query: Why is $\sin : \mathbb{R} \to [-5,5] $ different from $\sin : \mathbb{R} \to \mathbb{R}$?
positive_passages.docid: 155434
positive_passages.text: How is the codomain for a function defined?
negative_passages.docid: 113898
negative_passages.text: Why is that $\widehat{\mathbb{R}/\mathbb{Z}}\cong\mathbb{Z}$?
negative_passages.docid: 8352
negative_passages.text: When is $\sin x$ an algebraic number and when is it non-algebraic?
negative_passages.docid: 19754
negative_passages.text: $\mathbb{R}$ \ $\mathbb{Q}$ and $\mathbb{R}^2\setminus\mathbb{Q}^2$ disconnected?
negative_passages.docid: 126904
negative_passages.text: math.sin incorrect result
negative_passages.docid: 141273
negative_passages.text: When is $\sin(x)$ rational?
negative_passages.docid: 84189
negative_passages.text: Is $\mathbb{C}$ equal to $\mathbb{R}^2$?
negative_passages.docid: 88474
negative_passages.text: $\mathbb{A}^{2}$ not isomorphic to affine space minus the origin
negative_passages.docid: 35086
negative_passages.text: Why graph of $\sin(x+{45}^\circ$) shifts to left instead of right
negative_passages.docid: 157364
negative_passages.text: $\int_{-2}^{2} \sin(x^5)e^{(x^8\sin(x^4))} dx$
negative_passages.docid: 15909
negative_passages.text: Bijection from $\mathbb R$ to $\mathbb {R^N}$
negative_passages.docid: 27754
negative_passages.text: Is integration from $a$ to $b$ same or $b$ to $a$ or is negative?
negative_passages.docid: 101506
negative_passages.text: Why is $1, (x-5)^2, (x-5)^3$ a basis of $U=\{p \in \mathcal P_3(\mathbb R) \mid p'(5)=0\}$?
negative_passages.docid: 12110
negative_passages.text: How are $2\mathbb{Z}\ncong3\mathbb{Z}$ different as rings?
negative_passages.docid: 57037
negative_passages.text: How does $(a,b)$ where $a,b\in \mathbb{Q}$ form a basis for $\mathbb{R}$?
negative_passages.docid: 121729
negative_passages.text: Prove without using graphing calculators that $f: \mathbb R\to \mathbb R,\,f(x)=x+\sin x$ is both one-to-one, onto (bijective) function.
negative_passages.docid: 23837
negative_passages.text: Why is $\mathbb{R}/{\sim}$ not first countable at $[0]$, where $x \sim y \Leftrightarrow x = y\text{ or }x,y \in \mathbb{Z}$?