id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-7800 | Cut-Vertex divides Graph into Two or More Components | Let $G$ be a graph.
Let $v$ be a cut-vertex of $G$.
Then the vertex deletion $G - v$ contains $2$ or more components. | By definition of cut-vertex, $G - v$ contains at least $2$ components.
That it can contain more components than $2$ is best proved by illustration:
:520px
{{qed}} | Let $G$ be a [[Definition:Graph (Graph Theory)|graph]].
Let $v$ be a [[Definition:Cut-Vertex|cut-vertex]] of $G$.
Then the [[Definition:Vertex Deletion|vertex deletion]] $G - v$ contains $2$ or more [[Definition:Component (Graph Theory)|components]]. | By definition of [[Definition:Cut-Vertex|cut-vertex]], $G - v$ contains at least $2$ components.
That it can contain more components than $2$ is best proved by illustration:
:[[File:BigCutVertex.png|520px]]
{{qed}} | Cut-Vertex divides Graph into Two or More Components | https://proofwiki.org/wiki/Cut-Vertex_divides_Graph_into_Two_or_More_Components | https://proofwiki.org/wiki/Cut-Vertex_divides_Graph_into_Two_or_More_Components | [
"Graph Theory"
] | [
"Definition:Graph (Graph Theory)",
"Definition:Cut-Vertex",
"Definition:Vertex Deletion",
"Definition:Component of Graph"
] | [
"Definition:Cut-Vertex",
"File:BigCutVertex.png"
] |
proofwiki-7801 | Bridge divides Graph into Two Components | Let $G$ be a connected graph.
Let $e$ be a bridge of $G$.
Then the edge deletion $G - e$ contains exactly $2$ components. | Let $G$ be a connected graph and $e = u v$ be a bridge of $G$.
By definition of bridge, $G - e$ has to be of at least $2$ components.
{{AimForCont}} $G - e$ were of more than $2$ components.
Let $G_1, G_2, G_3$ be $3$ of those components such that $u \in G_1$ and $v \in G_2$.
Note that $u$ and $v$ cannot both be in the... | Let $G$ be a [[Definition:Connected Graph|connected graph]].
Let $e$ be a [[Definition:Bridge (Graph Theory)|bridge]] of $G$.
Then the [[Definition:Edge Deletion|edge deletion]] $G - e$ contains exactly $2$ [[Definition:Component (Graph Theory)|components]]. | Let $G$ be a [[Definition:Connected Graph|connected graph]] and $e = u v$ be a [[Definition:Bridge (Graph Theory)|bridge]] of $G$.
By definition of [[Definition:Bridge (Graph Theory)|bridge]], $G - e$ has to be of at least $2$ [[Definition:Component (Graph Theory)|components]].
{{AimForCont}} $G - e$ were of more th... | Bridge divides Graph into Two Components | https://proofwiki.org/wiki/Bridge_divides_Graph_into_Two_Components | https://proofwiki.org/wiki/Bridge_divides_Graph_into_Two_Components | [
"Graph Theory"
] | [
"Definition:Connected (Graph Theory)/Graph",
"Definition:Bridge (Graph Theory)",
"Definition:Edge Deletion",
"Definition:Component of Graph"
] | [
"Definition:Connected (Graph Theory)/Graph",
"Definition:Bridge (Graph Theory)",
"Definition:Bridge (Graph Theory)",
"Definition:Component of Graph",
"Definition:Component of Graph",
"Definition:Component of Graph",
"Definition:Component of Graph",
"Definition:Bridge (Graph Theory)",
"Definition:Con... |
proofwiki-7802 | Connected Graph with only Even Vertices has no Bridge | Let $G$ be a connected graph whose vertices are all even.
Then no edge of $G$ is a bridge. | Let the vertices of $G$ all be even.
Then by Characteristics of Eulerian Graph, $G$ is Eulerian.
By definition of Eulerian, $G$ therefore contains a Eulerian circuit.
Thus every edge of $G$ lies on a circuit of $G$.
From Condition for Edge to be Bridge, if an edge $e$ of $G$ is a bridge, then it does not lie on a circu... | Let $G$ be a [[Definition:Connected Graph|connected graph]] whose [[Definition:Vertex of Graph|vertices]] are all [[Definition:Even Vertex of Graph|even]].
Then no [[Definition:Edge of Graph|edge]] of $G$ is a [[Definition:Bridge (Graph Theory)|bridge]]. | Let the [[Definition:Vertex of Graph|vertices]] of $G$ all be [[Definition:Even Vertex of Graph|even]].
Then by [[Characteristics of Eulerian Graph]], $G$ is [[Definition:Eulerian Graph|Eulerian]].
By definition of [[Definition:Eulerian Graph|Eulerian]], $G$ therefore contains a [[Definition:Eulerian Circuit|Eulerian... | Connected Graph with only Even Vertices has no Bridge | https://proofwiki.org/wiki/Connected_Graph_with_only_Even_Vertices_has_no_Bridge | https://proofwiki.org/wiki/Connected_Graph_with_only_Even_Vertices_has_no_Bridge | [
"Graph Theory"
] | [
"Definition:Connected (Graph Theory)/Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Even Vertex of Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Bridge (Graph Theory)"
] | [
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Even Vertex of Graph",
"Characteristics of Eulerian Graph",
"Definition:Eulerian Graph",
"Definition:Eulerian Graph",
"Definition:Eulerian Circuit",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Circuit (Graph Theory)",
"Condition for Edge ... |
proofwiki-7803 | Graph of Cube is Hamiltonian | The graph of the cube is Hamiltonian. | Proof by demonstration:
:400px
A Hamiltonian cycle is indicated in {{color|blue}}.
{{qed}} | The [[Definition:Platonic Graph of Cube|graph of the cube]] is [[Definition:Hamiltonian Graph|Hamiltonian]]. | Proof by demonstration:
:[[File:CubeGraphHamiltonian.png|400px]]
A [[Definition:Hamiltonian Cycle|Hamiltonian cycle]] is indicated in {{color|blue}}.
{{qed}} | Graph of Cube is Hamiltonian | https://proofwiki.org/wiki/Graph_of_Cube_is_Hamiltonian | https://proofwiki.org/wiki/Graph_of_Cube_is_Hamiltonian | [
"Hamiltonian Graphs"
] | [
"Definition:Platonic Graph/Cube",
"Definition:Hamiltonian Graph"
] | [
"File:CubeGraphHamiltonian.png",
"Definition:Hamilton Cycle"
] |
proofwiki-7804 | Graph of Icosahedron is Hamiltonian | The graph of the icosahedron is Hamiltonian. | Proof by demonstration:
:400px
A Hamiltonian cycle is indicated in <span style="color:blue">blue</span>. | The [[Definition:Platonic Graph/Icosahedron|graph of the icosahedron]] is [[Definition:Hamiltonian Graph|Hamiltonian]]. | Proof by demonstration:
:[[File:IcosahedronGraphHamiltonian.png|400px]]
A [[Definition:Hamiltonian Cycle|Hamiltonian cycle]] is indicated in <span style="color:blue">blue</span>. | Graph of Icosahedron is Hamiltonian | https://proofwiki.org/wiki/Graph_of_Icosahedron_is_Hamiltonian | https://proofwiki.org/wiki/Graph_of_Icosahedron_is_Hamiltonian | [
"Hamiltonian Graphs"
] | [
"Definition:Platonic Graph/Icosahedron",
"Definition:Hamiltonian Graph"
] | [
"File:IcosahedronGraphHamiltonian.png",
"Definition:Hamilton Cycle"
] |
proofwiki-7805 | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Necessary Condition | Let $T$ be a tree of order $n$.
Then the size of $T$ is $n-1$. | Suppose $T$ is a tree with $n$ nodes. We need to show that $T$ has $n - 1$ edges.
The proof proceeds by strong induction.
Let $T_n$ be a tree with $n$ nodes.
For all $n \in \N_{>0}$, let $\map P n$ be the proposition that a tree with $n$ nodes has $n - 1$ edges.
=== Basis for the Induction ===
$\map P 1$ says that a tr... | Let $T$ be a [[Definition:Tree (Graph Theory)|tree]] of [[Definition:Order of Graph|order]] $n$.
Then the [[Definition:Size of Graph|size]] of $T$ is $n-1$. | Suppose $T$ is a [[Definition:Tree (Graph Theory)|tree]] with $n$ [[Definition:Node of Tree|nodes]]. We need to show that $T$ has $n - 1$ [[Definition:Edge of Graph|edges]].
The proof proceeds by [[Principle of Strong Induction|strong induction]].
Let $T_n$ be a [[Definition:Tree (Graph Theory)|tree]] with $n$ [[De... | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Necessary Condition | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Necessary_Condition | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Necessary_Condition | [
"Finite Connected Simple Graph is Tree iff Size is One Less than Order"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Tree (Graph Theory)/Node",
"Definition:Graph (Graph Theory)/Edge",
"Second Principle of Mathematical Induction",
"Definition:Tree (Graph Theory)",
"Definition:Tree (Graph Theory)/Node",
"Definition:Proposition",
"Definition:Tree (Graph Theory)/Node",
"De... |
proofwiki-7806 | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Necessary Condition | Let $T$ be a tree of order $n$.
Then the size of $T$ is $n-1$. | Let $T_{k + 1}$ be an arbitrary tree with $k + 1$ nodes.
Take any node $v$ of $T_{k + 1}$ of degree $1$.
Such a node exists from Finite Tree has Leaf Nodes.
Consider $T_k$, the subgraph of $T_{k + 1}$ created by removing $v$ and the edge connecting it to the rest of the tree.
By Connected Subgraph of Tree is Tree, $T_k... | Let $T$ be a [[Definition:Tree (Graph Theory)|tree]] of [[Definition:Order of Graph|order]] $n$.
Then the [[Definition:Size of Graph|size]] of $T$ is $n-1$. | Let $T_{k + 1}$ be an arbitrary [[Definition:Tree (Graph Theory)|tree]] with $k + 1$ [[Definition:Node of Tree|nodes]].
Take any [[Definition:Node of Tree|node]] $v$ of $T_{k + 1}$ of [[Definition:Degree of Vertex|degree]] $1$.
Such a node exists from [[Finite Tree has Leaf Nodes]].
Consider $T_k$, the [[Definition:... | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Necessary Condition/Induction Step/Proof 1 | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Necessary_Condition | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Necessary_Condition/Induction_Step/Proof_1 | [
"Finite Connected Simple Graph is Tree iff Size is One Less than Order"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Tree (Graph Theory)/Node",
"Definition:Tree (Graph Theory)/Node",
"Definition:Degree of Vertex",
"Finite Tree has Leaf Nodes",
"Definition:Subgraph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Tree (Graph Theory)",
"Connected Subgraph of Tree is... |
proofwiki-7807 | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Necessary Condition | Let $T$ be a tree of order $n$.
Then the size of $T$ is $n-1$. | Let $T_{k + 1}$ be an arbitrary tree with $k + 1$ nodes.
Remove any edge $e$ of $T_{k + 1}$.
By definition of tree $T_{k + 1}$ has no circuits.
Therefore from Condition for Edge to be Bridge it follows that $e$ must be a bridge.
So removing $e$ disconnects $T_{k + 1}$ into two trees $T_1$ and $T_2$, with $k_1$ and $k_2... | Let $T$ be a [[Definition:Tree (Graph Theory)|tree]] of [[Definition:Order of Graph|order]] $n$.
Then the [[Definition:Size of Graph|size]] of $T$ is $n-1$. | Let $T_{k + 1}$ be an arbitrary [[Definition:Tree (Graph Theory)|tree]] with $k + 1$ [[Definition:Node of Tree|nodes]].
Remove any [[Definition:Edge of Graph|edge]] $e$ of $T_{k + 1}$.
By definition of [[Definition:Tree (Graph Theory)|tree]] $T_{k + 1}$ has no [[Definition:Circuit (Graph Theory)|circuits]].
Therefor... | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Necessary Condition/Induction Step/Proof 2 | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Necessary_Condition | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Necessary_Condition/Induction_Step/Proof_2 | [
"Finite Connected Simple Graph is Tree iff Size is One Less than Order"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Tree (Graph Theory)/Node",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Tree (Graph Theory)",
"Definition:Circuit (Graph Theory)",
"Condition for Edge to be Bridge",
"Definition:Bridge (Graph Theory)",
"Definition:Connected (Graph Theory)/Graph/Disc... |
proofwiki-7808 | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Sufficient Condition | Let $T$ be a finite connected simple graph of order $n$.
Let the size of $T$ be $n - 1$.
Then $T$ is a (finite) tree. | Let $T$ is a connected simple graph of order $n$ with size $n - 1$.
From Finite Connected Simple Graph with Size One Less than Order has no Circuits:
:$T$ has no circuits.
Hence $T$ is a tree by definition.
{{qed}} | Let $T$ be a [[Definition:Finite Graph|finite]] [[Definition:Connected Graph|connected]] [[Definition:Simple Graph|simple graph]] of [[Definition:Order of Graph|order]] $n$.
Let the [[Definition:Size of Graph|size]] of $T$ be $n - 1$.
Then $T$ is a [[Definition:Finite Tree|(finite) tree]]. | Let $T$ is a [[Definition:Connected Graph|connected]] [[Definition:Simple Graph|simple graph]] of [[Definition:Order of Graph|order]] $n$ with [[Definition:Size of Graph|size]] $n - 1$.
From [[Finite Connected Simple Graph with Size One Less than Order has no Circuits]]:
:$T$ has no [[Definition:Circuit (Graph Theory)... | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Sufficient Condition | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Sufficient_Condition | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Sufficient_Condition | [
"Finite Connected Simple Graph is Tree iff Size is One Less than Order"
] | [
"Definition:Finite Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size",
"Definition:Tree (Graph Theory)/Finite"
] | [
"Definition:Connected (Graph Theory)/Graph",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size",
"Finite Connected Simple Graph with Size One Less than Order has no Circuits",
"Definition:Circuit (Graph Theory)",
"Definition:Tree (Graph Theory)"
] |
proofwiki-7809 | 1+1 = 2 | Define $0$ as the unique element in the set $P \setminus \map s P$, where:
:$P$ is the Peano Structure
:$\map s P$ is the image of the mapping $s$ defined in Peano structure
:$\setminus$ denotes the set difference.
Then:
:$1 + 1 = 2$
where:
:$1 := \map s 0$
:$2 := \map s 1 = \map s {\map s 0}$
:$+$ denotes addition
:$=... | $1$ is defined {{hypothesis}} as $\map s 0$ and $2$ as $\map s {\map s 0}$.
Hence the statement to be proven becomes:
:$\map s 0 + \map s 0 = \map s {\map s 0}$
Thus:
{{begin-eqn}}
{{eqn | q = \forall m, n \in P
| l = m + \map s n
| r = \map s {m + n}
| c = {{Defof|Addition in Peano Structure}}
}}
{{e... | Define $0$ as the [[Non-Successor Element of Peano Structure is Unique|unique]] [[Definition:Element|element]] in the [[Definition:Set|set]] $P \setminus \map s P$, where:
:$P$ is the [[Definition:Peano Structure|Peano Structure]]
:$\map s P$ is the [[Definition:Image of Mapping|image of the mapping]] $s$ defined in [[... | $1$ is defined {{hypothesis}} as $\map s 0$ and $2$ as $\map s {\map s 0}$.
Hence the statement to be proven becomes:
:$\map s 0 + \map s 0 = \map s {\map s 0}$
Thus:
{{begin-eqn}}
{{eqn | q = \forall m, n \in P
| l = m + \map s n
| r = \map s {m + n}
| c = {{Defof|Addition in Peano Structure}}
}}
... | 1+1 = 2/Proof 1 | https://proofwiki.org/wiki/1+1_=_2 | https://proofwiki.org/wiki/1+1_=_2/Proof_1 | [
"Mathematical Logic",
"1+1 = 2"
] | [
"Non-Successor Element of Peano Structure is Unique",
"Definition:Element",
"Definition:Set",
"Definition:Peano Structure",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Peano Structure",
"Definition:Set Difference",
"Definition:Addition/Peano Structure",
"Definition:Equals",
"Defin... | [
"Definition:Successor Mapping"
] |
proofwiki-7810 | 1+1 = 2 | Define $0$ as the unique element in the set $P \setminus \map s P$, where:
:$P$ is the Peano Structure
:$\map s P$ is the image of the mapping $s$ defined in Peano structure
:$\setminus$ denotes the set difference.
Then:
:$1 + 1 = 2$
where:
:$1 := \map s 0$
:$2 := \map s 1 = \map s {\map s 0}$
:$+$ denotes addition
:$=... | Defining $1$ as $\map s 0$ and $2$ as $\map s {\map s 0}$, the statement to be proven becomes:
:$\map s 0 + \map s 0 = \map s {\map s 0}$
By the definition of addition:
:$\forall m \in P: \forall n \in P: m + \map s n = \map s {m + n}$
Letting $m = \map s 0$ and $n = 0$:
{{begin-eqn}}
{{eqn | n = 1
| l = \map s 0... | Define $0$ as the [[Non-Successor Element of Peano Structure is Unique|unique]] [[Definition:Element|element]] in the [[Definition:Set|set]] $P \setminus \map s P$, where:
:$P$ is the [[Definition:Peano Structure|Peano Structure]]
:$\map s P$ is the [[Definition:Image of Mapping|image of the mapping]] $s$ defined in [[... | Defining $1$ as $\map s 0$ and $2$ as $\map s {\map s 0}$, the statement to be proven becomes:
:$\map s 0 + \map s 0 = \map s {\map s 0}$
By the definition of [[Definition:Addition in Peano Structure|addition]]:
:$\forall m \in P: \forall n \in P: m + \map s n = \map s {m + n}$
Letting $m = \map s 0$ and $n = 0$:
... | 1+1 = 2/Proof 2 | https://proofwiki.org/wiki/1+1_=_2 | https://proofwiki.org/wiki/1+1_=_2/Proof_2 | [
"Mathematical Logic",
"1+1 = 2"
] | [
"Non-Successor Element of Peano Structure is Unique",
"Definition:Element",
"Definition:Set",
"Definition:Peano Structure",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Peano Structure",
"Definition:Set Difference",
"Definition:Addition/Peano Structure",
"Definition:Equals",
"Defin... | [
"Definition:Addition/Peano Structure",
"Definition:Addition/Peano Structure",
"Definition:Successor Mapping on Natural Numbers",
"Equality is Transitive"
] |
proofwiki-7811 | Kruskal's Algorithm produces Minimum Spanning Tree | Kruskal's Algorithm produces a minimum spanning tree. | Let $N = \struct {V, E, f}$ be an undirected network.
Let $H$ be a subgraph of the underlying graph $G = \struct {V, E}$ of $N$.
Let $\ds \map f H = \sum_{e \mathop \in H} \map f e$.
Let $N$ be of order $p$.
Let $T$ be a spanning tree of $N$, created using Kruskal's Algorithm.
Then from Finite Connected Simple Graph is... | [[Kruskal's Algorithm]] produces a [[Definition:Minimum Spanning Tree|minimum spanning tree]]. | Let $N = \struct {V, E, f}$ be an [[Definition:Undirected Network|undirected network]].
Let $H$ be a [[Definition:Subgraph|subgraph]] of the [[Definition:Underlying Graph|underlying graph]] $G = \struct {V, E}$ of $N$.
Let $\ds \map f H = \sum_{e \mathop \in H} \map f e$.
Let $N$ be of [[Definition:Order of Graph|o... | Kruskal's Algorithm produces Minimum Spanning Tree | https://proofwiki.org/wiki/Kruskal's_Algorithm_produces_Minimum_Spanning_Tree | https://proofwiki.org/wiki/Kruskal's_Algorithm_produces_Minimum_Spanning_Tree | [
"Tree Theory"
] | [
"Kruskal's Algorithm",
"Definition:Minimum Spanning Tree"
] | [
"Definition:Network (Graph Theory)/Undirected",
"Definition:Subgraph",
"Definition:Underlying Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Spanning Tree",
"Kruskal's Algorithm",
"Finite Connected Simple Graph is Tree iff Size is One Less than Order",
"Definition:Graph (Graph Theory)/Ed... |
proofwiki-7812 | Regular Graph is Tree iff Complete Graph of Order 2 | Let $G$ be a non-edgeless regular graph.
Then $G$ is a tree {{iff}} $G$ is $K_2$, the complete graph of order $2$. | === Necessary Condition ===
Let $G$ be a non-edgeless regular graph which is also a tree.
From Finite Tree has Leaf Nodes it follows that $G$ has at least two vertices of degree $1$.
Therefore, for $G$ to be regular it need to be $1$-regular.
Suppose $G$ has $3$ or more vertices.
Let $u, v, w$ be such vertices of $G$.
... | Let $G$ be a [[Definition:Edgeless Graph|non-edgeless]] [[Definition:Regular Graph|regular graph]].
Then $G$ is a [[Definition:Tree (Graph Theory)|tree]] {{iff}} $G$ is $K_2$, the [[Definition:Complete Graph|complete graph of order $2$]]. | === Necessary Condition ===
Let $G$ be a [[Definition:Edgeless Graph|non-edgeless]] [[Definition:Regular Graph|regular graph]] which is also a [[Definition:Tree (Graph Theory)|tree]].
From [[Finite Tree has Leaf Nodes]] it follows that $G$ has at least two [[Definition:Vertex of Graph|vertices]] of [[Definition:Degre... | Regular Graph is Tree iff Complete Graph of Order 2 | https://proofwiki.org/wiki/Regular_Graph_is_Tree_iff_Complete_Graph_of_Order_2 | https://proofwiki.org/wiki/Regular_Graph_is_Tree_iff_Complete_Graph_of_Order_2 | [
"Tree Theory",
"Regular Graphs",
"Complete Graphs"
] | [
"Definition:Edgeless Graph",
"Definition:Regular Graph",
"Definition:Tree (Graph Theory)",
"Definition:Complete Graph"
] | [
"Definition:Edgeless Graph",
"Definition:Regular Graph",
"Definition:Tree (Graph Theory)",
"Finite Tree has Leaf Nodes",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Degree of Vertex",
"Definition:Regular Graph",
"Definition:Regular Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Defin... |
proofwiki-7813 | Perimeter of Circle | The perimeter $C$ of a circle with radius $r$ is given by:
:$C = 2 \pi r$ | By definition, the perimeter is the length of the circumference of the circle.
Let $C$ be the perimeter.
Then:
{{begin-eqn}}
{{eqn | l = \pi
| r = \frac C {2 r}
| c = {{Defof|Pi|$\pi$ (pi)}}
}}
{{eqn | ll= \leadsto
| l = C
| r = 2 \pi r
}}
{{end-eqn}}
{{qed}} | The [[Definition:Perimeter|perimeter]] $C$ of a [[Definition:Circle|circle]] with [[Definition:Radius of Circle|radius]] $r$ is given by:
:$C = 2 \pi r$ | By definition, the [[Definition:Perimeter|perimeter]] is the [[Definition:Arc Length|length]] of the [[Definition:Circumference of Circle|circumference of the circle]].
Let $C$ be the [[Definition:Perimeter|perimeter]].
Then:
{{begin-eqn}}
{{eqn | l = \pi
| r = \frac C {2 r}
| c = {{Defof|Pi|$\pi$ (pi)}... | Perimeter of Circle | https://proofwiki.org/wiki/Perimeter_of_Circle | https://proofwiki.org/wiki/Perimeter_of_Circle | [
"Circles",
"Perimeter Formulas"
] | [
"Definition:Perimeter",
"Definition:Circle",
"Definition:Circle/Radius"
] | [
"Definition:Perimeter",
"Definition:Arc Length",
"Definition:Circle/Circumference",
"Definition:Perimeter"
] |
proofwiki-7814 | Euler's Criterion/Quadratic Residue | Let $p$ be an odd prime.
