query: Let $f(r,\theta)=(r\cos\theta ,r\sin\theta)$ for $(r,\theta)$ $\in \mathbb R^2$ with $r\ne0$. then which are true statements
positive_passages.docid: 143200
positive_passages.text: $f(r,\theta)=(r\cos\theta,r\sin\theta)$ Then
negative_passages.docid: 73047
negative_passages.text: Proving surjectivity of $\cos(z)$ and $\sin(z)$ and find all $z : \cos(z) \in \mathbb R$ and all $z: \sin(z) \in \mathbb R$
negative_passages.docid: 33791
negative_passages.text: Is it generally true that $\arcsin \theta + \arccos \theta = \frac{\pi}{2}$?
negative_passages.docid: 141423
negative_passages.text: The continuity of $f(x,y) = \frac {\sin (xy)} {\sqrt {x^2 + y^2}}$ at $(0,0)$
negative_passages.docid: 24128
negative_passages.text: How do i show that $w=z^2$ maps $|z-1|\leq 1$ onto $R=2(1+\cos \theta)$?
negative_passages.docid: 86193
negative_passages.text: Rigorous proof that $dx dy=r\ dr\ d\theta$
negative_passages.docid: 60230
negative_passages.text: Prove that $\frac{1+\sin\theta+i\cos\theta}{1+\sin\theta-i\cos\theta}=\sin\theta+i\cos\theta$
negative_passages.docid: 50012
negative_passages.text: $f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
negative_passages.docid: 136056
negative_passages.text: $ f:\mathbb C\rightarrow \mathbb C$ defined by $f(z)=0$, if $Re(z)=0$ or $Im(z)=0$ and $f(z)=z$, otherwise.
negative_passages.docid: 143354
negative_passages.text: Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
negative_passages.docid: 60761
negative_passages.text: Proof: $\cos^p (\theta) \le \cos(p\theta)$
negative_passages.docid: 143900
negative_passages.text: how can be prove that $\max(f(n),g(n)) = \Theta(f(n)+g(n))$
negative_passages.docid: 138864
negative_passages.text: If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ and $f'(0)$ exists, then $f$ is differentiable on $\mathbb R$
negative_passages.docid: 98570
negative_passages.text: Prove whether $f(n)$ is $O$, $o$, $\Omega$, $\omega$ or $\Theta$ of $g(n)$. $f(n) = n + (\log n)^{2}$, $g(n) = n + \log(n^{2})$.
negative_passages.docid: 79969
negative_passages.text: Proof that a formula $\theta$ is valid if and only if $\forall x \theta$ is valid
negative_passages.docid: 112648
negative_passages.text: For every twice differentiable function $f : \bf R \rightarrow [–2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$, which of the following statements are TRUE?
negative_passages.docid: 19916
negative_passages.text: $f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$