Tagged Questions

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Expand trigonometric expression

I am supposed to expand this expression $${\frac {\sin \left( x \right) b \left( 4\,b\cos \left( x \right) + \sqrt {16\,{b}^{2}+1}+5 \right) }{4\,b\cos \left( x \right) +\sqrt {16 \,{b}^{2}+1}+1}}$$ ...
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Integral $\int_{0}^{\pi/2} \arctan \left(2\tan^2 x\right) \mathrm{d}x$

The following integral may seem easy to evaluate ... $$\int_{0}^{\Large\frac{\pi}{2}} \arctan \left(2 \tan^2 x\right) \mathrm{d}x = \pi \arctan \left( \frac{1}{2} \right).$$ Could you prove ...
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Pick a smart function

Our teacher wants us to find a function $f$ on $(0,\pi)$ such that $$\sqrt{\sin(x)} f(x)^{\frac{1}{4}} =k_1 + \cos(x)$$ and $$\sqrt{\sin(x)} f(x)^{-\frac{1}{4}} = k_2 + \cos(x).$$ The two constants ...
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Problem related to Mean Value Theorem

I found out a question that I can't figure out a way to solve it. Plz can anyone help me. Question is, Prove that $\exists\,C\in(0,\pi/4)\,\mathrm{s.t.}\,\tan(\pi/4+C)=3/C$ I know this should be ...
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Is $\sin (\mathbb N)$ dense in $[-1,1]$? [duplicate]

Let $\mathbb N$ be the set of positive integers, then is it true that $\sin (\mathbb N)$ is dense in $[-1,1]$ i.e. is it true that for every $x,y \in [-1,1]$ with $x<y$ , $\exists m \in \mathbb N$ ...
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A cycloid that goes through the beginning and through a general point

Parametric equations of the general cycloid through the beginning $(0,0)$ are $$x(t)=\frac{2t-\sin2t}{2d}$$ $$y(t)=\frac{1-\cos 2t}{2d}$$ How can we determine $d$ such that the cycloid goes through ...
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Evaluate position of first secondary maximum of $\frac{\sin N (x/2)}{\sin (x/2)}$

The function $$f(x) = \displaystyle \left | \frac{\sin \left( N \displaystyle \frac{x}{2} \right)}{\sin \left( \displaystyle \frac{x}{2} \right)} \right |$$ when evaluated for $x > 0$, has its ...
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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?
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Proof for $\forall x$, $\exists y$ s.t $1/2 \le |x-y|\le 1$ and $|\cos \pi x - \cos \pi y|\ge 1$

I am trying to understand a certain proof for the indifferentiability of the Weierstrass function, that strongly uses the following lemma: For all $x \in \mathbb{R}$ there exists $y \in \mathbb{R}$ ...
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Trigonometry Inequality Involving Powers of Sin, Cos.

Prove that for $0 < x < \frac{\pi}4$, $$\sin x^ {\sin x} < \cos x ^{\cos x}.$$ I don't have any nice ideas. I was thinking about taking the natural log and looking at the taylor series of ...
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