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-9900 | Primitive of x squared over Root of x squared minus a squared/Logarithm Form | :$\ds \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} } = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C$ | With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Power Rule for Derivatives
}}
{{end-eqn}}
and ... | :$\ds \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} } = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C$ | With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Power Rule for Derivatives]]
}}
{{end-eqn}}... | Primitive of x squared over Root of x squared minus a squared/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_x_squared_minus_a_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_x_squared_minus_a_squared/Logarithm_Form | [
"Primitive of x squared over Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Primitive of x over Root of x squared minus a squared",
"Integration by Parts",
"Primitive of Root of x squared minus a squared/Logarithm Form"
] |
proofwiki-9901 | Primitive of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form | :$\ds \int \sqrt {x^2 - a^2} \rd x = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \cosh^{-1} \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sinh \theta
| c = Derivative of Hyperbolic Cosine
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| l = x^... | :$\ds \int \sqrt {x^2 - a^2} \rd x = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \cosh^{-1} \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sinh \theta
| c = [[Derivative of Hyperbolic Cosine]]
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| ... | Primitive of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared/Inverse_Hyperbolic_Cosine_Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared/Inverse_Hyperbolic_Cosine_Form | [
"Primitive of Root of x squared minus a squared"
] | [] | [
"Derivative of Hyperbolic Cosine",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Square of Hyperbolic Sine Function/Corollary"
] |
proofwiki-9902 | Primitive of Root of x squared minus a squared/Logarithm Form | :$\ds \int \sqrt {x^2 - a^2} \rd x = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C$ | We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 \ge a^2$, that is, either:
:$x \ge a$
or:
:$x \le -a$
where it is assumed that $a > 0$.
First let $x \ge a$.
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh u
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d u}
| r = a \sinh u
| c ... | :$\ds \int \sqrt {x^2 - a^2} \rd x = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C$ | We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 \ge a^2$, that is, either:
:$x \ge a$
or:
:$x \le -a$
where it is assumed that $a > 0$.
First let $x \ge a$.
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh u
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d u}
| r = a \sinh u
... | Primitive of Root of x squared minus a squared/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared/Logarithm_Form | [
"Primitive of Root of x squared minus a squared"
] | [] | [
"Derivative of Hyperbolic Cosine",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Square of Hyperbolic Sine Function/Corollary",
"Real Area Hyperbolic Cosine of x over a in Logarithm Form",
"Definition:Pri... |
proofwiki-9903 | Primitive of x squared over Root of x squared minus a squared/Inverse Hyperbolic Cosine Form | :$\ds \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} } = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \cosh^{-1} \frac x a + C$
for $x > a$. | With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Power Rule for Derivatives
}}
{{end-eqn}}
and ... | :$\ds \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} } = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \cosh^{-1} \frac x a + C$
for $x > a$. | With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Power Rule for Derivatives]]
}}
{{end-eqn}}... | Primitive of x squared over Root of x squared minus a squared/Inverse Hyperbolic Cosine Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_x_squared_minus_a_squared/Inverse_Hyperbolic_Cosine_Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_x_squared_minus_a_squared/Inverse_Hyperbolic_Cosine_Form | [
"Primitive of x squared over Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Primitive of x over Root of x squared minus a squared",
"Integration by Parts",
"Primitive of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form"
] |
proofwiki-9904 | Primitive of x by Inverse Hyperbolic Cosine of x over a | :$\ds \int x \arcosh \frac x a \rd x = \paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \arcosh \dfrac x a - \dfrac {x \sqrt {x^2 - a^2} } 4 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcosh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {x^2 - a^2} }
| c = D... | :$\ds \int x \arcosh \frac x a \rd x = \paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \arcosh \dfrac x a - \dfrac {x \sqrt {x^2 - a^2} } 4 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcosh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {... | Primitive of x by Inverse Hyperbolic Cosine of x over a | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Cosine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Cosine_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Cosine Function",
"Primitive of x by Inverse Hyperbolic Cosine of x over a"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Cosine of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of x squared over Root of x squared minus a squared/Inverse Hyperbolic Cosine Form"
] |
proofwiki-9905 | Primitive of x squared over x squared minus a squared/Logarithm Form | :$\ds \int \frac {x^2 \rd x} {x^2 - a^2} = x + \frac a 2 \map \ln {\frac {x - a} {x + a} } + C$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {x^2 - a^2}
| r = \int \frac {x^2 - a^2 + a^2} {x^2 - a^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x^2 - a^2} {x^2 - a^2} \rd x + \int \frac {a^2} {x^2 - a^2} \rd x
| c =
}}
{{eqn | r = \int \d x + a^2 \int \frac {\d x} {x^2 - a^2}
| c = Pri... | :$\ds \int \frac {x^2 \rd x} {x^2 - a^2} = x + \frac a 2 \map \ln {\frac {x - a} {x + a} } + C$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {x^2 - a^2}
| r = \int \frac {x^2 - a^2 + a^2} {x^2 - a^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x^2 - a^2} {x^2 - a^2} \rd x + \int \frac {a^2} {x^2 - a^2} \rd x