Let $a \not \equiv 0 \pmod p$.
Then:
{{begin-eqn}}
{{eqn | l = a^{\frac {p-1} 2}
| o = \equiv
| r = 1
| rr= \pmod p
| c = {{iff}} $a$ is a quadratic residue of $p$
}}
{{eqn | l=a ^{\frac {p-1} 2}
| o = \equiv
| r = -1
| rr= \pmod p
| c = {{iff}} $... | Trivially, any $a \not \equiv 0 \pmod p$ is either a quadratic residue or a quadratic non-residue, modulo $p$.
Therefore, it suffices to check the sufficient condition for both of the equations (i.e., the ''if'' parts from the ''iff''s).
So let $a$ be a quadratic non-residue of $p$.
Also, let $b \in \set {1, 2, \ldots,... | Let $p$ be an [[Definition:Odd Prime|odd prime]].
Let $a \not \equiv 0 \pmod p$.
Then:
{{begin-eqn}}
{{eqn | l = a^{\frac {p-1} 2}
| o = \equiv
| r = 1
| rr= \pmod p
| c = {{iff}} $a$ is a [[Definition:Quadratic Residue|quadratic residue]] of $p$
}}
{{eqn | l=a ^{\frac {p-1} 2}
| o = \e... | Trivially, any $a \not \equiv 0 \pmod p$ is either a [[Definition:Quadratic Residue|quadratic residue]] or a [[Definition:Quadratic Non-Residue|quadratic non-residue]], modulo $p$.
Therefore, it suffices to check the sufficient condition for both of the equations (i.e., the ''if'' parts from the ''iff''s).
So let $a... | Euler's Criterion/Quadratic Residue/Proof 1 | https://proofwiki.org/wiki/Euler's_Criterion/Quadratic_Residue | https://proofwiki.org/wiki/Euler's_Criterion/Quadratic_Residue/Proof_1 | [
"Euler's Criterion",
"Quadratic Residues"
] | [
"Definition:Odd Prime",
"Definition:Quadratic Residue",
"Definition:Quadratic Residue/Non-Residue"
] | [
"Definition:Quadratic Residue",
"Definition:Quadratic Residue/Non-Residue",
"Definition:Quadratic Residue/Non-Residue",
"Definition:Congruence (Number Theory)",
"Solution of Linear Congruence",
"Definition:Quadratic Residue",
"Definition:Residue Class",
"Wilson's Theorem",
"Definition:Quadratic Resi... |
proofwiki-7815 | Euler's Criterion/Quadratic Residue | Let $p$ be an odd prime.
Let $a \not \equiv 0 \pmod p$.
Then:
{{begin-eqn}}
{{eqn | l = a^{\frac {p-1} 2}
| o = \equiv
| r = 1
| rr= \pmod p
| c = {{iff}} $a$ is a quadratic residue of $p$
}}
{{eqn | l=a ^{\frac {p-1} 2}
| o = \equiv
| r = -1
| rr= \pmod p
| c = {{iff}} $... | First note that the square roots of $1$ are $1, -1 \pmod p$.
Also, we have that $a^{p - 1} \equiv 1 \pmod p$ by Fermat's Little Theorem.
Combining these two observations, we find:
:$a^{\frac {p - 1} 2} \equiv 1 \text{ or } -1 \pmod p$
The theorem is therefore equivalent to stating that $a$ is a quadratic residue modulo... | Let $p$ be an [[Definition:Odd Prime|odd prime]].
Let $a \not \equiv 0 \pmod p$.
Then:
{{begin-eqn}}
{{eqn | l = a^{\frac {p-1} 2}
| o = \equiv
| r = 1
| rr= \pmod p
| c = {{iff}} $a$ is a [[Definition:Quadratic Residue|quadratic residue]] of $p$
}}
{{eqn | l=a ^{\frac {p-1} 2}
| o = \e... | First note that the [[Square Root of 1 Mod Prime|square roots of $1$]] are $1, -1 \pmod p$.
Also, we have that $a^{p - 1} \equiv 1 \pmod p$ by [[Fermat's Little Theorem]].
Combining these two observations, we find:
:$a^{\frac {p - 1} 2} \equiv 1 \text{ or } -1 \pmod p$
The theorem is therefore equivalent to statin... | Euler's Criterion/Quadratic Residue/Proof 2 | https://proofwiki.org/wiki/Euler's_Criterion/Quadratic_Residue | https://proofwiki.org/wiki/Euler's_Criterion/Quadratic_Residue/Proof_2 | [
"Euler's Criterion",
"Quadratic Residues"
] | [
"Definition:Odd Prime",
"Definition:Quadratic Residue",
"Definition:Quadratic Residue/Non-Residue"
] | [
"Square Root of 1 Mod Prime",
"Fermat's Little Theorem",
"Definition:Quadratic Residue",
"Definition:Quadratic Residue/Non-Residue",
"Definition:Congruence (Number Theory)",
"Definition:Quadratic Residue",
"Congruence of Powers",
"Fermat's Little Theorem",
"Definition:Primitive Root (Number Theory)"... |
proofwiki-7816 | Two Paths between Vertices in Cycle Graph | Let $G$ be a simple graph.
Let $u, v$ be vertices in $G$ such that $u \ne v$.
Then:
:for any two vertices $u, v$ in $G$ such that $u \ne v$ there exists exactly two paths between $u$ and $v$
{{iff}}:
:$G$ is a cycle graph. | === Necessary Condition ===
{{proof wanted}} | Let $G$ be a [[Definition:Simple Graph|simple graph]].
Let $u, v$ be [[Definition:Vertex of Graph|vertices]] in $G$ such that $u \ne v$.
Then:
:for any two [[Definition:Vertex of Graph|vertices]] $u, v$ in $G$ such that $u \ne v$ there exists exactly two [[Definition:Path (Graph Theory)|paths]] between $u$ and $v$
{... | === Necessary Condition ===
{{proof wanted}} | Two Paths between Vertices in Cycle Graph | https://proofwiki.org/wiki/Two_Paths_between_Vertices_in_Cycle_Graph | https://proofwiki.org/wiki/Two_Paths_between_Vertices_in_Cycle_Graph | [
"Cycle Graphs"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Path (Graph Theory)",
"Definition:Cycle Graph"
] | [] |
proofwiki-7817 | Value of Radian in Degrees | The value of a radian in degrees is given by:
:$1 \radians = \dfrac {180 \degrees} {\pi} \approx 57 \cdotp 29577 \, 95130 \ 82320 \, 87679 \, 8154 \ldots \degrees$
{{OEIS|A072097}} | By Measurement of Full Angle, a full angle measures $2 \pi$ radians.
By definition of degree of angle, a full angle measures $360$ degrees.
Thus $1$ radian is given by:
:$1 \radians = \dfrac {360 \degrees} {2 \pi} = \dfrac {180 \degrees} {\pi}$
{{qed}} | The value of a [[Definition:Radian|radian]] in [[Definition:Degree of Angle|degrees]] is given by:
:$1 \radians = \dfrac {180 \degrees} {\pi} \approx 57 \cdotp 29577 \, 95130 \ 82320 \, 87679 \, 8154 \ldots \degrees$
{{OEIS|A072097}} | By [[Measurement of Full Angle]], a [[Definition:Full Angle|full angle]] measures $2 \pi$ [[Definition:Radian|radians]].
By definition of [[Definition:Degree of Angle|degree of angle]], a [[Definition:Full Angle|full angle]] measures $360$ [[Definition:Degree of Angle|degrees]].
Thus $1$ [[Definition:Radian|radian]] ... | Value of Radian in Degrees | https://proofwiki.org/wiki/Value_of_Radian_in_Degrees | https://proofwiki.org/wiki/Value_of_Radian_in_Degrees | [
"Degrees of Angle",
"Radians",
"Angles",
"Unit Conversion"
] | [
"Definition:Angular Measure/Radian",
"Definition:Angular Measure/Degree"
] | [
"Measurements of Common Angles/Full Angle",
"Definition:Full Angle",
"Definition:Angular Measure/Radian",
"Definition:Angular Measure/Degree",
"Definition:Full Angle",
"Definition:Angular Measure/Degree",
"Definition:Angular Measure/Radian"
] |
proofwiki-7818 | Value of Degree in Radians | The value of a degree in radians is given by:
:$1 \degrees = \dfrac {\pi} {180} \radians \approx 0 \cdotp 01745 \, 32925 \, 19943 \, 29576 \, 9236 \ldots \radians$
{{OEIS|A019685}} | By Measurement of Full Angle, a full angle measures $2 \pi$ radians.
By definition of degree of angle, a full angle measures $360$ degrees.
Thus $1$ degree of angle is given by:
:$1 \degrees = \dfrac {2 \pi} {360} = \dfrac \pi {180}$
{{qed}} | The value of a [[Definition:Degree of Angle|degree]] in [[Definition:Radian|radians]] is given by:
:$1 \degrees = \dfrac {\pi} {180} \radians \approx 0 \cdotp 01745 \, 32925 \, 19943 \, 29576 \, 9236 \ldots \radians$
{{OEIS|A019685}} | By [[Measurement of Full Angle]], a [[Definition:Full Angle|full angle]] measures $2 \pi$ [[Definition:Radian|radians]].
By definition of [[Definition:Degree of Angle|degree of angle]], a [[Definition:Full Angle|full angle]] measures $360$ [[Definition:Degree of Angle|degrees]].
Thus $1$ [[Definition:Degree of Angle|... | Value of Degree in Radians | https://proofwiki.org/wiki/Value_of_Degree_in_Radians | https://proofwiki.org/wiki/Value_of_Degree_in_Radians | [
"Degrees of Angle",
"Radians",
"Angles",
"Unit Conversion"
] | [
"Definition:Angular Measure/Degree",
"Definition:Angular Measure/Radian"
] | [
"Measurements of Common Angles/Full Angle",
"Definition:Full Angle",
"Definition:Angular Measure/Radian",
"Definition:Angular Measure/Degree",
"Definition:Full Angle",
"Definition:Angular Measure/Degree",
"Definition:Angular Measure/Degree"
] |
proofwiki-7819 | Difference of Two Squares/Geometric Proof 1 | :$\forall x, y \in \R: x^2 - y^2 = \paren {x + y} \paren {x - y}$ | {{:Euclid:Proposition/II/5}}
:400px
Let $AB$ be cut into equal segments at $C$ and unequal segments at $D$.
Then the rectangle contained by $AD$ and $DB$ together with the square on $CD$ equals the square on $BC$.
(That is, let $x = AC, y = CD$. Then $\paren {x + y} \paren {x - y} + y^2 = x^2$.)
This is proved as follo... | :$\forall x, y \in \R: x^2 - y^2 = \paren {x + y} \paren {x - y}$ | {{:Euclid:Proposition/II/5}}
:[[File:Euclid-II-5.png|400px]]
Let $AB$ be cut into equal segments at $C$ and unequal segments at $D$.
Then the [[Definition:Containment of Rectangle|rectangle contained]] by $AD$ and $DB$ together with the square on $CD$ equals the square on $BC$.
(That is, let $x = AC, y = CD$. Then ... | Difference of Two Squares/Geometric Proof 1 | https://proofwiki.org/wiki/Difference_of_Two_Squares/Geometric_Proof_1 | https://proofwiki.org/wiki/Difference_of_Two_Squares/Geometric_Proof_1 | [
"Difference of Two Squares"
] | [] | [
"File:Euclid-II-5.png",
"Definition:Quadrilateral/Rectangle/Containment",
"Construction of Square on Given Straight Line",
"Construction of Parallel Line",
"Construction of Parallel Line",
"Construction of Parallel Line",
"Complements of Parallelograms are Equal",
"Parallelograms with Equal Base and S... |
proofwiki-7820 | Area of Parallelogram/Square | The area of a square equals the product of one of its bases and the associated altitude. | :190px
From Area of Square:
:$\paren {ABCD} = a^2$
where $a$ is the length of one of the sides of the square.
The altitude of a square is the same as its base.
Hence the result.
{{Qed}} | The [[Definition:Area|area]] of a [[Definition:Square (Geometry)|square]] equals the [[Definition:Real Multiplication|product]] of one of its [[Definition:Base of Parallelogram|bases]] and the associated [[Definition:Altitude of Parallelogram|altitude]]. | :[[File:AreaOfParallelogram-Square.png|190px]]
From [[Area of Square]]:
:$\paren {ABCD} = a^2$
where $a$ is the [[Definition:Length of Line|length]] of one of the [[Definition:Side of Polygon|sides]] of the [[Definition:Square (Geometry)|square]].
The [[Definition:Altitude of Parallelogram|altitude]] of a [[Definitio... | Area of Parallelogram/Square | https://proofwiki.org/wiki/Area_of_Parallelogram/Square | https://proofwiki.org/wiki/Area_of_Parallelogram/Square | [
"Area of Parallelogram"
] | [
"Definition:Area",
"Definition:Quadrilateral/Square",
"Definition:Multiplication/Real Numbers",
"Definition:Parallelogram/Base",
"Definition:Parallelogram/Altitude"
] | [
"File:AreaOfParallelogram-Square.png",
"Area of Square",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Quadrilateral/Square",
"Definition:Parallelogram/Altitude",
"Definition:Quadrilateral/Square",
"Definition:Parallelogram/Base"
] |
proofwiki-7821 | Area of Parallelogram/Rectangle | The area of a rectangle equals the product of one of its bases and the associated altitude. | Let $ABCD$ be a rectangle.
:300px
Then construct the square with side length:
:$\map \Area {AB + BI}$
where $BI = BC$, as shown in the figure above.
Note that $\square CDEF$ and $\square BCHI$ are squares.
Thus:
:$\square ABCD \cong \square CHGF$
Since congruent shapes have the same area:
:$\map \Area {ABCD} = \map \Ar... | The [[Definition:Area|area]] of a [[Definition:Rectangle|rectangle]] equals the product of one of its [[Definition:Base of Parallelogram|bases]] and the associated [[Definition:Altitude of Parallelogram|altitude]]. | Let $ABCD$ be a [[Definition:Rectangle|rectangle]].
:[[File:Area-of-Rectangle.png|300px]]
Then construct the [[Definition:Square (Geometry)|square]] with [[Definition:Side of Polygon|side]] [[Definition:Length of Line|length]]:
:$\map \Area {AB + BI}$
where $BI = BC$, as shown in the figure above.
Note that $\square... | Area of Parallelogram/Rectangle | https://proofwiki.org/wiki/Area_of_Parallelogram/Rectangle | https://proofwiki.org/wiki/Area_of_Parallelogram/Rectangle | [
"Area of Parallelogram"
] | [
"Definition:Area",
"Definition:Quadrilateral/Rectangle",
"Definition:Parallelogram/Base",
"Definition:Parallelogram/Altitude"
] | [
"Definition:Quadrilateral/Rectangle",
"File:Area-of-Rectangle.png",
"Definition:Quadrilateral/Square",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Quadrilateral/Square",
"Definition:Congruence (Geometry)",
"Definition:Area",
"Definition:Area",
"Definition:Geometric F... |
proofwiki-7822 | Area of Parallelogram/Parallelogram | Let $ABCD$ be a parallelogram whose adjacent sides are of length $a$ and $b$ enclosing an angle $\theta$.
The area of $ABCD$ equals the product of one of its bases and the associated altitude:
{{begin-eqn}}
{{eqn | l = \map \Area {ABCD}
| r = b h
| c =
}}
{{eqn | r = a b \sin \theta
| c =
}}
{{end-e... | :425px
Let $ABCD$ be the parallelogram whose area is being sought.
Let $F$ be the foot of the altitude from $C$
Also construct the point $E$ such that $DE$ is the altitude from $D$ (see figure above).
Extend $AB$ to $F$.
Then:
{{begin-eqn}}
{{eqn | l = AD
| o = \cong
| r = BC
}}
{{eqn | l = \angle AED
... | Let $ABCD$ be a [[Definition:Parallelogram|parallelogram]] whose [[Definition:Adjacent Sides|adjacent sides]] are of [[Definition:Length of Line|length]] $a$ and $b$ enclosing an [[Definition:Interior Angle of Polygon|angle]] $\theta$.
The [[Definition:Area|area]] of $ABCD$ equals the [[Definition:Real Multiplication|... | :[[File:Area-of-Parallelogram.png|425px]]
Let $ABCD$ be the [[Definition:Parallelogram|parallelogram]] whose [[Definition:Area|area]] is being sought.
Let $F$ be the foot of the [[Definition:Altitude of Parallelogram|altitude]] from $C$
Also construct the [[Definition:Point|point]] $E$ such that $DE$ is the [[Defini... | Area of Parallelogram/Parallelogram | https://proofwiki.org/wiki/Area_of_Parallelogram/Parallelogram | https://proofwiki.org/wiki/Area_of_Parallelogram/Parallelogram | [
"Area of Parallelogram"
] | [
"Definition:Quadrilateral/Parallelogram",
"Definition:Polygon/Adjacent/Sides",
"Definition:Linear Measure/Length",
"Definition:Polygon/Internal Angle",
"Definition:Area",
"Definition:Multiplication/Real Numbers",
"Definition:Parallelogram/Base",
"Definition:Parallelogram/Altitude",
"Definition:Polyg... | [
"File:Area-of-Parallelogram.png",
"Definition:Quadrilateral/Parallelogram",
"Definition:Area",
"Definition:Parallelogram/Altitude",
"Definition:Point",
"Definition:Parallelogram/Altitude"
] |
proofwiki-7823 | Perimeter of Rectangle | Let $ABCD$ be a rectangle whose side lengths are $a$ and $b$.
The perimeter of $ABCD$ is $2 a + 2 b$. | :300px
From Rectangle is Parallelogram, $ABCD$ is a parallelogram.
By Opposite Sides and Angles of Parallelogram are Equal it follows that:
:$AB = CD$
:$BC = AD$
The perimeter of $ABCD$ is $AB + BC + CD + AD$.
But $AB = CD = a$ and $BC = AD = b$.
Hence the result.
{{qed}} | Let $ABCD$ be a [[Definition:Rectangle|rectangle]] whose [[Definition:Side of Polygon|side]] [[Definition:Length (Linear Measure)|lengths]] are $a$ and $b$.
The [[Definition:Perimeter|perimeter]] of $ABCD$ is $2 a + 2 b$. | :[[File:PerimeterOfRectangle.png|300px]]
From [[Rectangle is Parallelogram]], $ABCD$ is a [[Definition:Parallelogram|parallelogram]].
By [[Opposite Sides and Angles of Parallelogram are Equal]] it follows that:
:$AB = CD$
:$BC = AD$
The [[Definition:Perimeter|perimeter]] of $ABCD$ is $AB + BC + CD + AD$.
But $AB = ... | Perimeter of Rectangle/Proof 1 | https://proofwiki.org/wiki/Perimeter_of_Rectangle | https://proofwiki.org/wiki/Perimeter_of_Rectangle/Proof_1 | [
"Perimeter of Rectangle",
"Rectangles",
"Perimeter Formulas"
] | [
"Definition:Quadrilateral/Rectangle",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Perimeter"
] | [
"File:PerimeterOfRectangle.png",
"Rectangle is Parallelogram",
"Definition:Quadrilateral/Parallelogram",
"Opposite Sides and Angles of Parallelogram are Equal",
"Definition:Perimeter"
] |
proofwiki-7824 | Perimeter of Rectangle | Let $ABCD$ be a rectangle whose side lengths are $a$ and $b$.
The perimeter of $ABCD$ is $2 a + 2 b$. | From Rectangle is Parallelogram, $ABCD$ is a parallelogram.
The result then follows from a direct application of Perimeter of Parallelogram.
{{qed}} | Let $ABCD$ be a [[Definition:Rectangle|rectangle]] whose [[Definition:Side of Polygon|side]] [[Definition:Length (Linear Measure)|lengths]] are $a$ and $b$.
The [[Definition:Perimeter|perimeter]] of $ABCD$ is $2 a + 2 b$. | From [[Rectangle is Parallelogram]], $ABCD$ is a [[Definition:Parallelogram|parallelogram]].
The result then follows from a direct application of [[Perimeter of Parallelogram]].
{{qed}} | Perimeter of Rectangle/Proof 2 | https://proofwiki.org/wiki/Perimeter_of_Rectangle | https://proofwiki.org/wiki/Perimeter_of_Rectangle/Proof_2 | [
"Perimeter of Rectangle",
"Rectangles",
"Perimeter Formulas"
] | [
"Definition:Quadrilateral/Rectangle",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Perimeter"
] | [
"Rectangle is Parallelogram",
"Definition:Quadrilateral/Parallelogram",
"Perimeter of Parallelogram"
] |
proofwiki-7825 | Perimeter of Parallelogram | Let $ABCD$ be a parallelogram whose side lengths are $a$ and $b$.
The perimeter of $ABCD$ is $2 a + 2 b$. | :400px
By Opposite Sides and Angles of Parallelogram are Equal it follows that:
:$AB = CD$
:$BC = AD$
The perimeter of $ABCD$ is $AB + BC + CD + AD$.
But $AB = CD = a$ and $BC = AD = b$.
Hence the result.
{{qed}} | Let $ABCD$ be a [[Definition:Parallelogram|parallelogram]] whose [[Definition:Side of Polygon|side]] [[Definition:Length of Line|lengths]] are $a$ and $b$.
The [[Definition:Perimeter|perimeter]] of $ABCD$ is $2 a + 2 b$. | :[[File:PerimeterOfParallelogram.png|400px]]
By [[Opposite Sides and Angles of Parallelogram are Equal]] it follows that:
:$AB = CD$
:$BC = AD$
The [[Definition:Perimeter|perimeter]] of $ABCD$ is $AB + BC + CD + AD$.
But $AB = CD = a$ and $BC = AD = b$.
Hence the result.
{{qed}} | Perimeter of Parallelogram | https://proofwiki.org/wiki/Perimeter_of_Parallelogram | https://proofwiki.org/wiki/Perimeter_of_Parallelogram | [
"Parallelograms",
"Perimeter Formulas"
] | [
"Definition:Quadrilateral/Parallelogram",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Perimeter"
] | [
"File:PerimeterOfParallelogram.png",
"Opposite Sides and Angles of Parallelogram are Equal",
"Definition:Perimeter"
] |
proofwiki-7826 | Perimeter of Triangle | Let $ABC$ be a triangle.
Then the perimeter $P$ of $ABC$ is given by:
:$P = a + b + c$
where $a, b, c$ are the lengths of the sides of $ABC$. | The perimeter of a plane geometric figure is defined as the total length of the boundary.
By definition, the boundary of a triangle comprises the three sides of that triangle.
Hence the result.
{{qed}} | Let $ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Then the [[Definition:Perimeter|perimeter]] $P$ of $ABC$ is given by:
:$P = a + b + c$
where $a, b, c$ are the [[Definition:Length (Linear Measure)|lengths]] of the [[Definition:Side of Polygon|sides]] of $ABC$. | The [[Definition:Perimeter|perimeter]] of a [[Definition:Plane Figure|plane geometric figure]] is defined as the total [[Definition:Length (Linear Measure)|length]] of the [[Definition:Boundary (Geometry)|boundary]].
