# Tagged Questions

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 $m, n, l >0$.
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 ...
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.
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 ...
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: ...
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 ...
79 views

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 $k\in\mathbb{Z}$. ...
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.