer
34 views

How find all $n\in \mathbb N$ such that $\cot \left(\frac{x}{2^{n+1}}\right)-\cot(x)>2^n$?

How find all $n\in \mathbb N$ such that $\cot \left(\frac{x}{2^{n+1}}\right)-\cot(x)>2^n$ for $x \in (0,\pi)$?
2
votes
1answer
111 views

How prove that $\max(|f(1)|,|f(2)|,|f(3)|,|f(4)|)\geq \frac{1}{2}$ if $f(x) = \cos(Ax)+\cos(Bx)$?

Let $ A, B$ be real numbers and $ f(x) =\cos(Ax) + \cos(Bx)$. How prove that $ \max(|f(1)|,|f(2)|,|f(3)|,|f(4)|)\geq \frac{1}{2}$?
2
votes
6answers
234 views

Algebraic proof of $\tan x>x$

I'm looking for a non-calculus proof of the statement that $\tan x>x$ on $(0,\pi/2)$, meaning "not using derivatives or integrals." (The calculus proof: if $f(x)=\tan x-x$ then $f'(x)=\sec^2 ...
0
votes
0answers
40 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
59 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
58 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
113 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 ...
2
votes
3answers
80 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
112 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
37 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
27 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
37 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
29 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
119 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
87 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
49 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
114 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
97 views

Proving $\cos 36° > \tan 36° $

How do we prove that $\cos 36° > \tan 36° $ ? Please help . Thank you.
0
votes
2answers
41 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
29 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
83 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
49 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
220 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
100 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 ...
2
votes
2answers
104 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
60 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
64 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
67 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
408 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
32 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 $$ ...