By definition, the [[Definition:Boundary (Geometry)|boundary]] of a [[Definition:Triangle (Geometry)|t... | Perimeter of Triangle | https://proofwiki.org/wiki/Perimeter_of_Triangle | https://proofwiki.org/wiki/Perimeter_of_Triangle | [
"Triangles",
"Perimeter Formulas"
] | [
"Definition:Triangle (Geometry)",
"Definition:Perimeter",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side"
] | [
"Definition:Perimeter",
"Definition:Geometric Figure/Plane Figure",
"Definition:Linear Measure/Length",
"Definition:Boundary (Geometry)",
"Definition:Boundary (Geometry)",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)"
] |
proofwiki-7827 | Area of Trapezium | :410px
Let $ABCD$ be a trapezium:
:whose parallel sides are of lengths $a$ and $b$
and
:whose height is $h$.
Then the area of $ABCD$ is given by:
:$\Box ABCD = \dfrac {h \paren {a + b} } 2$ | :600px
Extend line $AB$ to $E$ by length $a$.
Extend line $DC$ to $F$ by length $b$.
Then $BEFC$ is another trapezium whose parallel sides are of lengths $a$ and $b$ and whose height is $h$.
Also, $AEFD$ is a parallelogram which comprises the two trapezia $ABCD$ and $BEFC$.
So $\Box ABCD + \Box BEFC = \Box AEFD$ and $\... | :[[File:TrapezoidArea.png|410px]]
Let $ABCD$ be a [[Definition:Trapezium|trapezium]]:
:whose [[Definition:Parallel Lines|parallel]] [[Definition:Side of Polygon|sides]] are of [[Definition:Length of Line|lengths]] $a$ and $b$
and
:whose [[Definition:Height of Trapezium|height]] is $h$.
Then the [[Definition:Area|are... | :[[File:TrapezoidAreaProof.png|600px]]
Extend line $AB$ to $E$ by [[Definition:Length (Linear Measure)|length]] $a$.
Extend line $DC$ to $F$ by [[Definition:Length (Linear Measure)|length]] $b$.
Then $BEFC$ is another [[Definition:Trapezium|trapezium]] whose [[Definition:Parallel Lines|parallel]] [[Definition:Side o... | Area of Trapezium | https://proofwiki.org/wiki/Area_of_Trapezium | https://proofwiki.org/wiki/Area_of_Trapezium | [
"Areas of Quadrilaterals",
"Trapezia"
] | [
"File:TrapezoidArea.png",
"Definition:Quadrilateral/Trapezium",
"Definition:Parallel (Geometry)/Lines",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Quadrilateral/Trapezium/Height",
"Definition:Area"
] | [
"File:TrapezoidAreaProof.png",
"Definition:Linear Measure/Length",
"Definition:Linear Measure/Length",
"Definition:Quadrilateral/Trapezium",
"Definition:Parallel (Geometry)/Lines",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Quadrilateral/Trapezium/Height",
"Definition... |
proofwiki-7828 | Perimeter of Trapezium | :400px
Let $ABCD$ be a trapezium:
:whose parallel sides are of lengths $a$ and $b$
:whose height is $h$.
and
:whose non-parallel sides are at angles $\theta$ and $\phi$ with the parallels.
The perimeter $P$ of $ABCD$ is given by:
:$P = a + b + h \paren {\csc \theta + \csc \phi}$
where $\csc$ denotes cosecant. | The perimeter $P$ of $ABCD$ is given by:
:$P = AB + BC + CD + AD$
where the lines are used to indicate their length.
Thus:
{{begin-eqn}}
{{eqn | n = 1
| l = AB
| r = b
}}
{{eqn | n = 2
| l = CD
| r = a
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = h
| r = AD \sin \theta
| c = {{Defof|Sine... | :[[File:TrapezoidPerimeter.png|400px]]
Let $ABCD$ be a [[Definition:Trapezium|trapezium]]:
:whose [[Definition:Parallel Lines|parallel]] [[Definition:Side of Polygon|sides]] are of [[Definition:Length (Linear Measure)|lengths]] $a$ and $b$
:whose [[Definition:Height of Trapezium|height]] is $h$.
and
:whose non-[[Defin... | The [[Definition:Perimeter|perimeter]] $P$ of $ABCD$ is given by:
:$P = AB + BC + CD + AD$
where the lines are used to indicate their [[Definition:Length (Linear Measure)|length]].
Thus:
{{begin-eqn}}
{{eqn | n = 1
| l = AB
| r = b
}}
{{eqn | n = 2
| l = CD
| r = a
}}
{{end-eqn}}
{{begin-eqn}... | Perimeter of Trapezium | https://proofwiki.org/wiki/Perimeter_of_Trapezium | https://proofwiki.org/wiki/Perimeter_of_Trapezium | [
"Trapezia",
"Perimeter Formulas"
] | [
"File:TrapezoidPerimeter.png",
"Definition:Quadrilateral/Trapezium",
"Definition:Parallel (Geometry)/Lines",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Quadrilateral/Trapezium/Height",
"Definition:Parallel (Geometry)/Lines",
"Definition:Polygon/Side",
"Definition:Peri... | [
"Definition:Perimeter",
"Definition:Linear Measure/Length",
"Cosecant is Reciprocal of Sine",
"Cosecant is Reciprocal of Sine"
] |
proofwiki-7829 | Area of Regular Polygon | Let $P$ be a regular $n$-sided polygon whose side length is $b$.
Then the area of $P$ is given by:
:$\Box P = \dfrac 1 4 n b^2 \cot \dfrac \pi n$
where $\cot$ denotes cotangent. | :400px
Let $H$ be the center of the regular $n$-sided polygon $P$.
Let one of its sides be $AB$.
Consider the triangle $\triangle ABH$.
As $P$ is regular and $H$ is the center, $AH = BH$ and so $\triangle ABH$ is isosceles.
Thus $b = AB$ is the base of $\triangle ABH$.
Let $h = GH$ be its altitude.
See the diagram.
The... | Let $P$ be a [[Definition:Regular Polygon|regular]] [[Definition:N-Gon|$n$-sided polygon]] whose [[Definition:Side of Polygon|side]] [[Definition:Length (Linear Measure)|length]] is $b$.
Then the [[Definition:Area|area]] of $P$ is given by:
:$\Box P = \dfrac 1 4 n b^2 \cot \dfrac \pi n$
where $\cot$ denotes [[Definiti... | :[[File:RegularPolygonArea.png|400px]]
Let $H$ be the center of the [[Definition:Regular Polygon|regular]] [[Definition:N-Gon|$n$-sided polygon]] $P$.
Let one of its [[Definition:Side of Polygon|sides]] be $AB$.
Consider the [[Definition:Triangle (Geometry)|triangle]] $\triangle ABH$.
As $P$ is [[Definition:Regular... | Area of Regular Polygon | https://proofwiki.org/wiki/Area_of_Regular_Polygon | https://proofwiki.org/wiki/Area_of_Regular_Polygon | [
"Regular Polygons",
"Area Formulas"
] | [
"Definition:Polygon/Regular",
"Definition:Polygon/Multilateral",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Area",
"Definition:Cotangent"
] | [
"File:RegularPolygonArea.png",
"Definition:Polygon/Regular",
"Definition:Polygon/Multilateral",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Regular",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Triangle (Geometry)/Isosceles/Base",
"Definition:Altitude... |
proofwiki-7830 | Perimeter of Regular Polygon | Let $P$ be a regular $n$-sided polygon whose side length is $b$.
Then the perimeter $L$ of $P$ is given by:
:$L = n b$ | By definition, an $n$-sided polygon has $n$ sides.
By definition, a regular polygon has sides all the same length.
By definition, the perimeter of a polygon is the total length of all its sides.
Thus $P$ has $n$ sides all of length $b$.
Hence $L = n b$.
{{qed}} | Let $P$ be a [[Definition:Regular Polygon|regular]] [[Definition:N-Gon|$n$-sided polygon]] whose [[Definition:Side of Polygon|side]] [[Definition:Length of Line|length]] is $b$.
Then the [[Definition:Perimeter|perimeter]] $L$ of $P$ is given by:
:$L = n b$ | By definition, an [[Definition:N-Gon|$n$-sided polygon]] has $n$ [[Definition:Side of Polygon|sides]].
By definition, a [[Definition:Regular Polygon|regular polygon]] has [[Definition:Side of Polygon|sides]] all the same [[Definition:Length of Line|length]].
By definition, the [[Definition:Perimeter|perimeter]] of a ... | Perimeter of Regular Polygon | https://proofwiki.org/wiki/Perimeter_of_Regular_Polygon | https://proofwiki.org/wiki/Perimeter_of_Regular_Polygon | [
"Regular Polygons",
"Perimeter Formulas"
] | [
"Definition:Polygon/Regular",
"Definition:Polygon/Multilateral",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Perimeter"
] | [
"Definition:Polygon/Multilateral",
"Definition:Polygon/Side",
"Definition:Polygon/Regular",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Perimeter",
"Definition:Polygon",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Defi... |
proofwiki-7831 | Commensurability is Transitive | Let $a$, $b$, $c$ be three real numbers.
Let $a$ and $b$ be commensurable, and $b$ and $c$ be commensurable.
Then $a$ and $c$ are commensurable. | From the definition of commensurablility:
:$\dfrac a b, \dfrac b c \in \Q$
where $\Q$ denotes the set of all rational numbers.
From Rational Multiplication is Closed:
:$\dfrac a b \times \dfrac b c \in \Q$
Cancelling $b$, we have:
:$\dfrac a c \in \Q$
Hence the result.
{{qed}}
Category:Number Theory
1iqhdq9qiwy16kc734v... | Let $a$, $b$, $c$ be three [[Definition:Real Number|real numbers]].
Let $a$ and $b$ be [[Definition:Commensurable|commensurable]], and $b$ and $c$ be [[Definition:Commensurable|commensurable]].
Then $a$ and $c$ are [[Definition:Commensurable|commensurable]]. | From the definition of [[Definition:Commensurable|commensurablility]]:
:$\dfrac a b, \dfrac b c \in \Q$
where $\Q$ denotes the [[Definition:Set|set]] of all [[Definition:Rational Number|rational numbers]].
From [[Rational Multiplication is Closed]]:
:$\dfrac a b \times \dfrac b c \in \Q$
Cancelling $b$, we have:
:$\d... | Commensurability is Transitive | https://proofwiki.org/wiki/Commensurability_is_Transitive | https://proofwiki.org/wiki/Commensurability_is_Transitive | [
"Number Theory"
] | [
"Definition:Real Number",
"Definition:Commensurable",
"Definition:Commensurable",
"Definition:Commensurable"
] | [
"Definition:Commensurable",
"Definition:Set",
"Definition:Rational Number",
"Rational Multiplication is Closed",
"Category:Number Theory"
] |
proofwiki-7832 | Arc Length of Sector | Let $\CC = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$.
Let $BAC$ be the sector of $\CC$ whose angle between $AB$ and $AC$ is $\theta$.
:300px
Then the length $s$ of arc $BC$ is given by:
:$s = r \theta$
where:
:$r = AB$ is the length of the radius of the circle
:$\theta$ is measured in radians. | From Perimeter of Circle, the perimeter of $\CC$ is $2 \pi r$.
From Measurement of Full Angle, the angle within $\CC$ is $2 \pi$.
{{explain|Why is the density of the arc length uniform? i.e. why does equal rotation sweeps out equal arc length?}}
The fraction of the perimeter of $\CC$ within the sector $BAC$ is therefor... | Let $\CC = ABC$ be a [[Definition:Circle|circle]] whose [[Definition:Center of Circle|center]] is $A$ and with [[Definition:Radius of Circle|radii]] $AB$ and $AC$.
Let $BAC$ be the [[Definition:Sector of Circle|sector]] of $\CC$ whose [[Definition:Angle of Sector|angle]] between $AB$ and $AC$ is $\theta$.
:[[File:Sec... | From [[Perimeter of Circle]], the [[Definition:Perimeter|perimeter]] of $\CC$ is $2 \pi r$.
From [[Measurement of Full Angle]], the angle within $\CC$ is $2 \pi$.
{{explain|Why is the density of the arc length uniform? i.e. why does equal rotation sweeps out equal arc length?}}
The fraction of the [[Definition:Perim... | Arc Length of Sector | https://proofwiki.org/wiki/Arc_Length_of_Sector | https://proofwiki.org/wiki/Arc_Length_of_Sector | [
"Arc Length",
"Sectors of Circles"
] | [
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Definition:Sector of Circle",
"Definition:Sector of Circle/Angle",
"File:Sector.png",
"Definition:Arc Length",
"Definition:Circle/Arc",
"Definition:Linear Measure/Length",
"Definition:Circle/Radius",
"Definition:Circle... | [
"Perimeter of Circle",
"Definition:Perimeter",
"Measurements of Common Angles/Full Angle",
"Definition:Perimeter",
"Definition:Sector of Circle"
] |
proofwiki-7833 | Length of Inradius of Triangle | Let $\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$.
Then the length of the inradius $r$ of $\triangle ABC$ is given by:
:$r = \dfrac {\sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} } } s$
where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$. | :410px
Let $\AA$ be the area of $\triangle ABC$.
From Area of Triangle in Terms of Inradius:
:$\AA = r s$
where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.
From Heron's Formula:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac {a + b + c} 2$ is the semiperimeter... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Side of Polygon|sides]] are of [[Definition:Length of Line|lengths]] $a, b, c$.
Then the [[Definition:Length of Line|length]] of the [[Definition:Inradius of Triangle|inradius]] $r$ of $\triangle ABC$ is given by:
:$r = \dfrac {\s... | :[[File:Incircle.png|410px]]
Let $\AA$ be the [[Definition:Area|area]] of $\triangle ABC$.
From [[Area of Triangle in Terms of Inradius]]:
:$\AA = r s$
where $s = \dfrac {a + b + c} 2$ is the [[Definition:Semiperimeter|semiperimeter]] of $\triangle ABC$.
From [[Heron's Formula]]:
:$\AA = \sqrt {s \paren {s - a} \p... | Length of Inradius of Triangle | https://proofwiki.org/wiki/Length_of_Inradius_of_Triangle | https://proofwiki.org/wiki/Length_of_Inradius_of_Triangle | [
"Incircles of Triangles",
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Linear Measure/Length",
"Definition:Incircle of Triangle/Inradius",
"Definition:Semiperimeter"
] | [
"File:Incircle.png",
"Definition:Area",
"Area of Triangle in Terms of Inradius",
"Definition:Semiperimeter",
"Heron's Formula",
"Definition:Semiperimeter"
] |
proofwiki-7834 | Length of Circumradius of Triangle | Let $\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$.
Then the length of the circumradius $R$ of $\triangle ABC$ is given by:
:$R = \dfrac {abc} {4 \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} } }$
where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$. | :320px
Let $\AA$ be the area of $\triangle ABC$.
From Area of Triangle in Terms of Circumradius:
:$\AA = \dfrac {a b c} {4 R}$
From Heron's Formula:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.
Hence the result:
:$R = \dfrac {... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Side of Polygon|sides]] are of [[Definition:Length of Line|lengths]] $a, b, c$.
Then the [[Definition:Length of Line|length]] of the [[Definition:Circumradius of Triangle|circumradius]] $R$ of $\triangle ABC$ is given by:
:$R = \d... | :[[File:Circumcircle.png|320px]]
Let $\AA$ be the [[Definition:Area|area]] of $\triangle ABC$.
From [[Area of Triangle in Terms of Circumradius]]:
:$\AA = \dfrac {a b c} {4 R}$
From [[Heron's Formula]]:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac {a + b + c} 2$ is the [[Defin... | Length of Circumradius of Triangle | https://proofwiki.org/wiki/Length_of_Circumradius_of_Triangle | https://proofwiki.org/wiki/Length_of_Circumradius_of_Triangle | [
"Circumradii",
"Circumcircles of Triangles",
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Linear Measure/Length",
"Definition:Circumcircle of Triangle/Circumradius",
"Definition:Semiperimeter"
] | [
"File:Circumcircle.png",
"Definition:Area",
"Area of Triangle in Terms of Circumradius",
"Heron's Formula",
"Definition:Semiperimeter"
] |
proofwiki-7835 | Area of Regular Polygon by Circumradius | Let $P$ be a regular $n$-gon.
Let the circumradius of $P$ be $r$.
Then the area $\AA$ of $P$ is given by:
:$\AA = \dfrac 1 2 n r^2 \sin \dfrac {2 \pi} n$ | :400px
From Regular Polygon is composed of Isosceles Triangles, let $\triangle OAB$ be one of the $n$ isosceles triangles that compose $P$.
Then $\AA$ is equal to $n$ times the area of $\triangle OAB$.
Let $d$ be the length of one side of $P$.
Then $d$ is the length of the base of $\triangle OAB$.
Let $h$ be the altitu... | Let $P$ be a [[Definition:Regular Polygon|regular]] [[Definition:N-Gon|$n$-gon]].
Let the [[Definition:Circumradius of Polygon|circumradius]] of $P$ be $r$.
Then the [[Definition:Area|area]] $\AA$ of $P$ is given by:
:$\AA = \dfrac 1 2 n r^2 \sin \dfrac {2 \pi} n$ | :[[File:RegularPolygonAreaInscribed.png|400px]]
From [[Regular Polygon is composed of Isosceles Triangles]], let $\triangle OAB$ be one of the $n$ [[Definition:Isosceles Triangle|isosceles triangles]] that compose $P$.
Then $\AA$ is equal to $n$ times the [[Definition:Area|area]] of $\triangle OAB$.
Let $d$ be the [... | Area of Regular Polygon by Circumradius | https://proofwiki.org/wiki/Area_of_Regular_Polygon_by_Circumradius | https://proofwiki.org/wiki/Area_of_Regular_Polygon_by_Circumradius | [
"Regular Polygons",
"Circumradii",
"Area Formulas"
] | [
"Definition:Polygon/Regular",
"Definition:Polygon/Multilateral",
"Definition:Circumradius/Polygon",
"Definition:Area"
] | [
"File:RegularPolygonAreaInscribed.png",
"Regular Polygon is composed of Isosceles Triangles",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Area",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Triangle (Geometry)/Isosceles/Base... |
proofwiki-7836 | Perimeter of Regular Polygon by Circumradius | Let $P$ be a regular $n$-gon.
Let $C$ be a circumcircle of $P$.
Let the radius of $C$ be $r$.
Then the perimeter $\PP$ of $P$ is given by:
:$\PP = 2 n r \sin \dfrac \pi n$ | :400px
From Regular Polygon is composed of Isosceles Triangles, let $\triangle OAB$ be one of the $n$ isosceles triangles that compose $P$.
Then $\PP$ is equal to $n$ times the base of $\triangle OAB$.
Let $d$ be the length of one side of $P$.
Then $d$ is the length of the base of $\triangle OAB$.
The angle $\angle AOB... | Let $P$ be a [[Definition:Regular Polygon|regular]] [[Definition:N-Gon|$n$-gon]].
Let $C$ be a [[Definition:Circumcircle|circumcircle]] of $P$.
Let the [[Definition:Radius of Circle|radius]] of $C$ be $r$.
Then the [[Definition:Perimeter|perimeter]] $\PP$ of $P$ is given by:
:$\PP = 2 n r \sin \dfrac \pi n$ | :[[File:RegularPolygonAreaInscribed.png|400px]]
From [[Regular Polygon is composed of Isosceles Triangles]], let $\triangle OAB$ be one of the $n$ [[Definition:Isosceles Triangle|isosceles triangles]] that compose $P$.
Then $\PP$ is equal to $n$ times the [[Definition:Base of Isosceles Triangle|base]] of $\triangle O... | Perimeter of Regular Polygon by Circumradius | https://proofwiki.org/wiki/Perimeter_of_Regular_Polygon_by_Circumradius | https://proofwiki.org/wiki/Perimeter_of_Regular_Polygon_by_Circumradius | [
"Circumradii",
"Regular Polygons",
"Perimeter Formulas"
] | [
"Definition:Polygon/Regular",
"Definition:Polygon/Multilateral",
"Definition:Circumcircle",
"Definition:Circle/Radius",
"Definition:Perimeter"
] | [
"File:RegularPolygonAreaInscribed.png",
"Regular Polygon is composed of Isosceles Triangles",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Triangle (Geometry)/Isosceles/Base",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Tria... |
proofwiki-7837 | Area of Regular Polygon by Inradius | Let $P$ be a regular $n$-gon.
Let $C$ be an incircle of $P$.
Let the radius of $C$ be $r$.
Then the area $\AA$ of $P$ is given by:
:$\AA = n r^2 \tan \dfrac \pi n$ | :400px
From Regular Polygon is composed of Isosceles Triangles, let $\triangle OAB$ be one of the $n$ isosceles triangles that compose $P$.
Then $\AA$ is equal to $n$ times the area of $\triangle OAB$.
Also, $r$ is the length of the altitude of $\triangle OAB$.
Let $d$ be the length of one side of $P$.
Then $d$ is the ... | Let $P$ be a [[Definition:Regular Polygon|regular]] [[Definition:N-Gon|$n$-gon]].
Let $C$ be an [[Definition:Incircle|incircle]] of $P$.
Let the [[Definition:Radius of Circle|radius]] of $C$ be $r$.
Then the [[Definition:Area|area]] $\AA$ of $P$ is given by:
:$\AA = n r^2 \tan \dfrac \pi n$ | :[[File:RegularPolygonAreaCircumscribed.png|400px]]
From [[Regular Polygon is composed of Isosceles Triangles]], let $\triangle OAB$ be one of the $n$ [[Definition:Isosceles Triangle|isosceles triangles]] that compose $P$.
Then $\AA$ is equal to $n$ times the [[Definition:Area|area]] of $\triangle OAB$.
Also, $r$ is... | Area of Regular Polygon by Inradius | https://proofwiki.org/wiki/Area_of_Regular_Polygon_by_Inradius | https://proofwiki.org/wiki/Area_of_Regular_Polygon_by_Inradius | [
"Regular Polygons",
"Area Formulas"
] | [
"Definition:Polygon/Regular",
"Definition:Polygon/Multilateral",
"Definition:Incircle",
"Definition:Circle/Radius",
"Definition:Area"
] | [
"File:RegularPolygonAreaCircumscribed.png",
"Regular Polygon is composed of Isosceles Triangles",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Area",
"Definition:Linear Measure/Length",
"Definition:Altitude of Triangle",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Defi... |
proofwiki-7838 | Perimeter of Regular Polygon by Inradius | Let $P$ be a regular $n$-gon.
Let $C$ be an incircle of $P$.
Let the radius of $C$ be $r$.
Then the perimeter $\PP$ of $P$ is given by:
:$\PP = 2 n r \tan \dfrac \pi n$ | :400px
From Regular Polygon is composed of Isosceles Triangles, let $\triangle OAB$ be one of the $n$ isosceles triangles that compose $P$.
Then $\AA$ is equal to $n$ times the area of $\triangle OAB$.
Also, $r$ is the length of the altitude of $\triangle OAB$.
Let $d$ be the length of one side of $P$.
Then $d$ is the ... | Let $P$ be a [[Definition:Regular Polygon|regular]] [[Definition:N-Gon|$n$-gon]].
Let $C$ be an [[Definition:Incircle|incircle]] of $P$.
Let the [[Definition:Radius of Circle|radius]] of $C$ be $r$.
Then the [[Definition:Perimeter|perimeter]] $\PP$ of $P$ is given by:
:$\PP = 2 n r \tan \dfrac \pi n$ | :[[File:RegularPolygonAreaCircumscribed.png|400px]]
From [[Regular Polygon is composed of Isosceles Triangles]], let $\triangle OAB$ be one of the $n$ [[Definition:Isosceles Triangle|isosceles triangles]] that compose $P$.
Then $\AA$ is equal to $n$ times the [[Definition:Area|area]] of $\triangle OAB$.
Also, $r$ is... | Perimeter of Regular Polygon by Inradius | https://proofwiki.org/wiki/Perimeter_of_Regular_Polygon_by_Inradius | https://proofwiki.org/wiki/Perimeter_of_Regular_Polygon_by_Inradius | [
"Regular Polygons",
"Perimeter Formulas"
] | [
"Definition:Polygon/Regular",
"Definition:Polygon/Multilateral",
"Definition:Incircle",
"Definition:Circle/Radius",
"Definition:Perimeter"
] | [
"File:RegularPolygonAreaCircumscribed.png",
"Regular Polygon is composed of Isosceles Triangles",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Area",
"Definition:Linear Measure/Length",
"Definition:Altitude of Triangle",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Defi... |
proofwiki-7839 | Area of Isosceles Triangle | Let $\triangle ABC$ be an isosceles triangle whose apex is $A$.
Let $\theta$ be the angle of the apex $A$.
Let $r$ be the length of a leg of $\triangle ABC$.
Then the area $\AA$ of $\triangle ABC$ is given by:
:$\AA = \dfrac 1 2 r^2 \sin \theta$ | :300px
{{begin-eqn}}
{{eqn | l = \AA
| r = \frac 1 2 b h
| c = Area of Triangle in Terms of Side and Altitude
}}
{{eqn | r = \frac 1 2 b \paren {r \cos \dfrac \theta 2}
| c = {{Defof|Cosine of Angle|Cosine}}
}}
{{eqn | r = \frac 1 2 2 \paren {r \sin \dfrac \theta 2} \paren {r \cos \dfrac \theta 2}
... | Let $\triangle ABC$ be an [[Definition:Isosceles Triangle|isosceles triangle]] whose [[Definition:Apex of Isosceles Triangle|apex]] is $A$.
Let $\theta$ be the [[Definition:Angle|angle]] of the [[Definition:Apex of Isosceles Triangle|apex]] $A$.
Let $r$ be the [[Definition:Length (Linear Measure)|length]] of a [[Defi... | :[[File:IsoscelesTriangleArea.png|300px]]
{{begin-eqn}}
{{eqn | l = \AA
| r = \frac 1 2 b h
| c = [[Area of Triangle in Terms of Side and Altitude]]
}}
{{eqn | r = \frac 1 2 b \paren {r \cos \dfrac \theta 2}
| c = {{Defof|Cosine of Angle|Cosine}}
}}
{{eqn | r = \frac 1 2 2 \paren {r \sin \dfrac \thet... | Area of Isosceles Triangle/Proof 1 | https://proofwiki.org/wiki/Area_of_Isosceles_Triangle | https://proofwiki.org/wiki/Area_of_Isosceles_Triangle/Proof_1 | [
"Areas of Triangles",
"Isosceles Triangles",
"Area of Isosceles Triangle"
] | [
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Triangle (Geometry)/Isosceles/Apex",
"Definition:Angle",
"Definition:Triangle (Geometry)/Isosceles/Apex",
"Definition:Linear Measure/Length",
"Definition:Triangle (Geometry)/Isosceles/Legs",
"Definition:Area"
] | [
"File:IsoscelesTriangleArea.png",
"Area of Triangle in Terms of Side and Altitude",
"Double Angle Formulas/Sine"
] |
proofwiki-7840 | Area of Isosceles Triangle | Let $\triangle ABC$ be an isosceles triangle whose apex is $A$.
Let $\theta$ be the angle of the apex $A$.
Let $r$ be the length of a leg of $\triangle ABC$.
Then the area $\AA$ of $\triangle ABC$ is given by:
:$\AA = \dfrac 1 2 r^2 \sin \theta$ | A direct application of Area of Triangle in Terms of Two Sides and Angle:
:$\AA = \dfrac 1 2 a b \sin \theta$
where $a = b = r$.
{{qed}} | Let $\triangle ABC$ be an [[Definition:Isosceles Triangle|isosceles triangle]] whose [[Definition:Apex of Isosceles Triangle|apex]] is $A$.
Let $\theta$ be the [[Definition:Angle|angle]] of the [[Definition:Apex of Isosceles Triangle|apex]] $A$.
Let $r$ be the [[Definition:Length (Linear Measure)|length]] of a [[Defi... | A direct application of [[Area of Triangle in Terms of Two Sides and Angle]]:
:$\AA = \dfrac 1 2 a b \sin \theta$
where $a = b = r$.
{{qed}} | Area of Isosceles Triangle/Proof 2 | https://proofwiki.org/wiki/Area_of_Isosceles_Triangle | https://proofwiki.org/wiki/Area_of_Isosceles_Triangle/Proof_2 | [
"Areas of Triangles",
"Isosceles Triangles",
"Area of Isosceles Triangle"
] | [
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Triangle (Geometry)/Isosceles/Apex",
"Definition:Angle",
"Definition:Triangle (Geometry)/Isosceles/Apex",
"Definition:Linear Measure/Length",
"Definition:Triangle (Geometry)/Isosceles/Legs",
"Definition:Area"
] | [
"Area of Triangle in Terms of Two Sides and Angle"
] |
proofwiki-7841 | Area of Segment of Circle | Let $C$ be a circle of radius $r$.
Let $S$ be a segment of $C$ such that its base subtends an angle of $\theta$ at the center of the circle.
Then the area $\AA$ of $S$ is given by:
:$\AA = \dfrac 1 2 r^2 \paren {\theta - \sin \theta}$
where $\theta$ is measured in radians. | :350px
Let $BDCE$ be the segment $S$.
Let $b$ be the length of the base of $S$.
Let $BACE$ be the sector of $C$ whose angle is $\theta$.
The $\AA$ is equal to the area of $BACE$ minus the area of the isosceles triangle $\triangle ABC$ whose base is $b$.
Let $h$ be the altitude of $\triangle ABC$.
From Area of Sector, t... | Let $C$ be a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$.
Let $S$ be a [[Definition:Segment of Circle|segment]] of $C$ such that its [[Definition:Base of Segment|base]] [[Definition:Subtend|subtends]] an [[Definition:Angle|angle]] of $\theta$ at the [[Definition:Center of Circle|center o... | :[[File:AreaOfSegment.png|350px]]
Let $BDCE$ be the [[Definition:Segment of Circle|segment]] $S$.
Let $b$ be the [[Definition:Length of Line|length]] of the [[Definition:Base of Segment|base]] of $S$.
Let $BACE$ be the [[Definition:Sector of Circle|sector]] of $C$ whose [[Definition:Angle of Sector|angle]] is $\thet... | Area of Segment of Circle | https://proofwiki.org/wiki/Area_of_Segment_of_Circle | https://proofwiki.org/wiki/Area_of_Segment_of_Circle | [
"Circles",
"Area Formulas"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Segment of Circle",
"Definition:Segment of Circle/Base",
"Definition:Subtend",
"Definition:Angle",
"Definition:Circle/Center",
"Definition:Area",
"Definition:Angular Measure/Radian"
] | [
"File:AreaOfSegment.png",
"Definition:Segment of Circle",
"Definition:Linear Measure/Length",
"Definition:Segment of Circle/Base",
"Definition:Sector of Circle",
"Definition:Sector of Circle/Angle",
"Definition:Area",
"Definition:Area",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Trian... |
proofwiki-7842 | Mills' Theorem | There exists a real number $A$ such that $\floor {A^{3^n} }$ is a prime number for all $n \in \N_{>0}$, where:
:$\floor x$ denotes the floor function of $x$
:$\N$ denotes the set of all natural numbers. | {{refactor|level = basic|extract the definition below into its own page}}
We define $\map f x$ as a '''prime-representing function''' {{iff}}:
:$\forall x \in \N: \map f x \in \Bbb P$
where:
:$\N$ denotes the set of all natural numbers
:$\Bbb P$ denotes the set of all prime numbers.
Let $p_n$ be the $n$th prime number.... | There exists a [[Definition:Real Number|real number]] $A$ such that $\floor {A^{3^n} }$ is a [[Definition:Prime Number|prime number]] for all $n \in \N_{>0}$, where:
:$\floor x$ denotes the [[Definition:Floor Function|floor function]] of $x$
:$\N$ denotes the [[Definition:Set|set]] of all [[Definition:Natural Number|n... | {{refactor|level = basic|extract the definition below into its own page}}
We define $\map f x$ as a '''prime-representing function''' {{iff}}:
:$\forall x \in \N: \map f x \in \Bbb P$
where:
:$\N$ denotes the [[Definition:Set|set]] of all [[Definition:Natural Number|natural numbers]]
:$\Bbb P$ denotes the [[Definition... | Mills' Theorem | https://proofwiki.org/wiki/Mills'_Theorem | https://proofwiki.org/wiki/Mills'_Theorem | [
"Mills' Theorem",
"Prime Numbers",
"Number Theory"
] | [
"Definition:Real Number",
"Definition:Prime Number",
"Definition:Floor Function",
"Definition:Set",
"Definition:Natural Numbers"
] | [
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Prime Number",
"Definition:Prime Number",
"Difference between Consecutive Primes",
"Definition:Positive/Integer",
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-7843 | Equation of Ellipse in Reduced Form | Let $K$ be an ellipse aligned in a cartesian plane in reduced form.
Let:
:the major axis of $K$ have length $2 a$
:the minor axis of $K$ have length $2 b$. | :500px
By definition, the foci $F_1$ and $F_2$ of $K$ are located at $\tuple {-c, 0}$ and $\tuple {c, 0}$ respectively.
Let the vertices of $K$ be $V_1$ and $V_2$.
By definition, these are located at $\tuple {-a, 0}$ and $\tuple {a, 0}$.
Let the covertices of $K$ be $C_1$ and $C_2$.
By definition, these are located at ... | Let $K$ be an [[Definition:Ellipse|ellipse]] aligned in a [[Definition:Cartesian Plane|cartesian plane]] in [[Definition:Reduced Form of Ellipse|reduced form]].
Let:
:the [[Definition:Major Axis of Ellipse|major axis]] of $K$ have [[Definition:Length (Linear Measure)|length]] $2 a$
:the [[Definition:Minor Axis of El... | :[[File:EllipseEquation.png|500px]]
By definition, the [[Definition:Focus of Ellipse|foci]] $F_1$ and $F_2$ of $K$ are located at $\tuple {-c, 0}$ and $\tuple {c, 0}$ respectively.
Let the [[Definition:Vertex of Ellipse|vertices]] of $K$ be $V_1$ and $V_2$.
By definition, these are located at $\tuple {-a, 0}$ and $\... | Equation of Ellipse in Reduced Form/Cartesian Frame/Proof 1 | https://proofwiki.org/wiki/Equation_of_Ellipse_in_Reduced_Form | https://proofwiki.org/wiki/Equation_of_Ellipse_in_Reduced_Form/Cartesian_Frame/Proof_1 | [
"Equation of Ellipse in Reduced Form",
"Reduced Form of Ellipse",
"Ellipses"
] | [
"Definition:Ellipse",
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Ellipse",
"Definition:Ellipse/Major Axis",
"Definition:Linear Measure/Length",
"Definition:Ellipse/Minor Axis",
"Definition:Linear Measure/Length"
] | [
"File:EllipseEquation.png",
"Definition:Ellipse/Focus",
"Definition:Ellipse/Vertex",
"Definition:Ellipse/Covertex",
"Definition:Point",
"Definition:Locus",
"Definition:Ellipse/Equidistance",
"Definition:Constant",
"Definition:Ellipse",
"Equidistance of Ellipse equals Major Axis",
"Linear Eccentr... |
proofwiki-7844 | Equation of Ellipse in Reduced Form | Let $K$ be an ellipse aligned in a cartesian plane in reduced form.
Let:
:the major axis of $K$ have length $2 a$
:the minor axis of $K$ have length $2 b$. | :500px
Let $P$ be an arbitrary point in the plane.
Let $PM$ be dropped perpendicular to $V_1 V_2$.
Hence $M = \tuple {x, 0}$.
From Intersecting Chord Theorem for Conic Sections:
:$PM^2 = k V_1 M \times M V_2$
for some constant $k$.
Hence:
{{begin-eqn}}
{{eqn | l = y^2
| r = k \paren {a + x} \paren {a - x}
|... | Let $K$ be an [[Definition:Ellipse|ellipse]] aligned in a [[Definition:Cartesian Plane|cartesian plane]] in [[Definition:Reduced Form of Ellipse|reduced form]].
Let:
:the [[Definition:Major Axis of Ellipse|major axis]] of $K$ have [[Definition:Length (Linear Measure)|length]] $2 a$
:the [[Definition:Minor Axis of El... | :[[File:EllipseEquation-2.png|500px]]
Let $P$ be an arbitrary [[Definition:Point|point]] in [[Definition:The Plane|the plane]].
Let $PM$ be dropped [[Definition:Perpendicular|perpendicular]] to $V_1 V_2$.
Hence $M = \tuple {x, 0}$.
From [[Intersecting Chord Theorem for Conic Sections]]:
:$PM^2 = k V_1 M \times M V... | Equation of Ellipse in Reduced Form/Cartesian Frame/Proof 2 | https://proofwiki.org/wiki/Equation_of_Ellipse_in_Reduced_Form | https://proofwiki.org/wiki/Equation_of_Ellipse_in_Reduced_Form/Cartesian_Frame/Proof_2 | [
"Equation of Ellipse in Reduced Form",
"Reduced Form of Ellipse",
"Ellipses"
] | [
"Definition:Ellipse",
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Ellipse",
"Definition:Ellipse/Major Axis",
"Definition:Linear Measure/Length",
"Definition:Ellipse/Minor Axis",
"Definition:Linear Measure/Length"
] | [
"File:EllipseEquation-2.png",
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Definition:Right Angle/Perpendicular",
"Intersecting Chord Theorem for Conic Sections",
"Definition:Constant",
"Definition:Point",
"Definition:Axis/Y-Axis"
] |
proofwiki-7845 | Area of Ellipse | Let $K$ be an ellipse whose major axis is of length $2 a$ and whose minor axis is of length $2 b$.
The area $\AA$ of $K$ is given by:
:$\AA = \pi a b$ | Let $K$ be an ellipse aligned in a cartesian plane in reduced form.
Then from Equation of Ellipse in Reduced Form:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
Thus:
:$y = \pm b \sqrt {1 - \dfrac {x^2} {a^2} }$
From the geometric interpretation of the definite integral:
{{begin-eqn}}
{{eqn | l = \AA
| r = b \in... | Let $K$ be an [[Definition:Ellipse|ellipse]] whose [[Definition:Major Axis of Ellipse|major axis]] is of [[Definition:Length of Line|length]] $2 a$ and whose [[Definition:Minor Axis of Ellipse|minor axis]] is of [[Definition:Length of Line|length]] $2 b$.
The [[Definition:Area|area]] $\AA$ of $K$ is given by:
:$\AA =... | Let $K$ be an [[Definition:Ellipse|ellipse]] aligned in a [[Definition:Cartesian Plane|cartesian plane]] in [[Definition:Reduced Form of Ellipse|reduced form]].
Then from [[Equation of Ellipse in Reduced Form]]:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
Thus:
:$y = \pm b \sqrt {1 - \dfrac {x^2} {a^2} }$
From t... | Area of Ellipse/Proof 1 | https://proofwiki.org/wiki/Area_of_Ellipse | https://proofwiki.org/wiki/Area_of_Ellipse/Proof_1 | [
"Area of Ellipse",
"Ellipses",
"Area Formulas"
] | [
"Definition:Ellipse",
"Definition:Ellipse/Major Axis",
"Definition:Linear Measure/Length",
"Definition:Ellipse/Minor Axis",
"Definition:Linear Measure/Length",
"Definition:Area"
] | [
"Definition:Ellipse",
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Ellipse",
"Equation of Ellipse in Reduced Form",
"Definition:Darboux Integral/Geometric Interpretation",
"Integration by Substitution",
"Sum of Squares of Sine and Cosine",
"Integral of Constant/Definite",
"Pr... |
proofwiki-7846 | Area of Ellipse | Let $K$ be an ellipse whose major axis is of length $2 a$ and whose minor axis is of length $2 b$.
The area $\AA$ of $K$ is given by:
:$\AA = \pi a b$ | Let $K$ be an ellipse aligned in a cartesian plane in reduced form.
Then from Equation of Ellipse in Reduced Form:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
Thus:
:$y = \pm \dfrac b a \sqrt {a^2 - x^2}$
Consider a circle of radius $a$ whose center is at the origin.
From Equation of Circle center Origin, its equati... | Let $K$ be an [[Definition:Ellipse|ellipse]] whose [[Definition:Major Axis of Ellipse|major axis]] is of [[Definition:Length of Line|length]] $2 a$ and whose [[Definition:Minor Axis of Ellipse|minor axis]] is of [[Definition:Length of Line|length]] $2 b$.
The [[Definition:Area|area]] $\AA$ of $K$ is given by:
:$\AA =... | Let $K$ be an [[Definition:Ellipse|ellipse]] aligned in a [[Definition:Cartesian Plane|cartesian plane]] in [[Definition:Reduced Form of Ellipse|reduced form]].
Then from [[Equation of Ellipse in Reduced Form]]:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
Thus:
:$y = \pm \dfrac b a \sqrt {a^2 - x^2}$
Consider a ... | Area of Ellipse/Proof 2 | https://proofwiki.org/wiki/Area_of_Ellipse | https://proofwiki.org/wiki/Area_of_Ellipse/Proof_2 | [
"Area of Ellipse",
"Ellipses",
"Area Formulas"
] | [
"Definition:Ellipse",
"Definition:Ellipse/Major Axis",
"Definition:Linear Measure/Length",
"Definition:Ellipse/Minor Axis",
"Definition:Linear Measure/Length",
"Definition:Area"
] | [
"Definition:Ellipse",
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Ellipse",
"Equation of Ellipse in Reduced Form",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Equation of Circle center Origin",
"Defin... |
proofwiki-7847 | Equation of Ellipse in Reduced Form/Cartesian Frame | The equation of $K$ is:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$ | :500px
By definition, the foci $F_1$ and $F_2$ of $K$ are located at $\tuple {-c, 0}$ and $\tuple {c, 0}$ respectively.
Let the vertices of $K$ be $V_1$ and $V_2$.
By definition, these are located at $\tuple {-a, 0}$ and $\tuple {a, 0}$.
Let the covertices of $K$ be $C_1$ and $C_2$.
By definition, these are located at ... | The equation of $K$ is:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$ | :[[File:EllipseEquation.png|500px]]
By definition, the [[Definition:Focus of Ellipse|foci]] $F_1$ and $F_2$ of $K$ are located at $\tuple {-c, 0}$ and $\tuple {c, 0}$ respectively.
Let the [[Definition:Vertex of Ellipse|vertices]] of $K$ be $V_1$ and $V_2$.
By definition, these are located at $\tuple {-a, 0}$ and $\... | Equation of Ellipse in Reduced Form/Cartesian Frame/Proof 1 | https://proofwiki.org/wiki/Equation_of_Ellipse_in_Reduced_Form/Cartesian_Frame | https://proofwiki.org/wiki/Equation_of_Ellipse_in_Reduced_Form/Cartesian_Frame/Proof_1 | [
"Equation of Ellipse in Reduced Form"
] | [] | [
"File:EllipseEquation.png",
"Definition:Ellipse/Focus",
"Definition:Ellipse/Vertex",
"Definition:Ellipse/Covertex",
"Definition:Point",
"Definition:Locus",
"Definition:Ellipse/Equidistance",
"Definition:Constant",
"Definition:Ellipse",
"Equidistance of Ellipse equals Major Axis",
"Linear Eccentr... |
proofwiki-7848 | Equation of Ellipse in Reduced Form/Cartesian Frame | The equation of $K$ is:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$ | :500px
Let $P$ be an arbitrary point in the plane.
Let $PM$ be dropped perpendicular to $V_1 V_2$.
Hence $M = \tuple {x, 0}$.
From Intersecting Chord Theorem for Conic Sections:
:$PM^2 = k V_1 M \times M V_2$
for some constant $k$.
Hence:
{{begin-eqn}}
{{eqn | l = y^2
| r = k \paren {a + x} \paren {a - x}
|... | The equation of $K$ is:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$ | :[[File:EllipseEquation-2.png|500px]]
Let $P$ be an arbitrary [[Definition:Point|point]] in [[Definition:The Plane|the plane]].
Let $PM$ be dropped [[Definition:Perpendicular|perpendicular]] to $V_1 V_2$.
Hence $M = \tuple {x, 0}$.
From [[Intersecting Chord Theorem for Conic Sections]]:
:$PM^2 = k V_1 M \times M V... | Equation of Ellipse in Reduced Form/Cartesian Frame/Proof 2 | https://proofwiki.org/wiki/Equation_of_Ellipse_in_Reduced_Form/Cartesian_Frame | https://proofwiki.org/wiki/Equation_of_Ellipse_in_Reduced_Form/Cartesian_Frame/Proof_2 | [
"Equation of Ellipse in Reduced Form"
] | [] | [
"File:EllipseEquation-2.png",
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Definition:Right Angle/Perpendicular",
"Intersecting Chord Theorem for Conic Sections",
"Definition:Constant",
"Definition:Point",
"Definition:Axis/Y-Axis"
] |
proofwiki-7849 | Equation of Ellipse in Reduced Form/Cartesian Frame/Parametric Form | The equation of $K$ in parametric form is:
{{begin-eqn}}
{{eqn | l = x
| r = a \cos \theta
}}
{{eqn | l = y
| r = b \sin \theta
}}
{{end-eqn}}
where $\theta$ is the eccentric angle of the point $P = \tuple {x, y}$ {{WRT}} $K$. | Let the point $\tuple {x, y}$ satisfy the equations:
{{begin-eqn}}
{{eqn | l = x
| r = a \cos \theta
}}
{{eqn | l = y
| r = b \sin \theta
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \frac {x^2} {a^2} + \frac {y^2} {b^2}
| r = \frac {\paren {a \cos \theta}^2} {a^2} + \frac {\paren {b \sin \theta}^2}... | The equation of $K$ in [[Definition:Parametric Equation|parametric form]] is:
{{begin-eqn}}
{{eqn | l = x
| r = a \cos \theta
}}
{{eqn | l = y
| r = b \sin \theta
}}
{{end-eqn}}
where $\theta$ is the [[Definition:Eccentric Angle of Ellipse|eccentric angle]] of the [[Definition:Point|point]] $P = \tuple {x... | Let the point $\tuple {x, y}$ satisfy the equations:
{{begin-eqn}}
{{eqn | l = x
| r = a \cos \theta
}}
{{eqn | l = y
| r = b \sin \theta
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \frac {x^2} {a^2} + \frac {y^2} {b^2}
| r = \frac {\paren {a \cos \theta}^2} {a^2} + \frac {\paren {b \sin \theta... | Equation of Ellipse in Reduced Form/Cartesian Frame/Parametric Form | https://proofwiki.org/wiki/Equation_of_Ellipse_in_Reduced_Form/Cartesian_Frame/Parametric_Form | https://proofwiki.org/wiki/Equation_of_Ellipse_in_Reduced_Form/Cartesian_Frame/Parametric_Form | [
"Equation of Ellipse in Reduced Form"
] | [
"Definition:Parametric Equation",
"Definition:Eccentric Angle/Ellipse",
"Definition:Point"
] | [
"Sum of Squares of Sine and Cosine"
] |
proofwiki-7850 | Perimeter of Ellipse | Let $K$ be an ellipse whose major axis is of length $2 a$ and whose minor axis is of length $2 b$.
The perimeter $\PP$ of $K$ is given by:
:$\PP = 4 a \map E e$
where:
:$\ds \map E e = \int_0^{\pi / 2} \sqrt{1 - e^2 \sin^2 \theta} \rd \theta$ is the complete elliptic integral of the second kind
:$e = \dfrac {\sqrt {a^2... | Let $K$ be aligned in a cartesian plane such that:
:the major axis of $K$ is aligned with the $x$-axis
:the minor axis of $K$ is aligned with the $y$-axis.
Then from Equation of Ellipse in Reduced Form: parametric form:
:$x = a \cos \theta, y = b \sin \theta$
Thus:
{{begin-eqn}}
{{eqn | l = \frac {\d x} {\d \theta}
... | Let $K$ be an [[Definition:Ellipse|ellipse]] whose [[Definition:Major Axis of Ellipse|major axis]] is of [[Definition:Length of Line|length]] $2 a$ and whose [[Definition:Minor Axis of Ellipse|minor axis]] is of [[Definition:Length of Line|length]] $2 b$.
The [[Definition:Perimeter|perimeter]] $\PP$ of $K$ is given b... | Let $K$ be aligned in a [[Definition:Cartesian Plane|cartesian plane]] such that:
:the [[Definition:Major Axis of Ellipse|major axis]] of $K$ is aligned with the [[Definition:X-Axis|$x$-axis]]
:the [[Definition:Minor Axis of Ellipse|minor axis]] of $K$ is aligned with the [[Definition:Y-Axis|$y$-axis]].
Then from [[... | Perimeter of Ellipse | https://proofwiki.org/wiki/Perimeter_of_Ellipse | https://proofwiki.org/wiki/Perimeter_of_Ellipse | [
"Perimeter of Ellipse",
"Ellipses",
"Perimeter Formulas",
"Complete Elliptic Integral of the Second Kind"
] | [
"Definition:Ellipse",
"Definition:Ellipse/Major Axis",
"Definition:Linear Measure/Length",
"Definition:Ellipse/Minor Axis",
"Definition:Linear Measure/Length",
"Definition:Perimeter",
"Definition:Elliptic Integral of the Second Kind/Complete",
"Definition:Ellipse/Eccentricity"
] | [
"Definition:Cartesian Plane",
"Definition:Ellipse/Major Axis",
"Definition:Axis/X-Axis",
"Definition:Ellipse/Minor Axis",
"Definition:Axis/Y-Axis",
"Equation of Ellipse in Reduced Form/Cartesian Frame/Parametric Form",
"Derivative of Cosine Function",
"Derivative of Sine Function",
"Arc Length for P... |
proofwiki-7851 | Rational Numbers under Multiplication form Commutative Monoid | The set of rational numbers under multiplication $\struct {\Q, \times}$ forms a countably infinite commutative monoid. | From Rational Numbers under Multiplication form Monoid, $\struct {\Q, \times}$ is a monoid.
Then:
:from Rational Multiplication is Commutative we have that $\times$ is commutative on $\Q$
:from Rational Numbers are Countably Infinite we have that $\Q$ is a countably infinite set.
{{Qed}} | The [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] under [[Definition:Rational Multiplication|multiplication]] $\struct {\Q, \times}$ forms a [[Definition:Countably Infinite Set|countably infinite]] [[Definition:Commutative Monoid|commutative monoid]]. | From [[Rational Numbers under Multiplication form Monoid]], $\struct {\Q, \times}$ is a [[Definition:Monoid|monoid]].
Then:
:from [[Rational Multiplication is Commutative]] we have that $\times$ is [[Definition:Commutative Operation|commutative]] on $\Q$
:from [[Rational Numbers are Countably Infinite]] we have that ... | Rational Numbers under Multiplication form Commutative Monoid | https://proofwiki.org/wiki/Rational_Numbers_under_Multiplication_form_Commutative_Monoid | https://proofwiki.org/wiki/Rational_Numbers_under_Multiplication_form_Commutative_Monoid | [
"Rational Multiplication",
"Examples of Commutative Monoids"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Multiplication/Rational Numbers",
"Definition:Countably Infinite/Set",
"Definition:Commutative Monoid"
] | [
"Rational Numbers under Multiplication form Monoid",
"Definition:Monoid",
"Rational Multiplication is Commutative",
"Definition:Commutative/Operation",
"Rational Numbers are Countably Infinite",
"Definition:Countably Infinite/Set"
] |
proofwiki-7852 | Distance of Point from Origin in Cartesian Coordinates | Let $P = \tuple {x, y}$ be a point in the cartesian plane.
Then $P$ is at a distance of $\sqrt {x^2 + y^2}$ from the origin. | :300px
By definition of the cartesian plane, the point $P$ is $x$ units from the $y$-axis and $y$ units from the $x$-axis.
The $y$-axis and $x$-axis are perpendicular to each other, also by definition.
Thus $x$, $y$ and $OP$ form a right-angled triangle.
By Pythagoras' Theorem:
:$OP^2 = x^2 + y^2$
Hence the result.
{{q... | Let $P = \tuple {x, y}$ be a [[Definition:Point|point]] in the [[Definition:Cartesian Plane|cartesian plane]].
Then $P$ is at a [[Definition:Length (Linear Measure)|distance]] of $\sqrt {x^2 + y^2}$ from the [[Definition:Origin|origin]]. | :[[File:DistanceFromOrigin.png|300px]]
By definition of the [[Definition:Cartesian Plane|cartesian plane]], the point $P$ is $x$ units from the [[Definition:Y-Axis|$y$-axis]] and $y$ units from the [[Definition:X-Axis|$x$-axis]].
The [[Definition:Y-Axis|$y$-axis]] and [[Definition:X-Axis|$x$-axis]] are [[Definition:... | Distance of Point from Origin in Cartesian Coordinates | https://proofwiki.org/wiki/Distance_of_Point_from_Origin_in_Cartesian_Coordinates | https://proofwiki.org/wiki/Distance_of_Point_from_Origin_in_Cartesian_Coordinates | [
"Cartesian Coordinate Systems"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Linear Measure/Length",
"Definition:Coordinate System/Origin"
] | [
"File:DistanceFromOrigin.png",
"Definition:Cartesian Plane",
"Definition:Axis/Y-Axis",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Axis/X-Axis",
"Definition:Right Angle/Perpendicular",
"Definition:Triangle (Geometry)/Right-Angled",
"Pythagoras's Theorem"
] |
proofwiki-7853 | Composite of Monomorphisms is Monomorphism | Let:
:$\struct {S_1, \circ_1, \circ_2, \ldots, \circ_n}$
:$\struct {S_2, *_1, *_2, \ldots, *_n}$
:$\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$
be algebraic structures.
Let:
:$\phi: \struct {S_1, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
:$\psi: \struct {S_2, *_1, *_2, \ldots,... | From Composite of Homomorphisms on Algebraic Structure is Homomorphism, $\psi \circ \phi$ is a homomorphism.
From Composite of Injections is Injection, $\psi \circ \phi$ is an injection.
A monomorphism is an injective homorphism.
Hence $\psi \circ \phi$ is a monomorphism.
{{qed}} | Let:
:$\struct {S_1, \circ_1, \circ_2, \ldots, \circ_n}$
:$\struct {S_2, *_1, *_2, \ldots, *_n}$
:$\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$
be [[Definition:Algebraic Structure|algebraic structures]].
Let:
:$\phi: \struct {S_1, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
:$\... | From [[Composite of Homomorphisms on Algebraic Structure is Homomorphism]], $\psi \circ \phi$ is a [[Definition:Homomorphism (Abstract Algebra)|homomorphism]].
From [[Composite of Injections is Injection]], $\psi \circ \phi$ is an [[Definition:Injection|injection]].
A [[Definition:Monomorphism (Abstract Algebra)|mono... | Composite of Monomorphisms is Monomorphism | https://proofwiki.org/wiki/Composite_of_Monomorphisms_is_Monomorphism | https://proofwiki.org/wiki/Composite_of_Monomorphisms_is_Monomorphism | [
"Monomorphisms (Abstract Algebra)",
"Composite Mappings"
] | [
"Definition:Algebraic Structure",
"Definition:Monomorphism (Abstract Algebra)",
"Definition:Composition of Mappings",
"Definition:Monomorphism (Abstract Algebra)"
] | [
"Composite of Homomorphisms is Homomorphism/Algebraic Structure",
"Definition:Homomorphism (Abstract Algebra)",
"Composite of Injections is Injection",
"Definition:Injection",
"Definition:Monomorphism (Abstract Algebra)",
"Definition:Injection",
"Definition:Homomorphism (Abstract Algebra)",
"Definitio... |
proofwiki-7854 | Composite of Epimorphisms is Epimorphism | Let:
:$\struct {S_1, \circ_1, \circ_2, \ldots, \circ_n}$
:$\struct {S_2, *_1, *_2, \ldots, *_n}$
:$\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$
be algebraic structures.
Let:
:$\phi: \struct {S_1, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
:$\psi: \struct {S_2, *_1, *_2, \ldots,... | From Composite of Homomorphisms on Algebraic Structure is Homomorphism, $\psi \circ \phi$ is a homomorphism.
From Composite of Surjections is Surjection, $\psi \circ \phi$ is a surjection.
An epimorphism is a surjective homomorphism.
Hence $\psi \circ \phi$ is an epimorphism.
{{qed}} | Let:
:$\struct {S_1, \circ_1, \circ_2, \ldots, \circ_n}$
:$\struct {S_2, *_1, *_2, \ldots, *_n}$
:$\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$
be [[Definition:Algebraic Structure|algebraic structures]].
Let:
:$\phi: \struct {S_1, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
:$\... | From [[Composite of Homomorphisms on Algebraic Structure is Homomorphism]], $\psi \circ \phi$ is a [[Definition:Homomorphism (Abstract Algebra)|homomorphism]].
From [[Composite of Surjections is Surjection]], $\psi \circ \phi$ is a [[Definition:Surjection|surjection]].
An [[Definition:Epimorphism (Abstract Algebra)|e... | Composite of Epimorphisms is Epimorphism | https://proofwiki.org/wiki/Composite_of_Epimorphisms_is_Epimorphism | https://proofwiki.org/wiki/Composite_of_Epimorphisms_is_Epimorphism | [
"Epimorphisms (Abstract Algebra)",
"Composite Mappings"
] | [
"Definition:Algebraic Structure",
"Definition:Epimorphism (Abstract Algebra)",
"Definition:Composition of Mappings",
"Definition:Epimorphism (Abstract Algebra)"
] | [
"Composite of Homomorphisms is Homomorphism/Algebraic Structure",
"Definition:Homomorphism (Abstract Algebra)",
"Composite of Surjections is Surjection",
"Definition:Surjection",
"Definition:Epimorphism (Abstract Algebra)",
"Definition:Surjection",
"Definition:Homomorphism (Abstract Algebra)",
"Defini... |
proofwiki-7855 | Composite of Automorphisms is Automorphism | Let $\struct {S, \odot_1, \odot_2, \ldots, \odot_n}$ be an algebraic structure.
Let:
:$\phi: \struct {S, \odot_1, \odot_2, \ldots, \odot_n} \to \struct {S, \odot_1, \odot_2, \ldots, \odot_n}$
:$\psi: \struct {S, \odot_1, \odot_2, \ldots, \odot_n} \to \struct {S, \odot_1, \odot_2, \ldots, \odot_n}$
be automorphisms.
The... | From Composite of Homomorphisms on Algebraic Structure is Homomorphism, $\psi \circ \phi$ is a homomorphism.
By the definition of a composite mapping, $\psi \circ \phi$ is a mapping from $S$ into $S$.
Hence $\psi \circ \phi$ is an automorphism.
{{qed}} | Let $\struct {S, \odot_1, \odot_2, \ldots, \odot_n}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let:
:$\phi: \struct {S, \odot_1, \odot_2, \ldots, \odot_n} \to \struct {S, \odot_1, \odot_2, \ldots, \odot_n}$
:$\psi: \struct {S, \odot_1, \odot_2, \ldots, \odot_n} \to \struct {S, \odot_1, \odot_2, \ld... | From [[Composite of Homomorphisms on Algebraic Structure is Homomorphism]], $\psi \circ \phi$ is a [[Definition:Homomorphism (Abstract Algebra)|homomorphism]].
By the definition of a [[Definition:Composition of Mappings|composite mapping]], $\psi \circ \phi$ is a [[Definition:Mapping|mapping]] from $S$ into $S$.
Henc... | Composite of Automorphisms is Automorphism | https://proofwiki.org/wiki/Composite_of_Automorphisms_is_Automorphism | https://proofwiki.org/wiki/Composite_of_Automorphisms_is_Automorphism | [
"Automorphisms (Abstract Algebra)",
"Composite Mappings"
] | [
"Definition:Algebraic Structure",
"Definition:Automorphism (Abstract Algebra)",
"Definition:Composition of Mappings",
"Definition:Automorphism (Abstract Algebra)"
] | [
"Composite of Homomorphisms is Homomorphism/Algebraic Structure",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Composition of Mappings",
"Definition:Mapping",
"Definition:Automorphism (Abstract Algebra)"
] |
proofwiki-7856 | Composite of Endomorphisms is Endomorphism | Let $\struct {S, \odot_1, \odot_2, \ldots, \odot_n}$ be an algebraic structure.
Let:
:$\phi: \struct {S, \odot_1, \odot_2, \ldots, \odot_n} \to \struct {S, \odot_1, \odot_2, \ldots, \odot_n}$
:$\psi: \struct {S, \odot_1, \odot_2, \ldots, \odot_n} \to \struct {S, \odot_1, \odot_2, \ldots, \odot_n}$
be endomorphisms.
The... | From Composite of Homomorphisms on Algebraic Structure is Homomorphism, $\psi \circ \phi$ is a homomorphism.
From the definition of composition of mappings, $\psi \circ \phi$ is a mapping from $S$ into $S$.
Hence $\psi \circ \phi$ is an endomorphism.
{{qed}} | Let $\struct {S, \odot_1, \odot_2, \ldots, \odot_n}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let:
:$\phi: \struct {S, \odot_1, \odot_2, \ldots, \odot_n} \to \struct {S, \odot_1, \odot_2, \ldots, \odot_n}$
:$\psi: \struct {S, \odot_1, \odot_2, \ldots, \odot_n} \to \struct {S, \odot_1, \odot_2, \ld... | From [[Composite of Homomorphisms on Algebraic Structure is Homomorphism]], $\psi \circ \phi$ is a [[Definition:Homomorphism (Abstract Algebra)|homomorphism]].
From the definition of [[Definition:Composition of Mappings|composition of mappings]], $\psi \circ \phi$ is a [[Definition:Mapping|mapping]] from $S$ into $S$.... | Composite of Endomorphisms is Endomorphism | https://proofwiki.org/wiki/Composite_of_Endomorphisms_is_Endomorphism | https://proofwiki.org/wiki/Composite_of_Endomorphisms_is_Endomorphism | [
"Endomorphisms",
"Composite Mappings"
] | [
"Definition:Algebraic Structure",
"Definition:Endomorphism",
"Definition:Composition of Mappings",
"Definition:Endomorphism"
] | [
"Composite of Homomorphisms is Homomorphism/Algebraic Structure",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Composition of Mappings",
"Definition:Mapping",
"Definition:Endomorphism"
] |
proofwiki-7857 | Sine of Angle in Cartesian Plane | Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$.
Let $\theta$ be the angle between the $x$-axis and the line $OP$.
Let $r$ be the length of $OP$.
Then:
:$\sin \theta = \dfrac y r$
where $\sin$ denotes the sine of $\theta$. | :500px
Let a unit circle $C$ be drawn with its center at the origin $O$.
Let $Q$ be the point on $C$ which intersects $OP$.
$\angle OSP = \angle ORQ$, as both are right angles.
Both $\triangle OSP$ and $\triangle ORQ$ share angle $\theta$.
By Triangles with Two Equal Angles are Similar it follows that $\triangle OSP$ a... | Let $P = \tuple {x, y}$ be a [[Definition:Point|point]] in the [[Definition:Cartesian Plane|cartesian plane]] whose [[Definition:Origin|origin]] is at $O$.
Let $\theta$ be the [[Definition:Angle|angle]] between the [[Definition:X-Axis|$x$-axis]] and the [[Definition:Line Segment|line]] $OP$.
Let $r$ be the [[Definiti... | :[[File:SineCartesian.png|500px]]
Let a [[Definition:Unit Circle|unit circle]] $C$ be drawn with its [[Definition:Center of Circle|center]] at the [[Definition:Origin|origin]] $O$.
Let $Q$ be the [[Definition:Point|point]] on $C$ which [[Definition:Intersection (Geometry)|intersects]] $OP$.
$\angle OSP = \angle ORQ$... | Sine of Angle in Cartesian Plane | https://proofwiki.org/wiki/Sine_of_Angle_in_Cartesian_Plane | https://proofwiki.org/wiki/Sine_of_Angle_in_Cartesian_Plane | [
"Sine Function"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Line/Segment",
"Definition:Linear Measure/Length",
"Definition:Sine/Definition from Triangle"
] | [
"File:SineCartesian.png",
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Right Angle",
"Definition:Angle",
"Triangles with Two Equal Angles are Similar",
"Definition:Similar Triangle... |
proofwiki-7858 | Equivalence of Definitions of Sine of Angle | Let $\theta$ be an angle.
{{TFAE|def = Sine of Angle|view = sine of $\theta$}} | === Definition from Triangle implies Definition from Circle ===
Let $\sin \theta$ be defined as $\dfrac {\text{Opposite}} {\text{Hypotenuse}}$ in a right triangle.
Consider the triangle $\triangle OAP$.
By construction, $\angle OAP$ is a right angle.
Thus:
{{begin-eqn}}
{{eqn | l = \sin \theta
| r = \frac {AP} {O... | Let $\theta$ be an [[Definition:Angle|angle]].
{{TFAE|def = Sine of Angle|view = sine of $\theta$}} | === Definition from Triangle implies Definition from Circle ===
Let $\sin \theta$ be defined as $\dfrac {\text{Opposite}} {\text{Hypotenuse}}$ in a [[Definition:Right Triangle|right triangle]].
Consider the [[Definition:Triangle (Geometry)|triangle]] $\triangle OAP$.
By construction, $\angle OAP$ is a [[Definition:R... | Equivalence of Definitions of Sine of Angle | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Sine_of_Angle | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Sine_of_Angle | [
"Sine Function"
] | [
"Definition:Angle"
] | [
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)",
"Definition:Right Angle",
"Definition:Circle/Radius",
"Definition:Unit Circle",
"Definition:Right Angle",
"Definition:Circle/Radius",
"Definition:Unit Circle"
] |
proofwiki-7859 | Equivalence of Definitions of Cosine of Angle | Let $\theta$ be an angle.
{{TFAE|def = Cosine of Angle|view = cosine}} | === Definition from Triangle implies Definition from Circle ===
Let $\cos \theta$ be defined as $\dfrac {\text {Adjacent}} {\text {Hypotenuse}}$ in a right triangle.
Consider the triangle $\triangle OAP$.
By construction, $\angle OAP$ is a right angle.
From Parallelism implies Equal Alternate Angles:
:$\angle OPA = \th... | Let $\theta$ be an [[Definition:Angle|angle]].
{{TFAE|def = Cosine of Angle|view = cosine}} | === Definition from Triangle implies Definition from Circle ===
Let $\cos \theta$ be defined as $\dfrac {\text {Adjacent}} {\text {Hypotenuse}}$ in a [[Definition:Right Triangle|right triangle]].
Consider the [[Definition:Triangle (Geometry)|triangle]] $\triangle OAP$.
By construction, $\angle OAP$ is a [[Definition... | Equivalence of Definitions of Cosine of Angle | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Cosine_of_Angle | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Cosine_of_Angle | [
"Cosine Function"
] | [
"Definition:Angle"
] | [
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)",
"Definition:Right Angle",
"Parallelism implies Equal Alternate Angles",
"Definition:Circle/Radius",
"Definition:Unit Circle",
"Parallelism implies Equal Alternate Angles",
"Definition:Right Angle",
"Definition:Circle/Ra... |
proofwiki-7860 | Equivalence of Definitions of Tangent of Angle | Let $\theta$ be an angle.
{{TFAE|def = Tangent of Angle|view = Tangent of $\theta$}} | === Definition from Triangle implies Definition from Circle ===
Let $\tan \theta$ be defined as $\dfrac {\text{Opposite}} {\text{Adjacent}}$ in a right triangle.
Consider the triangle $\triangle OAB$.
By construction, $\angle OAB$ is a right angle.
Thus:
{{begin-eqn}}
{{eqn | l = \tan \theta
| r = \frac {AB} {OA}... | Let $\theta$ be an [[Definition:Angle|angle]].
{{TFAE|def = Tangent of Angle|view = Tangent of $\theta$}} | === Definition from Triangle implies Definition from Circle ===
Let $\tan \theta$ be defined as $\dfrac {\text{Opposite}} {\text{Adjacent}}$ in a [[Definition:Right Triangle|right triangle]].
Consider the [[Definition:Triangle (Geometry)|triangle]] $\triangle OAB$.
By construction, $\angle OAB$ is a [[Definition:Rig... | Equivalence of Definitions of Tangent of Angle | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Tangent_of_Angle | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Tangent_of_Angle | [
"Tangent Function"
] | [
"Definition:Angle"
] | [
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)",
"Definition:Right Angle",
"Definition:Circle/Radius",
"Definition:Unit Circle",
"Definition:Right Angle",
"Definition:Circle/Radius",
"Definition:Unit Circle"
] |
proofwiki-7861 | Cardinality of Subset Relation on Power Set of Finite Set | Let $S$ be a set such that:
:$\card S = n$
where $\card S$ denotes the cardinality of $S$.
From Subset Relation on Power Set is Partial Ordering we have that $\struct {\powerset S, \subseteq}$ is an ordered set.
The cardinality of $\subseteq$ as a relation is $3^n$. | Let $X \in \powerset S$.
Since $X \subseteq S$, it follows that:
:$X' \subseteq X \implies X' \in \powerset S$
because the Subset Relation is Transitive.
From Cardinality of Power Set of Finite Set, it follows that for any $X \in \powerset S$:
:$\set {X' \in \powerset S: X' \subseteq X}$
has $2^{\card X}$ elements.
The... | Let $S$ be a [[Definition:Set|set]] such that:
:$\card S = n$
where $\card S$ denotes the [[Definition:Cardinality|cardinality]] of $S$.
From [[Subset Relation on Power Set is Partial Ordering]] we have that $\struct {\powerset S, \subseteq}$ is an [[Definition:Ordered Set|ordered set]].
The [[Definition:Cardinality... | Let $X \in \powerset S$.
Since $X \subseteq S$, it follows that:
:$X' \subseteq X \implies X' \in \powerset S$
because the [[Subset Relation is Transitive]].
From [[Cardinality of Power Set of Finite Set]], it follows that for any $X \in \powerset S$:
:$\set {X' \in \powerset S: X' \subseteq X}$
has $2^{\card X}... | Cardinality of Subset Relation on Power Set of Finite Set | https://proofwiki.org/wiki/Cardinality_of_Subset_Relation_on_Power_Set_of_Finite_Set | https://proofwiki.org/wiki/Cardinality_of_Subset_Relation_on_Power_Set_of_Finite_Set | [
"Set Theory"
] | [
"Definition:Set",
"Definition:Cardinality",
"Subset Relation on Power Set is Partial Ordering",
"Definition:Ordered Set",
"Definition:Cardinality",
"Definition:Relation"
] | [
"Subset Relation is Transitive",
"Cardinality of Power Set of Finite Set",
"Definition:Cardinality",
"Cardinality of Set of Subsets",
"Binomial Theorem/Integral Index"
] |
proofwiki-7862 | Quotient Structure of Semigroup is Semigroup | Let $\RR$ be a congruence relation on a semigroup $\struct {S, \circ}$.
Then the quotient structure $\struct {S / \RR, \circ_\RR}$ is a semigroup. | From Quotient Structure is Well-Defined we have that $\circ_\RR$ is closed on $S / \RR$.
Let $\eqclass x \RR, \eqclass y \RR, \eqclass z \RR \in S / \RR$ under $\circ_\RR$ be arbitrary.
We shall prove that $\circ_\RR$ is associative:
{{begin-eqn}}
{{eqn | l = \paren {\eqclass x \RR \circ_\RR \eqclass y \RR} \circ_\RR \... | Let $\RR$ be a [[Definition:Congruence Relation|congruence relation]] on a [[Definition:Semigroup|semigroup]] $\struct {S, \circ}$.
Then the [[Definition:Quotient Structure|quotient structure]] $\struct {S / \RR, \circ_\RR}$ is a [[Definition:Semigroup|semigroup]]. | From [[Quotient Structure is Well-Defined]] we have that $\circ_\RR$ is [[Definition:Closed Operation|closed]] on $S / \RR$.
Let $\eqclass x \RR, \eqclass y \RR, \eqclass z \RR \in S / \RR$ under $\circ_\RR$ be arbitrary.
We shall prove that $\circ_\RR$ is [[Definition:Associative Operation|associative]]:
{{begin-eq... | Quotient Structure of Semigroup is Semigroup | https://proofwiki.org/wiki/Quotient_Structure_of_Semigroup_is_Semigroup | https://proofwiki.org/wiki/Quotient_Structure_of_Semigroup_is_Semigroup | [
"Quotient Structures",
"Semigroups"
] | [
"Definition:Congruence Relation",
"Definition:Semigroup",
"Definition:Quotient Structure",
"Definition:Semigroup"
] | [
"Quotient Structure is Well-Defined",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Associative Operation",
"Definition:Semigroup"
] |
proofwiki-7863 | Quotient Structure of Monoid is Monoid | Let $\RR$ be a congruence relation on a monoid $\struct {S, \circ}$ with an identity $e$.
Then the quotient structure $\struct {S / \RR, \circ_\RR}$ is a monoid. | From Quotient Structure of Semigroup is Semigroup $\struct {S / \RR, \circ_\RR}$ is a semigroup.
Let $\eqclass x {\RR} \in S / \RR$.
Consider $\eqclass e \RR$:
{{begin-eqn}}
{{eqn | l = \eqclass x \RR \circ_\RR \eqclass e \RR
| r = \eqclass {x \circ e} \RR
| c = {{Defof|Operation Induced on Quotient Set|Ope... | Let $\RR$ be a [[Definition:Congruence Relation|congruence relation]] on a [[Definition:Monoid|monoid]] $\struct {S, \circ}$ with an [[Definition:Identity Element|identity]] $e$.
Then the [[Definition:Quotient Structure|quotient structure]] $\struct {S / \RR, \circ_\RR}$ is a [[Definition:Monoid|monoid]]. | From [[Quotient Structure of Semigroup is Semigroup]] $\struct {S / \RR, \circ_\RR}$ is a [[Definition:Semigroup|semigroup]].
Let $\eqclass x {\RR} \in S / \RR$.
Consider $\eqclass e \RR$:
{{begin-eqn}}
{{eqn | l = \eqclass x \RR \circ_\RR \eqclass e \RR
| r = \eqclass {x \circ e} \RR
| c = {{Defof|Opera... | Quotient Structure of Monoid is Monoid | https://proofwiki.org/wiki/Quotient_Structure_of_Monoid_is_Monoid | https://proofwiki.org/wiki/Quotient_Structure_of_Monoid_is_Monoid | [
"Quotient Structures",
"Monoids"
] | [
"Definition:Congruence Relation",
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Quotient Structure",
"Definition:Monoid"
] | [
"Quotient Structure of Semigroup is Semigroup",
"Definition:Semigroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Monoid"
] |
proofwiki-7864 | Quotient Structure of Group is Group | Let $\RR$ be a congruence relation on a group $\struct {G, \circ}$.
Then the quotient structure $\struct {G / \RR, \circ_\RR}$ is a group. | From Quotient Structure of Monoid is Monoid $\struct {G / \RR, \circ_\RR}$ is a monoid with $\eqclass e \RR$ as its identity.
Let $\eqclass x \RR \in G / \RR$.
Let $x \in G$ be arbitrary.
Let $-x$ denote the inverse of $x \in G$ under $\circ$.
Consider the equivalence class $\eqclass {-x} \RR \in G / \RR$.
We need to s... | Let $\RR$ be a [[Definition:Congruence Relation|congruence relation]] on a [[Definition:Group|group]] $\struct {G, \circ}$.
Then the [[Definition:Quotient Structure|quotient structure]] $\struct {G / \RR, \circ_\RR}$ is a [[Definition:Group|group]]. | From [[Quotient Structure of Monoid is Monoid]] $\struct {G / \RR, \circ_\RR}$ is a [[Definition:Monoid|monoid]] with $\eqclass e \RR$ as its [[Definition:Identity Element|identity]].
Let $\eqclass x \RR \in G / \RR$.
Let $x \in G$ be arbitrary.
Let $-x$ denote the [[Definition:Inverse Element|inverse]] of $x \in G... | Quotient Structure of Group is Group | https://proofwiki.org/wiki/Quotient_Structure_of_Group_is_Group | https://proofwiki.org/wiki/Quotient_Structure_of_Group_is_Group | [
"Quotient Structures",
"Group Theory"
] | [
"Definition:Congruence Relation",
"Definition:Group",
"Definition:Quotient Structure",
"Definition:Group"
] | [
"Quotient Structure of Monoid is Monoid",
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Equivalence Class",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Group"
] |
proofwiki-7865 | Quotient Structure of Abelian Group is Abelian Group | Let $\RR$ be a congruence relation on an abelian group $\struct {G, \circ}$.
Then the quotient structure $\struct {G / \RR, \circ_\RR}$ is an abelian group. | From Quotient Structure of Group is Group we have that $\struct {G / \RR, \circ_\RR}$ is a group.
Let $\eqclass x \RR, \eqclass y \RR \in S / \RR$.
{{begin-eqn}}
{{eqn | l = \eqclass x \RR \circ_\RR \eqclass y \RR
| r = \eqclass {x \circ y} \RR
| c = {{Defof|Operation Induced on Quotient Set|Operation Induc... | Let $\RR$ be a [[Definition:Congruence Relation|congruence relation]] on an [[Definition:Abelian Group|abelian group]] $\struct {G, \circ}$.
Then the [[Definition:Quotient Structure|quotient structure]] $\struct {G / \RR, \circ_\RR}$ is an [[Definition:Abelian Group|abelian group]]. | From [[Quotient Structure of Group is Group]] we have that $\struct {G / \RR, \circ_\RR}$ is a [[Definition:Group|group]].
Let $\eqclass x \RR, \eqclass y \RR \in S / \RR$.
{{begin-eqn}}
{{eqn | l = \eqclass x \RR \circ_\RR \eqclass y \RR
| r = \eqclass {x \circ y} \RR
| c = {{Defof|Operation Induced on Q... | Quotient Structure of Abelian Group is Abelian Group | https://proofwiki.org/wiki/Quotient_Structure_of_Abelian_Group_is_Abelian_Group | https://proofwiki.org/wiki/Quotient_Structure_of_Abelian_Group_is_Abelian_Group | [
"Quotient Structures",
"Abelian Groups"
] | [
"Definition:Congruence Relation",
"Definition:Abelian Group",
"Definition:Quotient Structure",
"Definition:Abelian Group"
] | [
"Quotient Structure of Group is Group",
"Definition:Group",
"Definition:Commutative/Operation",
"Definition:Commutative/Operation",
"Definition:Abelian Group"
] |
proofwiki-7866 | Quotient Structure is Similar to Structure | Let $\RR$ be a congruence relation on a algebraic structure $\struct {G, \circ}$.
{{MissingLinks|"similar" structure?}}
Then the quotient structure $\struct {G / \RR, \circ_\RR}$ is a similar structure to $\struct {G, \circ}$. | === Quotient Structure of Semigroup is Semigroup ===
{{:Quotient Structure of Semigroup is Semigroup}} | Let $\RR$ be a [[Definition:Congruence Relation|congruence relation]] on a [[Definition:Algebraic Structure|algebraic structure]] $\struct {G, \circ}$.
{{MissingLinks|"similar" structure?}}
Then the [[Definition:Quotient Structure|quotient structure]] $\struct {G / \RR, \circ_\RR}$ is a similar structure to $\struct ... | === [[Quotient Structure of Semigroup is Semigroup]] ===
{{:Quotient Structure of Semigroup is Semigroup}} | Quotient Structure is Similar to Structure | https://proofwiki.org/wiki/Quotient_Structure_is_Similar_to_Structure | https://proofwiki.org/wiki/Quotient_Structure_is_Similar_to_Structure | [
"Quotient Structures"
] | [
"Definition:Congruence Relation",
"Definition:Algebraic Structure",
"Definition:Quotient Structure"
] | [
"Quotient Structure of Semigroup is Semigroup"
] |
proofwiki-7867 | Shape of Secant Function | The nature of the secant function on the set of real numbers $\R$ is as follows:
:$(1): \quad \sec x$ is continuous and strictly increasing on the intervals $\hointr 0 {\dfrac \pi 2}$ and $\hointl {\dfrac \pi 2} \pi$
:$(2): \quad \sec x$ is continuous and strictly decreasing on the intervals $\hointr {-\pi} {-\dfrac \p... | From Derivative of Secant Function:
:$\map {D_x} {\sec x} = \dfrac {\sin x} {\cos^2 x}$
From Sine and Cosine are Periodic on Reals: Corollary:
:$\forall x \in \openint {-\pi} \pi \setminus \set {-\dfrac \pi 2, \dfrac \pi 2}: \cos x \ne 0$
Thus, from Square of Non-Zero Element of Ordered Integral Domain is Strictly Posi... | The nature of the [[Definition:Real Secant Function|secant function]] on the [[Definition:Real Number|set of real numbers]] $\R$ is as follows:
:$(1): \quad \sec x$ is [[Definition:Continuous on Interval|continuous]] and [[Definition:Strictly Increasing Real Function|strictly increasing]] on the [[Definition:Half-Open... | From [[Derivative of Secant Function]]:
:$\map {D_x} {\sec x} = \dfrac {\sin x} {\cos^2 x}$
From [[Sine and Cosine are Periodic on Reals/Corollary|Sine and Cosine are Periodic on Reals: Corollary]]:
:$\forall x \in \openint {-\pi} \pi \setminus \set {-\dfrac \pi 2, \dfrac \pi 2}: \cos x \ne 0$
Thus, from [[Square of ... | Shape of Secant Function | https://proofwiki.org/wiki/Shape_of_Secant_Function | https://proofwiki.org/wiki/Shape_of_Secant_Function | [
"Secant Function"
] | [
"Definition:Secant Function/Real",
"Definition:Real Number",
"Definition:Continuous Real Function/Interval",
"Definition:Strictly Increasing/Real Function",
"Definition:Real Interval/Half-Open",
"Definition:Continuous Real Function/Interval",
"Definition:Strictly Decreasing/Real Function",
"Definition... | [
"Derivative of Secant Function",
"Sine and Cosine are Periodic on Reals/Corollary",
"Square of Non-Zero Element of Ordered Integral Domain is Strictly Positive",
"Sine and Cosine are Periodic on Reals/Corollary",
"Definition:Real Interval/Open",
"Sine and Cosine are Periodic on Reals/Corollary",
"Defini... |
proofwiki-7868 | Shape of Cosecant Function | The nature of the cosecant function on the set of real numbers $\R$ is as follows:
:$(1): \quad$ strictly decreasing on the intervals $\hointr {-\dfrac \pi 2} 0$ and $\hointl 0 {\dfrac \pi 2}$
:$(2): \quad$ strictly increasing on the intervals $\hointr {\dfrac \pi 2} \pi$ and $\hointl \pi {\dfrac {3 \pi} 2}$
:$(3): \qu... | From Derivative of Cosecant Function::
:$\map {D_x} {\csc x} = -\dfrac {\cos x} {\sin^2 x}$
From Sine and Cosine are Periodic on Reals: Corollary:
:$\forall x \in \openint {-\dfrac \pi 2} {\dfrac {3 \pi} 2} \setminus \set {0, \pi}: \sin x \ne 0$
Thus, from Square of Non-Zero Element of Ordered Integral Domain is Strict... | The nature of the [[Definition:Real Cosecant Function|cosecant function]] on the set of [[Definition:Real Number|real numbers]] $\R$ is as follows:
:$(1): \quad$ [[Definition:Strictly Decreasing Real Function|strictly decreasing]] on the [[Definition:Half-Open Real Interval|intervals]] $\hointr {-\dfrac \pi 2} 0$ and ... | From [[Derivative of Cosecant Function]]::
:$\map {D_x} {\csc x} = -\dfrac {\cos x} {\sin^2 x}$
From [[Sine and Cosine are Periodic on Reals/Corollary|Sine and Cosine are Periodic on Reals: Corollary]]:
:$\forall x \in \openint {-\dfrac \pi 2} {\dfrac {3 \pi} 2} \setminus \set {0, \pi}: \sin x \ne 0$
Thus, from [[Squ... | Shape of Cosecant Function | https://proofwiki.org/wiki/Shape_of_Cosecant_Function | https://proofwiki.org/wiki/Shape_of_Cosecant_Function | [
"Cosecant Function"
] | [
"Definition:Cosecant/Real Function",
"Definition:Real Number",
"Definition:Strictly Decreasing/Real Function",
"Definition:Real Interval/Half-Open",
"Definition:Strictly Increasing/Real Function",
"Definition:Real Interval/Half-Open"
] | [
"Derivative of Cosecant Function",
"Sine and Cosine are Periodic on Reals/Corollary",
"Square of Non-Zero Element of Ordered Integral Domain is Strictly Positive",
"Sine and Cosine are Periodic on Reals/Corollary",
"Definition:Real Interval/Open",
"Sine and Cosine are Periodic on Reals/Corollary",
"Defi... |
proofwiki-7869 | Semigroup is Group Iff Latin Square Property Holds | Let $\struct {S, \circ}$ be a semigroup.
Then $\struct {S, \circ}$ is a group {{iff}} for all $a, b \in S$ the Latin square property holds in $S$:
:$a \circ x = b$
:$y \circ a = b$
for $x$ and $y$ each unique in $S$. | === Necessary Condition ===
Let $\struct {S, \circ}$ be a group.
$\struct {S, \circ}$ is a semigroup by the definition of a group.
By Group has Latin Square Property, the Latin square property holds in $S$.
{{qed|lemma}}
=== Sufficient Condition ===
Let $\struct {S, \circ}$ be a semigroup on which the Latin square pro... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Then $\struct {S, \circ}$ is a [[Definition:Group|group]] {{iff}} for all $a, b \in S$ the [[Definition:Latin Square Property|Latin square property]] holds in $S$:
:$a \circ x = b$
:$y \circ a = b$
for $x$ and $y$ each [[Definition:Unique|unique]] in $... | === Necessary Condition ===
Let $\struct {S, \circ}$ be a [[Definition:Group|group]].
$\struct {S, \circ}$ is a [[Definition:Semigroup|semigroup]] by the definition of a [[Definition:Group|group]].
By [[Group has Latin Square Property]], the [[Definition:Latin Square Property|Latin square property]] holds in $S$.
{{... | Semigroup is Group Iff Latin Square Property Holds/Proof 1 | https://proofwiki.org/wiki/Semigroup_is_Group_Iff_Latin_Square_Property_Holds | https://proofwiki.org/wiki/Semigroup_is_Group_Iff_Latin_Square_Property_Holds/Proof_1 | [
"Semigroup is Group Iff Latin Square Property Holds",
"Latin Square Property",
"Semigroups",
"Group Theory"
] | [
"Definition:Semigroup",
"Definition:Group",
"Definition:Latin Square Property",
"Definition:Unique"
] | [
"Definition:Group",
"Definition:Semigroup",
"Definition:Group",
"Group has Latin Square Property",
"Definition:Latin Square Property",
"Definition:Semigroup",
"Definition:Latin Square Property",
"Axiom:Group Axioms",
"Definition:Semigroup",
"Definition:Closure (Abstract Algebra)/Algebraic Structur... |
proofwiki-7870 | Semigroup is Group Iff Latin Square Property Holds | Let $\struct {S, \circ}$ be a semigroup.
Then $\struct {S, \circ}$ is a group {{iff}} for all $a, b \in S$ the Latin square property holds in $S$:
:$a \circ x = b$
:$y \circ a = b$
for $x$ and $y$ each unique in $S$. | === Necessary Condition ===
Let $\struct {S, \circ}$ be a group.
$\struct {S, \circ}$ is a semigroup by the definition of a group.
By Group has Latin Square Property, the Latin square property holds in $S$.
{{qed|lemma}}
=== Sufficient Condition ===
Let $a \in G$.
We have {{hypothesis}}:
:$\exists e \in S: a \circ e =... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Then $\struct {S, \circ}$ is a [[Definition:Group|group]] {{iff}} for all $a, b \in S$ the [[Definition:Latin Square Property|Latin square property]] holds in $S$:
:$a \circ x = b$
:$y \circ a = b$
for $x$ and $y$ each [[Definition:Unique|unique]] in $... | === Necessary Condition ===
Let $\struct {S, \circ}$ be a [[Definition:Group|group]].
$\struct {S, \circ}$ is a [[Definition:Semigroup|semigroup]] by the definition of a [[Definition:Group|group]].
By [[Group has Latin Square Property]], the [[Definition:Latin Square Property|Latin square property]] holds in $S$.
{{... | Semigroup is Group Iff Latin Square Property Holds/Proof 2 | https://proofwiki.org/wiki/Semigroup_is_Group_Iff_Latin_Square_Property_Holds | https://proofwiki.org/wiki/Semigroup_is_Group_Iff_Latin_Square_Property_Holds/Proof_2 | [
"Semigroup is Group Iff Latin Square Property Holds",
"Latin Square Property",
"Semigroups",
"Group Theory"
] | [
"Definition:Semigroup",
"Definition:Group",
"Definition:Latin Square Property",
"Definition:Unique"
] | [
"Definition:Group",
"Definition:Semigroup",
"Definition:Group",
"Group has Latin Square Property",
"Definition:Latin Square Property",
"Definition:Unique",
"Definition:Unique",
"Axiom:Group Axioms/Right",
"Definition:Group"
] |
proofwiki-7871 | Integer Multiples under Multiplication form Semigroup | Let $n \Z$ be the set of integer multiples of $n$.
Then $\struct {n \Z, \times}$ is a semigroup.
If $\size n > 1$ then $\struct {n \Z, \times}$ has no identity. | === Closure ===
Let $p, q \in n \Z$.
Then for some $p', q' \in \Z$:
:$p = n p'$
:$q = n q'$
Hence:
:$p q = \paren {n p'} \paren {n q'}$
By the commutativity and associativity of integer multiplication:
:$p q = n \paren {n \paren {p' q'} }$
Hence $p q \in n \Z$ and $n \Z$ is closed under $\times$.
{{qed|lemma}} | Let $n \Z$ be the [[Definition:Set of Integer Multiples|set of integer multiples]] of $n$.
Then $\struct {n \Z, \times}$ is a [[Definition:Semigroup|semigroup]].
If $\size n > 1$ then $\struct {n \Z, \times}$ has no [[Definition:Identity Element|identity]]. | === Closure ===
Let $p, q \in n \Z$.
Then for some $p', q' \in \Z$:
:$p = n p'$
:$q = n q'$
Hence:
:$p q = \paren {n p'} \paren {n q'}$
By the [[Integer Multiplication is Commutative|commutativity]] and [[Integer Multiplication is Associative|associativity]] of [[Definition:Integer Multiplication|integer multipl... | Integer Multiples under Multiplication form Semigroup | https://proofwiki.org/wiki/Integer_Multiples_under_Multiplication_form_Semigroup | https://proofwiki.org/wiki/Integer_Multiples_under_Multiplication_form_Semigroup | [
"Sets of Integer Multiples",
"Examples of Semigroups"
] | [
"Definition:Set of Integer Multiples",
"Definition:Semigroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Integer Multiplication is Commutative",
"Integer Multiplication is Associative",
"Definition:Multiplication/Integers"
] |
proofwiki-7872 | Order of External Direct Product | Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be algebraic structures.
Then the order of $\struct {S \times T, \circ}$ is $\card S \times \card T$. | By definition the order of $\struct {S \times T, \circ}$ is $\card S \times \card T$ is the cardinality of the underlying set $S \times T$.
The result follows directly from Cardinality of Cartesian Product of Finite Sets.
{{qed}} | Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be [[Definition:Algebraic Structure|algebraic structures]].
Then the [[Definition:Order of Structure|order]] of $\struct {S \times T, \circ}$ is $\card S \times \card T$. | By definition the [[Definition:Order of Structure|order]] of $\struct {S \times T, \circ}$ is $\card S \times \card T$ is the [[Definition:Cardinality|cardinality]] of the [[Definition:Underlying Set of Structure|underlying set]] $S \times T$.
The result follows directly from [[Cardinality of Cartesian Product of Fini... | Order of External Direct Product | https://proofwiki.org/wiki/Order_of_External_Direct_Product | https://proofwiki.org/wiki/Order_of_External_Direct_Product | [
"External Direct Products"
] | [
"Definition:Algebraic Structure",
"Definition:Order of Structure"
] | [
"Definition:Order of Structure",
"Definition:Cardinality",
"Definition:Underlying Set/Abstract Algebra",
"Cardinality of Cartesian Product of Finite Sets"
] |
proofwiki-7873 | Congruence (Number Theory) is Congruence Relation | Congruence modulo $m$ is a congruence relation on $\struct {\Z, +}$. | Suppose $a \equiv b \bmod m$ and $c \equiv d \bmod m$.
Then by the definition of congruence there exists $k, k' \in \Z$ such that:
:$\paren {a - b} = k m$
:$\paren {c - d} = k' m$
Hence:
:$\paren {a - b} + \paren {c - d} = k m + k' m$
Using the properties of the integers:
:$\paren {a + c} - \paren {b + d} = m \paren {k... | [[Definition:Congruence (Number Theory)|Congruence modulo $m$]] is a [[Definition:Congruence Relation|congruence relation]] on $\struct {\Z, +}$. | Suppose $a \equiv b \bmod m$ and $c \equiv d \bmod m$.
Then by the definition of [[Definition:Congruence (Number Theory)|congruence]] there exists $k, k' \in \Z$ such that:
:$\paren {a - b} = k m$
:$\paren {c - d} = k' m$
Hence:
:$\paren {a - b} + \paren {c - d} = k m + k' m$
Using the properties of the [[Definit... | Congruence (Number Theory) is Congruence Relation | https://proofwiki.org/wiki/Congruence_(Number_Theory)_is_Congruence_Relation | https://proofwiki.org/wiki/Congruence_(Number_Theory)_is_Congruence_Relation | [
"Integers"
] | [
"Definition:Congruence (Number Theory)",
"Definition:Congruence Relation"
] | [
"Definition:Congruence (Number Theory)",
"Definition:Integer",
"Definition:Congruence (Number Theory)",
"Definition:Congruence Relation"
] |
proofwiki-7874 | Group of Rationals Modulo One is Group | The set of equivalence classes $\Q / \Z$ with respect to the relation:
:$\forall a, b \in \Q: a \sim b \iff a - b \in \Z$
with the binary operation
:$\Q / \Z \times \Q / \Z \to \Q / \Z: \struct {\eqclass a {}, \eqclass b {} } \mapsto \eqclass {a + b} {}$
is an infinite abelian group. | By Rational Numbers under Addition form Infinite Abelian Group, $\Q$ is an infinite Abelian group.
By Subgroup of Abelian Group is Normal, $\Z$ is a normal subgroup of $\Q$.
It follows from Quotient Group is Group that $\Q / \Z$ is a group.
By Quotient Group of Abelian Group is Abelian, $\Q / \Z$ is an Abelian group.
{... | The set of [[Definition:Equivalence Class|equivalence classes]] $\Q / \Z$ with respect to the [[Definition:Relation|relation]]:
:$\forall a, b \in \Q: a \sim b \iff a - b \in \Z$
with the [[Definition:Operation/Binary Operation|binary operation]]
:$\Q / \Z \times \Q / \Z \to \Q / \Z: \struct {\eqclass a {}, \eqclass b... | By [[Rational Numbers under Addition form Infinite Abelian Group]], $\Q$ is an [[Definition:Infinite Group|infinite]] [[Definition:Abelian Group|Abelian group]].
By [[Subgroup of Abelian Group is Normal]], $\Z$ is a [[Definition:Normal Subgroup|normal subgroup]] of $\Q$.
It follows from [[Quotient Group is Group]] th... | Group of Rationals Modulo One is Group | https://proofwiki.org/wiki/Group_of_Rationals_Modulo_One_is_Group | https://proofwiki.org/wiki/Group_of_Rationals_Modulo_One_is_Group | [
"Examples of Groups",
"Rational Numbers"
] | [
"Definition:Equivalence Class",
"Definition:Relation",
"Definition:Operation/Binary Operation",
"Definition:Infinite Group",
"Definition:Abelian Group"
] | [
"Rational Numbers under Addition form Infinite Abelian Group",
"Definition:Infinite Group",
"Definition:Abelian Group",
"Subgroup of Abelian Group is Normal",
"Definition:Normal Subgroup",
"Quotient Group is Group",
"Definition:Group",
"Quotient Group of Abelian Group is Abelian",
"Definition:Abelia... |
proofwiki-7875 | Rational Numbers with Denominators Coprime to Prime under Addition form Group | Let $p$ be a prime number.
Let $\Q_p$ denote the set:
:$\set {\dfrac r s : s \perp p}$
where $s \perp p$ denotes that $s$ is coprime to $p$.
Then $\struct {\Q_p, +}$ is a group. | {{improve|For a neater proof, suggest the One-Step Subgroup Test or Two-Step Subgroup Test be used.}}
Taking each of the group axioms in turn: | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\Q_p$ denote the set:
:$\set {\dfrac r s : s \perp p}$
where $s \perp p$ denotes that $s$ is [[Definition:Coprime Integers|coprime]] to $p$.
Then $\struct {\Q_p, +}$ is a [[Definition:Group|group]]. | {{improve|For a neater proof, suggest the [[One-Step Subgroup Test]] or [[Two-Step Subgroup Test]] be used.}}
Taking each of the [[Axiom:Group Axioms|group axioms]] in turn: | Rational Numbers with Denominators Coprime to Prime under Addition form Group | https://proofwiki.org/wiki/Rational_Numbers_with_Denominators_Coprime_to_Prime_under_Addition_form_Group | https://proofwiki.org/wiki/Rational_Numbers_with_Denominators_Coprime_to_Prime_under_Addition_form_Group | [
"Examples of Groups",
"Rational Addition"
] | [
"Definition:Prime Number",
"Definition:Coprime/Integers",
"Definition:Group"
] | [
"One-Step Subgroup Test",
"Two-Step Subgroup Test",
"Axiom:Group Axioms"
] |
proofwiki-7876 | Equivalence of Definitions of Cotangent of Angle | Let $\theta$ be an angle.
{{TFAE|def = Cotangent of Angle|cotangent}} | === Definition from Triangle implies Definition from Circle ===
Let $\cot \theta$ be defined as $\dfrac {\text{Adjacent}} {\text{Opposite}}$ in a right triangle.
Consider the triangle $\triangle OAB$.
By construction, $\angle OAB$ is a right angle.
From Parallelism implies Equal Alternate Angles:
:$\angle OBA = \theta$... | Let $\theta$ be an [[Definition:Angle|angle]].
{{TFAE|def = Cotangent of Angle|cotangent}} | === Definition from Triangle implies Definition from Circle ===
Let $\cot \theta$ be defined as $\dfrac {\text{Adjacent}} {\text{Opposite}}$ in a [[Definition:Right Triangle|right triangle]].
Consider the [[Definition:Triangle (Geometry)|triangle]] $\triangle OAB$.
By construction, $\angle OAB$ is a [[Definition:Rig... | Equivalence of Definitions of Cotangent of Angle | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Cotangent_of_Angle | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Cotangent_of_Angle | [
"Cotangent Function"
] | [
"Definition:Angle"
] | [
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)",
"Definition:Right Angle",
"Parallelism implies Equal Alternate Angles",
"Definition:Circle/Radius",
"Definition:Unit Circle",
"Parallelism implies Equal Alternate Angles",
"Definition:Right Angle",
"Definition:Circle/Ra... |
proofwiki-7877 | Equivalence of Definitions of Secant of Angle | Let $\theta$ be an angle.
{{TFAE|def = Secant of Angle|view = secant}} | === Definition from Triangle implies Definition from Circle ===
Let $\sec \theta$ be defined as $\dfrac {\text{Hypotenuse}} {\text{Adjacent}}$ in a right triangle.
Consider the triangle $\triangle OAB$.
By construction, $\angle OAB$ is a right angle.
Thus:
{{begin-eqn}}
{{eqn | l = \sec \theta
| r = \frac {OB} {O... | Let $\theta$ be an [[Definition:Angle|angle]].
{{TFAE|def = Secant of Angle|view = secant}} | === Definition from Triangle implies Definition from Circle ===
Let $\sec \theta$ be defined as $\dfrac {\text{Hypotenuse}} {\text{Adjacent}}$ in a [[Definition:Right Triangle|right triangle]].
Consider the [[Definition:Triangle (Geometry)|triangle]] $\triangle OAB$.
By construction, $\angle OAB$ is a [[Definition:R... | Equivalence of Definitions of Secant of Angle | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Secant_of_Angle | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Secant_of_Angle | [
"Secant Function"
] | [
"Definition:Angle"
] | [
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)",
"Definition:Right Angle",
"Definition:Circle/Radius",
"Definition:Unit Circle",
"Definition:Right Angle",
"Definition:Circle/Radius",
"Definition:Unit Circle"
] |
proofwiki-7878 | Equivalence of Definitions of Cosecant of Angle | Let $\theta$ be an angle.
{{TFAE|def = Cosecant of Angle|view = cosecant}} | === Definition from Triangle implies Definition from Circle ===
Let $\csc \theta$ be defined as $\dfrac {\text{Hypotenuse}} {\text{Opposite}}$ in a right triangle.
Consider the triangle $\triangle OAB$.
By construction, $\angle OAB$ is a right angle.
From Parallelism implies Equal Alternate Angles:
:$\angle OBA = \thet... | Let $\theta$ be an [[Definition:Angle|angle]].
{{TFAE|def = Cosecant of Angle|view = cosecant}} | === Definition from Triangle implies Definition from Circle ===
Let $\csc \theta$ be defined as $\dfrac {\text{Hypotenuse}} {\text{Opposite}}$ in a [[Definition:Right Triangle|right triangle]].
Consider the [[Definition:Triangle (Geometry)|triangle]] $\triangle OAB$.
By construction, $\angle OAB$ is a [[Definition:R... | Equivalence of Definitions of Cosecant of Angle | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Cosecant_of_Angle | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Cosecant_of_Angle | [
"Cosecant Function"
] | [
"Definition:Angle"
] | [
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)",
"Definition:Right Angle",
"Parallelism implies Equal Alternate Angles",
"Definition:Circle/Radius",
"Definition:Unit Circle",
"Parallelism implies Equal Alternate Angles",
"Definition:Right Angle",
"Definition:Circle/Ra... |
proofwiki-7879 | Cosine of Angle in Cartesian Plane | Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$.
Let $\theta$ be the angle between the $x$-axis and the line $OP$.
Let $r$ be the length of $OP$.
Then:
:$\cos \theta = \dfrac x r$
where $\cos$ denotes the cosine of $\theta$. | :500px
Let a unit circle $C$ be drawn with its center at the origin $O$.
Let $Q$ be the point on $C$ which intersects $OP$.
From Parallelism implies Equal Alternate Angles, $\angle OQR = \theta$.
Thus:
:$(1): \quad \cos \theta = RQ$
by definition of cosine
$\angle OSP = \angle ORQ$, as both are right angles.
Both $\tri... | Let $P = \tuple {x, y}$ be a [[Definition:Point|point]] in the [[Definition:Cartesian Plane|cartesian plane]] whose [[Definition:Origin|origin]] is at $O$.
Let $\theta$ be the [[Definition:Angle|angle]] between the [[Definition:X-Axis|$x$-axis]] and the [[Definition:Line Segment|line]] $OP$.
Let $r$ be the [[Definiti... | :[[File:CosineCartesian.png|500px]]
Let a [[Definition:Unit Circle|unit circle]] $C$ be drawn with its [[Definition:Center of Circle|center]] at the [[Definition:Origin|origin]] $O$.
Let $Q$ be the [[Definition:Point|point]] on $C$ which [[Definition:Intersection (Geometry)|intersects]] $OP$.
From [[Parallelism impl... | Cosine of Angle in Cartesian Plane | https://proofwiki.org/wiki/Cosine_of_Angle_in_Cartesian_Plane | https://proofwiki.org/wiki/Cosine_of_Angle_in_Cartesian_Plane | [
"Cosine Function"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Line/Segment",
"Definition:Linear Measure/Length",
"Definition:Cosine/Definition from Triangle"
] | [
"File:CosineCartesian.png",
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Parallelism implies Equal Alternate Angles",
"Definition:Cosine/Definition from Circle",
"Definition:Right Angle",
"De... |
proofwiki-7880 | Tangent of Angle in Cartesian Plane | Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$.
Let $\theta$ be the angle between the $x$-axis and the line $OP$.
Let $r$ be the length of $OP$.
Then:
:$\tan \theta = \dfrac y x$
where $\tan$ denotes the tangent of $\theta$. | :500px
Let a unit circle $C$ be drawn with its center at the origin $O$.
Let a tangent line be drawn to $C$ parallel to $PS$ meeting $C$ at $R$.
Let $Q$ be the point on $OP$ which intersects this tangent line.
$\angle OSP = \angle ORQ$, as both are right angles.
Both $\triangle OSP$ and $\triangle ORQ$ share angle $\th... | Let $P = \tuple {x, y}$ be a [[Definition:Point|point]] in the [[Definition:Cartesian Plane|cartesian plane]] whose [[Definition:Origin|origin]] is at $O$.
Let $\theta$ be the [[Definition:Angle|angle]] between the [[Definition:X-Axis|$x$-axis]] and the [[Definition:Line Segment|line]] $OP$.
Let $r$ be the [[Definiti... | :[[File:TangentCartesian.png|500px]]
Let a [[Definition:Unit Circle|unit circle]] $C$ be drawn with its [[Definition:Center of Circle|center]] at the [[Definition:Origin|origin]] $O$.
Let a [[Definition:Tangent to Circle|tangent line]] be drawn to $C$ [[Definition:Parallel Lines|parallel]] to $PS$ meeting $C$ at $R$.... | Tangent of Angle in Cartesian Plane | https://proofwiki.org/wiki/Tangent_of_Angle_in_Cartesian_Plane | https://proofwiki.org/wiki/Tangent_of_Angle_in_Cartesian_Plane | [
"Tangent Function"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Line/Segment",
"Definition:Linear Measure/Length",
"Definition:Tangent Function/Definition from Triangle"
] | [
"File:TangentCartesian.png",
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Tangent Line/Circle",
"Definition:Parallel (Geometry)/Lines",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Tangent Line/Circle",
"Definitio... |
proofwiki-7881 | Cotangent of Angle in Cartesian Plane | Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$.
Let $\theta$ be the angle between the $x$-axis and the line $OP$.
Let $r$ be the length of $OP$.
Then:
:$\cot \theta = \dfrac x y$
where $\cot$ denotes the cotangent of $\theta$. | {{ProofWanted|Anybody want to take this on? I seem to have lost interest.}} | Let $P = \tuple {x, y}$ be a [[Definition:Point|point]] in the [[Definition:Cartesian Plane|cartesian plane]] whose [[Definition:Origin|origin]] is at $O$.
Let $\theta$ be the [[Definition:Angle|angle]] between the [[Definition:X-Axis|$x$-axis]] and the [[Definition:Line Segment|line]] $OP$.
Let $r$ be the [[Definit... | {{ProofWanted|Anybody want to take this on? I seem to have lost interest.}} | Cotangent of Angle in Cartesian Plane | https://proofwiki.org/wiki/Cotangent_of_Angle_in_Cartesian_Plane | https://proofwiki.org/wiki/Cotangent_of_Angle_in_Cartesian_Plane | [
"Cotangent Function"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Line/Segment",
"Definition:Linear Measure/Length",
"Definition:Cotangent/Definition from Triangle"
] | [] |
proofwiki-7882 | Secant of Angle in Cartesian Plane | Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$.
Let $\theta$ be the angle between the $x$-axis and the line $OP$.
Let $r$ be the length of $OP$.
Then:
:$\sec \theta = \dfrac r x$
where $\sec$ denotes the secant of $\theta$. | {{ProofWanted|Anybody want to take this on? I seem to have lost interest.}} | Let $P = \tuple {x, y}$ be a [[Definition:Point|point]] in the [[Definition:Cartesian Plane|cartesian plane]] whose [[Definition:Origin|origin]] is at $O$.
Let $\theta$ be the [[Definition:Angle|angle]] between the [[Definition:X-Axis|$x$-axis]] and the [[Definition:Line Segment|line]] $OP$.
Let $r$ be the [[Definit... | {{ProofWanted|Anybody want to take this on? I seem to have lost interest.}} | Secant of Angle in Cartesian Plane | https://proofwiki.org/wiki/Secant_of_Angle_in_Cartesian_Plane | https://proofwiki.org/wiki/Secant_of_Angle_in_Cartesian_Plane | [
"Secant Function"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Line/Segment",
"Definition:Linear Measure/Length",
"Definition:Secant Function/Definition from Triangle"
] | [] |
proofwiki-7883 | Cosecant of Angle in Cartesian Plane | Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$.
Let $\theta$ be the angle between the $x$-axis and the line $OP$.
Let $r$ be the length of $OP$.
Then:
:$\csc \theta = \dfrac r x$
where $\csc$ denotes the secant of $\theta$. | {{ProofWanted|Anybody want to take this on? I seem to have lost interest.}} | Let $P = \tuple {x, y}$ be a [[Definition:Point|point]] in the [[Definition:Cartesian Plane|cartesian plane]] whose [[Definition:Origin|origin]] is at $O$.
Let $\theta$ be the [[Definition:Angle|angle]] between the [[Definition:X-Axis|$x$-axis]] and the [[Definition:Line Segment|line]] $OP$.
Let $r$ be the [[Definit... | {{ProofWanted|Anybody want to take this on? I seem to have lost interest.}} | Cosecant of Angle in Cartesian Plane | https://proofwiki.org/wiki/Cosecant_of_Angle_in_Cartesian_Plane | https://proofwiki.org/wiki/Cosecant_of_Angle_in_Cartesian_Plane | [
"Cosecant Function"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Line/Segment",
"Definition:Linear Measure/Length",
"Definition:Cosecant/Definition from Triangle"
] | [] |
proofwiki-7884 | Tangent is Sine divided by Cosine | Let $\theta$ be an angle such that $\cos \theta \ne 0$.
Then:
:$\tan \theta = \dfrac {\sin \theta} {\cos \theta}$
where $\tan$, $\sin$ and $\cos$ mean tangent, sine and cosine respectively. | Let a point $P = \tuple {x, y}$ be placed in a cartesian plane with origin $O$ such that $OP$ forms an angle $\theta$ with the $x$-axis.
Then:
{{begin-eqn}}
{{eqn | l = \frac {\sin \theta} {\cos \theta}
| r = \frac {y / r} {x / r}
| c = Sine of Angle in Cartesian Plane and Cosine of Angle in Cartesian Plane... | Let $\theta$ be an [[Definition:Angle|angle]] such that $\cos \theta \ne 0$.
Then:
:$\tan \theta = \dfrac {\sin \theta} {\cos \theta}$
where $\tan$, $\sin$ and $\cos$ mean [[Definition:Tangent of Angle|tangent]], [[Definition:Sine of Angle|sine]] and [[Definition:Cosine of Angle|cosine]] respectively. | Let a [[Definition:Point|point]] $P = \tuple {x, y}$ be placed in a [[Definition:Cartesian Plane|cartesian plane]] with [[Definition:Origin|origin]] $O$ such that $OP$ forms an [[Definition:Angle|angle]] $\theta$ with the [[Definition:X-Axis|$x$-axis]].
Then:
{{begin-eqn}}
{{eqn | l = \frac {\sin \theta} {\cos \theta... | Tangent is Sine divided by Cosine | https://proofwiki.org/wiki/Tangent_is_Sine_divided_by_Cosine | https://proofwiki.org/wiki/Tangent_is_Sine_divided_by_Cosine | [
"Sine Function",
"Cosine Function",
"Tangent Function"
] | [
"Definition:Angle",
"Definition:Tangent Function/Definition from Triangle",
"Definition:Sine/Definition from Triangle",
"Definition:Cosine/Definition from Triangle"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Sine of Angle in Cartesian Plane",
"Cosine of Angle in Cartesian Plane",
"Tangent of Angle in Cartesian Plane"
] |
proofwiki-7885 | Cotangent is Cosine divided by Sine | Let $\theta$ be an angle such that $\sin \theta \ne 0$.
Then:
:$\cot \theta = \dfrac {\cos \theta} {\sin \theta}$
where $\cot$, $\sin$ and $\cos$ mean cotangent, sine and cosine respectively. | Let a point $P = \tuple {x, y}$ be placed in a cartesian plane with origin $O$ such that $OP$ forms an angle $\theta$ with the $x$-axis.
Then:
{{begin-eqn}}
{{eqn | l = \frac {\cos \theta} {\sin \theta}
| r = \frac {x / r} {y / r}
| c = Cosine of Angle in Cartesian Plane and Sine of Angle in Cartesian Plane... | Let $\theta$ be an [[Definition:Angle|angle]] such that $\sin \theta \ne 0$.
Then:
:$\cot \theta = \dfrac {\cos \theta} {\sin \theta}$
where $\cot$, $\sin$ and $\cos$ mean [[Definition:Cotangent of Angle|cotangent]], [[Definition:Sine of Angle|sine]] and [[Definition:Cosine of Angle|cosine]] respectively. | Let a [[Definition:Point|point]] $P = \tuple {x, y}$ be placed in a [[Definition:Cartesian Plane|cartesian plane]] with [[Definition:Origin|origin]] $O$ such that $OP$ forms an [[Definition:Angle|angle]] $\theta$ with the [[Definition:X-Axis|$x$-axis]].
Then:
{{begin-eqn}}
{{eqn | l = \frac {\cos \theta} {\sin \theta... | Cotangent is Cosine divided by Sine | https://proofwiki.org/wiki/Cotangent_is_Cosine_divided_by_Sine | https://proofwiki.org/wiki/Cotangent_is_Cosine_divided_by_Sine | [
"Sine Function",
"Cosine Function",
"Cotangent Function"
] | [
"Definition:Angle",
"Definition:Cotangent/Definition from Triangle",
"Definition:Sine/Definition from Triangle",
"Definition:Cosine/Definition from Triangle"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Cosine of Angle in Cartesian Plane",
"Sine of Angle in Cartesian Plane",
"Cotangent of Angle in Cartesian Plane"
] |
proofwiki-7886 | Cotangent is Reciprocal of Tangent | :$\cot \theta = \dfrac 1 {\tan \theta}$ | Let a point $P = \tuple {x, y}$ be placed in a cartesian plane with origin $O$ such that $OP$ forms an angle $\theta$ with the $x$-axis.
Then:
{{begin-eqn}}
{{eqn | l = \cot \theta
| r = \frac x y
| c = Cotangent of Angle in Cartesian Plane
}}
{{eqn | r = \frac 1 {y / x}
| c =
}}
{{eqn | r = \frac 1 ... | :$\cot \theta = \dfrac 1 {\tan \theta}$ | Let a [[Definition:Point|point]] $P = \tuple {x, y}$ be placed in a [[Definition:Cartesian Plane|cartesian plane]] with [[Definition:Origin|origin]] $O$ such that $OP$ forms an [[Definition:Angle|angle]] $\theta$ with the [[Definition:X-Axis|$x$-axis]].
Then:
{{begin-eqn}}
{{eqn | l = \cot \theta
| r = \frac x ... | Cotangent is Reciprocal of Tangent | https://proofwiki.org/wiki/Cotangent_is_Reciprocal_of_Tangent | https://proofwiki.org/wiki/Cotangent_is_Reciprocal_of_Tangent | [
"Cotangent Function",
"Tangent Function"
] | [] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Cotangent of Angle in Cartesian Plane",
"Tangent of Angle in Cartesian Plane"
] |
proofwiki-7887 | Secant is Reciprocal of Cosine | :$\sec \theta = \dfrac 1 {\cos \theta}$ | Let a point $P = \tuple {x, y}$ be placed in a cartesian plane with origin $O$ such that $OP$ forms an angle $\theta$ with the $x$-axis.
Then:
{{begin-eqn}}
{{eqn | l = \sec \theta
| r = \frac r x
| c = Secant of Angle in Cartesian Plane
}}
{{eqn | r = \frac 1 {x / r}
| c =
}}
{{eqn | r = \frac 1 {\c... | :$\sec \theta = \dfrac 1 {\cos \theta}$ | Let a [[Definition:Point|point]] $P = \tuple {x, y}$ be placed in a [[Definition:Cartesian Plane|cartesian plane]] with [[Definition:Origin|origin]] $O$ such that $OP$ forms an [[Definition:Angle|angle]] $\theta$ with the [[Definition:X-Axis|$x$-axis]].
Then:
{{begin-eqn}}
{{eqn | l = \sec \theta
| r = \frac r ... | Secant is Reciprocal of Cosine | https://proofwiki.org/wiki/Secant_is_Reciprocal_of_Cosine | https://proofwiki.org/wiki/Secant_is_Reciprocal_of_Cosine | [
"Cosine Function",
"Secant Function"
] | [] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Secant of Angle in Cartesian Plane",
"Cosine of Angle in Cartesian Plane"
] |
proofwiki-7888 | Cosecant is Reciprocal of Sine | :$\csc \theta = \dfrac 1 {\sin \theta}$ | Let a point $P = \tuple {x, y}$ be placed in a cartesian plane with origin $O$ such that $OP$ forms an angle $\theta$ with the $x$-axis.
Then:
{{begin-eqn}}
{{eqn | l = \csc \theta
| r = \frac r y
| c = Cosecant of Angle in Cartesian Plane
}}
{{eqn | r = \frac 1 {y / r}
| c =
}}
{{eqn | r = \frac 1 {... | :$\csc \theta = \dfrac 1 {\sin \theta}$ | Let a [[Definition:Point|point]] $P = \tuple {x, y}$ be placed in a [[Definition:Cartesian Plane|cartesian plane]] with [[Definition:Origin|origin]] $O$ such that $OP$ forms an [[Definition:Angle|angle]] $\theta$ with the [[Definition:X-Axis|$x$-axis]].
Then:
{{begin-eqn}}
{{eqn | l = \csc \theta
| r = \frac r ... | Cosecant is Reciprocal of Sine | https://proofwiki.org/wiki/Cosecant_is_Reciprocal_of_Sine | https://proofwiki.org/wiki/Cosecant_is_Reciprocal_of_Sine | [
"Sine Function",
"Cosecant Function"
] | [] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Cosecant of Angle in Cartesian Plane",
"Sine of Angle in Cartesian Plane"
] |
proofwiki-7889 | Positive-Term Generalized Sum Converges iff Supremum | Let $\struct {G, \circ, \le}$ be an abelian totally ordered group, considered under the order topology.
Let $\set {x_i: i \in I}$ be an indexed set of positive elements of $G$.
{{explain|What actually does "positive" mean in this context?}}
Then:
:the generalized sum $\ds \sum \set {x_i: i \in I}$ converges to a point ... | {{explain|There are plenty of details to fill in.}} | Let $\struct {G, \circ, \le}$ be an [[Definition:Abelian Group|abelian]] [[Definition:Totally Ordered Group|totally ordered group]], considered under the [[Definition:Order Topology|order topology]].
Let $\set {x_i: i \in I}$ be an [[Definition:Indexed Set|indexed set]] of positive elements of $G$.
{{explain|What act... | {{explain|There are plenty of details to fill in.}} | Positive-Term Generalized Sum Converges iff Supremum | https://proofwiki.org/wiki/Positive-Term_Generalized_Sum_Converges_iff_Supremum | https://proofwiki.org/wiki/Positive-Term_Generalized_Sum_Converges_iff_Supremum | [
"Topology",
"Group Theory"
] | [
"Definition:Abelian Group",
"Definition:Totally Ordered Group",
"Definition:Order Topology",
"Definition:Indexing Set/Indexed Set",
"Definition:Generalized Sum",
"Definition:Supremum of Set"
] | [] |
proofwiki-7890 | Half Angle Formulas/Sine | {{begin-eqn}}
{{eqn | l = \sin \frac \theta 2
| r = +\sqrt {\frac {1 - \cos \theta} 2}
| c = for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {II}$
}}
{{eqn | l = \sin \frac \theta 2
| r = -\sqrt {\dfrac {1 - \cos \theta} 2}
| c = for $\dfrac \theta 2$ in quadrant $\text {III}$ or ... | {{begin-eqn}}
{{eqn | l = \cos \theta
| r = 1 - 2 \ \sin^2 \frac \theta 2
| c = {{Corollary|Double Angle Formula for Cosine|2}}
}}
{{eqn | ll= \leadsto
| l = 2 \ \sin^2 \frac \theta 2
| r = 1 - \cos \theta
}}
{{eqn | ll= \leadsto
| l = \sin \frac \theta 2
| r = \pm \sqrt {\frac {1 - ... | {{begin-eqn}}
{{eqn | l = \sin \frac \theta 2
| r = +\sqrt {\frac {1 - \cos \theta} 2}
| c = for $\dfrac \theta 2$ in [[Definition:First Quadrant|quadrant $\text I$]] or [[Definition:Second Quadrant|quadrant $\text {II}$]]
}}
{{eqn | l = \sin \frac \theta 2
| r = -\sqrt {\dfrac {1 - \cos \theta} 2}
... | {{begin-eqn}}
{{eqn | l = \cos \theta
| r = 1 - 2 \ \sin^2 \frac \theta 2
| c = {{Corollary|Double Angle Formula for Cosine|2}}
}}
{{eqn | ll= \leadsto
| l = 2 \ \sin^2 \frac \theta 2
| r = 1 - \cos \theta
}}
{{eqn | ll= \leadsto
| l = \sin \frac \theta 2
| r = \pm \sqrt {\frac {1 - ... | Half Angle Formulas/Sine/Proof 1 | https://proofwiki.org/wiki/Half_Angle_Formulas/Sine | https://proofwiki.org/wiki/Half_Angle_Formulas/Sine/Proof_1 | [
"Sine Function",
"Half Angle Formula for Sine"
] | [
"Definition:Cartesian Plane/Quadrants/First",
"Definition:Cartesian Plane/Quadrants/Second",
"Definition:Cartesian Plane/Quadrants/Third",
"Definition:Cartesian Plane/Quadrants/Fourth"
] | [
"Definition:Sine/Definition from Circle/First Quadrant",
"Definition:Sine/Definition from Circle/Second Quadrant",
"Definition:Sine/Definition from Circle/Third Quadrant",
"Definition:Sine/Definition from Circle/Fourth Quadrant"
] |
proofwiki-7891 | Half Angle Formulas/Sine | {{begin-eqn}}
{{eqn | l = \sin \frac \theta 2
| r = +\sqrt {\frac {1 - \cos \theta} 2}
| c = for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {II}$
}}
{{eqn | l = \sin \frac \theta 2
| r = -\sqrt {\dfrac {1 - \cos \theta} 2}
| c = for $\dfrac \theta 2$ in quadrant $\text {III}$ or ... | Define:
:$u = \dfrac \theta 2$
Then:
{{begin-eqn}}
{{eqn | l = \sin^2 u
| r = \frac {1 - \cos 2 u} 2
| c = Power Reduction Formulas
}}
{{eqn | ll= \leadsto
| l = \sin \frac \theta 2
| r = \pm \sqrt {\frac {1 - \cos \theta} 2}
}}
{{end-eqn}}
We also have that:
:In quadrant $\text I$, and quadrant... | {{begin-eqn}}
{{eqn | l = \sin \frac \theta 2
| r = +\sqrt {\frac {1 - \cos \theta} 2}
| c = for $\dfrac \theta 2$ in [[Definition:First Quadrant|quadrant $\text I$]] or [[Definition:Second Quadrant|quadrant $\text {II}$]]
}}
{{eqn | l = \sin \frac \theta 2
| r = -\sqrt {\dfrac {1 - \cos \theta} 2}
... | Define:
:$u = \dfrac \theta 2$
Then:
{{begin-eqn}}
{{eqn | l = \sin^2 u
| r = \frac {1 - \cos 2 u} 2
| c = [[Power Reduction Formulas]]
}}
{{eqn | ll= \leadsto
| l = \sin \frac \theta 2
| r = \pm \sqrt {\frac {1 - \cos \theta} 2}
}}
{{end-eqn}}
We also have that:
:In [[Definition:Sine/Defin... | Half Angle Formulas/Sine/Proof 2 | https://proofwiki.org/wiki/Half_Angle_Formulas/Sine | https://proofwiki.org/wiki/Half_Angle_Formulas/Sine/Proof_2 | [
"Sine Function",
"Half Angle Formula for Sine"
] | [
"Definition:Cartesian Plane/Quadrants/First",
"Definition:Cartesian Plane/Quadrants/Second",
"Definition:Cartesian Plane/Quadrants/Third",
"Definition:Cartesian Plane/Quadrants/Fourth"
] | [
"Power Reduction Formulas",
"Definition:Sine/Definition from Circle/First Quadrant",
"Definition:Sine/Definition from Circle/Second Quadrant",
"Definition:Sine/Definition from Circle/Third Quadrant",
"Definition:Sine/Definition from Circle/Fourth Quadrant"
] |
proofwiki-7892 | Half Angle Formulas/Cosine | {{begin-eqn}}
{{eqn | l = \cos \frac \theta 2
| r = +\sqrt {\frac {1 + \cos \theta} 2}
| c = for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {IV}$
}}
{{eqn | l = \cos \frac \theta 2
| r = -\sqrt {\frac {1 + \cos \theta} 2}
| c = for $\dfrac \theta 2$ in quadrant $\text {II}$ or qu... | {{begin-eqn}}
{{eqn | l = \cos \theta
| r = 2 \cos^2 \frac \theta 2 - 1
| c = {{Corollary|Double Angle Formula for Cosine|1}}
}}
{{eqn | ll= \leadsto
| l = 2 \cos^2 \frac \theta 2
| r = 1 + \cos \theta
}}
{{eqn | ll= \leadsto
| l = \cos \frac \theta 2
| r = \pm \sqrt {\frac {1 + \cos... | {{begin-eqn}}
{{eqn | l = \cos \frac \theta 2
| r = +\sqrt {\frac {1 + \cos \theta} 2}
| c = for $\dfrac \theta 2$ in [[Definition:First Quadrant|quadrant $\text I$]] or [[Definition:Fourth Quadrant|quadrant $\text {IV}$]]
}}
{{eqn | l = \cos \frac \theta 2
| r = -\sqrt {\frac {1 + \cos \theta} 2}
... | {{begin-eqn}}
{{eqn | l = \cos \theta
| r = 2 \cos^2 \frac \theta 2 - 1
| c = {{Corollary|Double Angle Formula for Cosine|1}}
}}
{{eqn | ll= \leadsto
| l = 2 \cos^2 \frac \theta 2
| r = 1 + \cos \theta
}}
{{eqn | ll= \leadsto
| l = \cos \frac \theta 2
| r = \pm \sqrt {\frac {1 + \cos... | Half Angle Formulas/Cosine/Proof 1 | https://proofwiki.org/wiki/Half_Angle_Formulas/Cosine | https://proofwiki.org/wiki/Half_Angle_Formulas/Cosine/Proof_1 | [
"Cosine Function",
"Half Angle Formula for Cosine"
] | [
"Definition:Cartesian Plane/Quadrants/First",
"Definition:Cartesian Plane/Quadrants/Fourth",
"Definition:Cartesian Plane/Quadrants/Second",
"Definition:Cartesian Plane/Quadrants/Third"
] | [
"Definition:Cosine/Definition from Circle/First Quadrant",
"Definition:Cosine/Definition from Circle/Fourth Quadrant",
"Definition:Cosine/Definition from Circle/Second Quadrant",
"Definition:Cosine/Definition from Circle/Third Quadrant"
] |
proofwiki-7893 | Half Angle Formulas/Cosine | {{begin-eqn}}
{{eqn | l = \cos \frac \theta 2
| r = +\sqrt {\frac {1 + \cos \theta} 2}
| c = for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {IV}$
}}
{{eqn | l = \cos \frac \theta 2
| r = -\sqrt {\frac {1 + \cos \theta} 2}
| c = for $\dfrac \theta 2$ in quadrant $\text {II}$ or qu... | Define:
:$u = \dfrac \theta 2$
Then:
{{begin-eqn}}
{{eqn | l = \cos^2 u
| r = \frac {1 + \cos 2 u} 2
| c = Power Reduction Formulas
}}
{{eqn | ll= \leadsto
| l = \cos \frac \theta 2
| r = \pm \sqrt {\frac {1 + \cos \theta} 2}
}}
{{end-eqn}}
We also have that:
:In quadrant $\text I$, and quadran... | {{begin-eqn}}
{{eqn | l = \cos \frac \theta 2
| r = +\sqrt {\frac {1 + \cos \theta} 2}
| c = for $\dfrac \theta 2$ in [[Definition:First Quadrant|quadrant $\text I$]] or [[Definition:Fourth Quadrant|quadrant $\text {IV}$]]
}}
{{eqn | l = \cos \frac \theta 2
| r = -\sqrt {\frac {1 + \cos \theta} 2}
... | Define:
:$u = \dfrac \theta 2$
Then:
{{begin-eqn}}
{{eqn | l = \cos^2 u
| r = \frac {1 + \cos 2 u} 2
| c = [[Power Reduction Formulas]]
}}
{{eqn | ll= \leadsto
| l = \cos \frac \theta 2
| r = \pm \sqrt {\frac {1 + \cos \theta} 2}
}}
{{end-eqn}}
We also have that:
:In [[Definition:Cosine/Def... | Half Angle Formulas/Cosine/Proof 2 | https://proofwiki.org/wiki/Half_Angle_Formulas/Cosine | https://proofwiki.org/wiki/Half_Angle_Formulas/Cosine/Proof_2 | [
"Cosine Function",
"Half Angle Formula for Cosine"
] | [
"Definition:Cartesian Plane/Quadrants/First",
"Definition:Cartesian Plane/Quadrants/Fourth",
"Definition:Cartesian Plane/Quadrants/Second",
"Definition:Cartesian Plane/Quadrants/Third"
] | [
"Power Reduction Formulas",
"Definition:Cosine/Definition from Circle/First Quadrant",
"Definition:Cosine/Definition from Circle/Fourth Quadrant",
"Definition:Cosine/Definition from Circle/Second Quadrant",
"Definition:Cosine/Definition from Circle/Third Quadrant"
] |
proofwiki-7894 | Half Angle Formulas/Tangent | {{begin-eqn}}
{{eqn | l = \tan \frac \theta 2
| r = +\sqrt {\dfrac {1 - \cos \theta} {1 + \cos \theta} }
| c = for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {III}$
}}
{{eqn | l = \tan \frac \theta 2
| r = -\sqrt {\dfrac {1 - \cos \theta} {1 + \cos \theta} }
| c = for $\dfrac \th... | {{begin-eqn}}
{{eqn | l = \tan \frac \theta 2
| r = \frac {\sin \frac \theta 2} {\cos \frac \theta 2}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\pm \sqrt {\frac {1 - \cos \theta} 2} } {\pm \sqrt {\frac {1 + \cos \theta} 2} }
| c = Half Angle Formula for Sine and Half Angle Formula ... | {{begin-eqn}}
{{eqn | l = \tan \frac \theta 2
| r = +\sqrt {\dfrac {1 - \cos \theta} {1 + \cos \theta} }
| c = for $\dfrac \theta 2$ in [[Definition:First Quadrant|quadrant $\text I$]] or [[Definition:Third Quadrant|quadrant $\text {III}$]]
}}
{{eqn | l = \tan \frac \theta 2
| r = -\sqrt {\dfrac {1 - ... | {{begin-eqn}}
{{eqn | l = \tan \frac \theta 2
| r = \frac {\sin \frac \theta 2} {\cos \frac \theta 2}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\pm \sqrt {\frac {1 - \cos \theta} 2} } {\pm \sqrt {\frac {1 + \cos \theta} 2} }
| c = [[Half Angle Formula for Sine]] and [[Half Angl... | Half Angle Formulas/Tangent | https://proofwiki.org/wiki/Half_Angle_Formulas/Tangent | https://proofwiki.org/wiki/Half_Angle_Formulas/Tangent | [
"Half Angle Formula for Tangent",
"Tangent Function"
] | [
"Definition:Cartesian Plane/Quadrants/First",
"Definition:Cartesian Plane/Quadrants/Third",
"Definition:Cartesian Plane/Quadrants/Second",
"Definition:Cartesian Plane/Quadrants/Fourth",
"Definition:Tangent Function/Definition from Triangle",
"Definition:Cosine/Definition from Triangle"
] | [
"Tangent is Sine divided by Cosine",
"Half Angle Formulas/Sine",
"Half Angle Formulas/Cosine"
] |
proofwiki-7895 | Kernel of Normal Operator is Kernel of Adjoint | Let $H$ be a Hilbert space.
Let $A \in \map B H$ be a normal operator.
Then:
:$\ker A = \ker A^*$
where:
:$\ker$ denotes kernel
:$A^*$ denotes the adjoint of $A$. | Let $x \in H$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \ker A
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = A x
| r = \mathbf 0_H
| c = {{Defof|Kernel of Linear Transformation}}
}}
{{eqn | ll= \leadstoandfrom
| l = \innerprod {A x} {A x}
| r = 0
... | Let $H$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $A \in \map B H$ be a [[Definition:Normal Operator|normal operator]].
Then:
:$\ker A = \ker A^*$
where:
:$\ker$ denotes [[Definition:Kernel of Linear Transformation|kernel]]
:$A^*$ denotes the [[Definition:Adjoint Linear Transformation|adjoint]] of $A$. | Let $x \in H$ be [[Definition:Arbitrary|arbitrary]].
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \ker A
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = A x
| r = \mathbf 0_H
| c = {{Defof|Kernel of Linear Transformation}}
}}
{{eqn | ll= \leadstoandfrom
| l = \innerprod {A x... | Kernel of Normal Operator is Kernel of Adjoint/Proof 1 | https://proofwiki.org/wiki/Kernel_of_Normal_Operator_is_Kernel_of_Adjoint | https://proofwiki.org/wiki/Kernel_of_Normal_Operator_is_Kernel_of_Adjoint/Proof_1 | [
"Kernel of Normal Operator is Kernel of Adjoint",
"Normal Operators",
"Kernels of Linear Transformations",
"Adjoints",
"Linear Transformations on Hilbert Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Normal Operator",
"Definition:Kernel of Linear Transformation",
"Definition:Adjoint Linear Transformation"
] | [
"Definition:Arbitrary",
"Adjoint is Involutive",
"Definition:Inner Product",
"Definition:Positiveness",
"Definition:Set Equality/Definition 1"
] |
proofwiki-7896 | Half Angle Formula for Tangent/Corollary 1 | :$\tan \dfrac \theta 2 = \dfrac {\sin \theta} {1 + \cos \theta}$ | {{begin-eqn}}
{{eqn | l = \tan \frac \theta 2
| r = \pm \sqrt {\frac {1 - \cos \theta} {1 + \cos \theta} }
| c = Half Angle Formula for Tangent
}}
{{eqn | r = \pm \sqrt {\frac {\paren {1 - \cos \theta} \paren {1 + \cos \theta} } {\paren {1 + \cos \theta}^2} }
| c = multiplying top and bottom by $\sqrt... | :$\tan \dfrac \theta 2 = \dfrac {\sin \theta} {1 + \cos \theta}$ | {{begin-eqn}}
{{eqn | l = \tan \frac \theta 2
| r = \pm \sqrt {\frac {1 - \cos \theta} {1 + \cos \theta} }
| c = [[Half Angle Formula for Tangent]]
}}
{{eqn | r = \pm \sqrt {\frac {\paren {1 - \cos \theta} \paren {1 + \cos \theta} } {\paren {1 + \cos \theta}^2} }
| c = multiplying top and bottom by $\... | Half Angle Formula for Tangent/Corollary 1 | https://proofwiki.org/wiki/Half_Angle_Formula_for_Tangent/Corollary_1 | https://proofwiki.org/wiki/Half_Angle_Formula_for_Tangent/Corollary_1 | [
"Half Angle Formula for Tangent"
] | [] | [
"Half Angle Formulas/Tangent",
"Difference of Two Squares",
"Sum of Squares of Sine and Cosine",
"L'Hôpital's Rule",
"Definition:Sine/Definition from Circle/First Quadrant",
"Definition:Sine/Definition from Circle/Second Quadrant",
"Definition:Sine/Definition from Circle/Third Quadrant",
"Definition:S... |
proofwiki-7897 | Half Angle Formula for Tangent/Corollary 2 | :$\tan \dfrac \theta 2 = \dfrac {1 - \cos \theta} {\sin \theta}$ | {{begin-eqn}}
{{eqn | l = \tan \frac \theta 2
| r = \pm \sqrt {\frac {1 - \cos \theta} {1 + \cos \theta} }
| c = Half Angle Formula for Tangent
}}
{{eqn | r = \pm \sqrt {\frac {\paren {1 - \cos \theta}^2} {\paren {1 + \cos \theta} \paren {1 - \cos \theta} } }
| c = multiplying both numerator and denom... | :$\tan \dfrac \theta 2 = \dfrac {1 - \cos \theta} {\sin \theta}$ | {{begin-eqn}}
{{eqn | l = \tan \frac \theta 2
| r = \pm \sqrt {\frac {1 - \cos \theta} {1 + \cos \theta} }
| c = [[Half Angle Formulas/Tangent|Half Angle Formula for Tangent]]
}}
{{eqn | r = \pm \sqrt {\frac {\paren {1 - \cos \theta}^2} {\paren {1 + \cos \theta} \paren {1 - \cos \theta} } }
| c = mult... | Half Angle Formula for Tangent/Corollary 2/Proof 1 | https://proofwiki.org/wiki/Half_Angle_Formula_for_Tangent/Corollary_2 | https://proofwiki.org/wiki/Half_Angle_Formula_for_Tangent/Corollary_2/Proof_1 | [
"Half Angle Formula for Tangent"
] | [] | [
"Half Angle Formulas/Tangent",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares",
"Sum of Squares of Sine and Cosine",
"L'Hôpital's Rule",
"Definition:Sine/Definition from Circle/First Quadrant",
"Definition:Sine/Definition from Circle/Second Quadrant",
... |
proofwiki-7898 | Half Angle Formula for Tangent/Corollary 2 | :$\tan \dfrac \theta 2 = \dfrac {1 - \cos \theta} {\sin \theta}$ | {{begin-eqn}}
{{eqn | l = \tan \frac \theta 2
| r = \frac {\sin \frac \theta 2} {\cos \frac \theta 2}
| c = {{Defof|Real Tangent Function}}
}}
{{eqn | r = \frac {\sin \frac \theta 2} {\cos \frac \theta 2} \frac {2 \sin \frac \theta 2} {2 \sin \frac \theta 2}
| c = multiplying both numerator and denomi... | :$\tan \dfrac \theta 2 = \dfrac {1 - \cos \theta} {\sin \theta}$ | {{begin-eqn}}
{{eqn | l = \tan \frac \theta 2
| r = \frac {\sin \frac \theta 2} {\cos \frac \theta 2}
| c = {{Defof|Real Tangent Function}}
}}
{{eqn | r = \frac {\sin \frac \theta 2} {\cos \frac \theta 2} \frac {2 \sin \frac \theta 2} {2 \sin \frac \theta 2}
| c = multiplying both [[Definition:Numerat... | Half Angle Formula for Tangent/Corollary 2/Proof 2 | https://proofwiki.org/wiki/Half_Angle_Formula_for_Tangent/Corollary_2 | https://proofwiki.org/wiki/Half_Angle_Formula_for_Tangent/Corollary_2/Proof_2 | [
"Half Angle Formula for Tangent"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Double Angle Formulas/Sine"
] |
proofwiki-7899 | Half Angle Formula for Tangent/Corollary 3 | :$\tan \dfrac \theta 2 = \csc \theta - \cot \theta$ | {{begin-eqn}}
{{eqn | l = \tan \frac \theta 2
| r = \frac {1 - \cos \theta} {\sin \theta}
| c = {{Corollary|Half Angle Formula for Tangent|2}}
}}
{{eqn | r = \frac 1 {\sin \theta} - \frac {\cos \theta} {\sin \theta}
| c =
}}
{{eqn | r = \csc \theta - \cot \theta
| c = Cosecant is Reciprocal of ... | :$\tan \dfrac \theta 2 = \csc \theta - \cot \theta$ | {{begin-eqn}}
{{eqn | l = \tan \frac \theta 2
| r = \frac {1 - \cos \theta} {\sin \theta}
| c = {{Corollary|Half Angle Formula for Tangent|2}}
}}
{{eqn | r = \frac 1 {\sin \theta} - \frac {\cos \theta} {\sin \theta}
| c =
}}
{{eqn | r = \csc \theta - \cot \theta
| c = [[Cosecant is Reciprocal o... | Half Angle Formula for Tangent/Corollary 3 | https://proofwiki.org/wiki/Half_Angle_Formula_for_Tangent/Corollary_3 | https://proofwiki.org/wiki/Half_Angle_Formula_for_Tangent/Corollary_3 | [
"Half Angle Formula for Tangent"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Cotangent is Cosine divided by Sine"
] |
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