| c =
}}
{{eqn | r = \int \d x + a^2 \int \frac {\d x} {x^2 - a^2}
| c = [[P... | Primitive of x squared over x squared minus a squared/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_minus_a_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_minus_a_squared/Logarithm_Form | [
"Primitive of x squared over x squared minus a squared"
] | [] | [
"Primitive of Constant Multiple of Function",
"Primitive of Constant",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form"
] |
proofwiki-9906 | Primitive of x squared over x squared minus a squared/Inverse Hyperbolic Cotangent Form | :$\ds \int \frac {x^2 \rd x} {x^2 - a^2} = x - a \coth^{-1} \frac x a + C$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {x^2 - a^2}
| r = \int \frac {x^2 - a^2 + a^2} {x^2 - a^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x^2 - a^2} {x^2 - a^2} \rd x + \int \frac {a^2} {x^2 - a^2} \rd x
| c =
}}
{{eqn | r = \int \d x + a^2 \int \frac {\d x} {x^2 - a^2}
| c = Pri... | :$\ds \int \frac {x^2 \rd x} {x^2 - a^2} = x - a \coth^{-1} \frac x a + C$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {x^2 - a^2}
| r = \int \frac {x^2 - a^2 + a^2} {x^2 - a^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x^2 - a^2} {x^2 - a^2} \rd x + \int \frac {a^2} {x^2 - a^2} \rd x
| c =
}}
{{eqn | r = \int \d x + a^2 \int \frac {\d x} {x^2 - a^2}
| c = [[P... | Primitive of x squared over x squared minus a squared/Inverse Hyperbolic Cotangent Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_minus_a_squared/Inverse_Hyperbolic_Cotangent_Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_minus_a_squared/Inverse_Hyperbolic_Cotangent_Form | [
"Primitive of x squared over x squared minus a squared"
] | [] | [
"Primitive of Constant Multiple of Function",
"Primitive of Constant",
"Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form",
"Category:Primitive of x squared over x squared minus a squared"
] |
proofwiki-9907 | Primitive of x squared over a squared minus x squared/Logarithm Form | :$\ds \int \frac {x^2 \rd x} {a^2 - x^2} = -x + \frac a 2 \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {a^2 - x^2}
| r = \int \frac {x^2 - a^2 + a^2} {a^2 - x^2} \rd x
| c =
}}
{{eqn | r = \int \frac {-\paren {a^2 - x^2} } {a^2 - x^2} \rd x + \int \frac {a^2} {a^2 - x^2} \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = -\int \d x + a^2 \int \... | :$\ds \int \frac {x^2 \rd x} {a^2 - x^2} = -x + \frac a 2 \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {a^2 - x^2}
| r = \int \frac {x^2 - a^2 + a^2} {a^2 - x^2} \rd x
| c =
}}
{{eqn | r = \int \frac {-\paren {a^2 - x^2} } {a^2 - x^2} \rd x + \int \frac {a^2} {a^2 - x^2} \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = -\int \d x + a^2 \i... | Primitive of x squared over a squared minus x squared/Logarithm Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_squared_minus_x_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_squared_minus_x_squared/Logarithm_Form | [
"Primitive of x squared over a squared minus x squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form"
] |
proofwiki-9908 | Primitive of x squared over a squared minus x squared/Inverse Hyperbolic Tangent Form | :$\ds \int \frac {x^2 \rd x} {a^2 - x^2} = -x + a \tanh^{-1} \frac x a + C$
for $x^2 < a^2$. | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {a^2 - x^2}
| r = \int \frac {x^2 - a^2 + a^2} {a^2 - x^2} \rd x
| c =
}}
{{eqn | r = \int \frac {-\paren {a^2 - x^2} } {a^2 - x^2} \rd x + \int \frac {a^2} {a^2 - x^2} \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = -\int \rd x + a^2 ... | :$\ds \int \frac {x^2 \rd x} {a^2 - x^2} = -x + a \tanh^{-1} \frac x a + C$
for $x^2 < a^2$. | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {a^2 - x^2}
| r = \int \frac {x^2 - a^2 + a^2} {a^2 - x^2} \rd x
| c =
}}
{{eqn | r = \int \frac {-\paren {a^2 - x^2} } {a^2 - x^2} \rd x + \int \frac {a^2} {a^2 - x^2} \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = -\int \rd x + ... | Primitive of x squared over a squared minus x squared/Inverse Hyperbolic Tangent Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_squared_minus_x_squared/Inverse_Hyperbolic_Tangent_Form | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_squared_minus_x_squared/Inverse_Hyperbolic_Tangent_Form | [
"Primitive of x squared over a squared minus x squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form",
"Category:Primitive of x squared over a squared minus x squared"
] |
proofwiki-9909 | Primitive of x by Inverse Hyperbolic Tangent of x over a | :$\ds \int x \artanh \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \artanh \frac x a + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \artanh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {a^2 - x^2}
| c = Derivative... | :$\ds \int x \artanh \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \artanh \frac x a + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \artanh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {a^2 - x... | Primitive of x by Inverse Hyperbolic Tangent of x over a | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Tangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Tangent_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Tangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Tangent of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x squared over a squared minus x squared/Inverse Hyperbolic Tangent Form"
] |
proofwiki-9910 | Primitive of x by Inverse Hyperbolic Cotangent of x over a | :$\ds \int x \arcoth \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \arcoth \frac x a + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcoth \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {a^2 - x^2}
| c = Derivative... | :$\ds \int x \arcoth \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \arcoth \frac x a + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcoth \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {a^2 - x... | Primitive of x by Inverse Hyperbolic Cotangent of x over a | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Cotangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Cotangent_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Cotangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Cotangent of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x squared over x squared minus a squared/Inverse Hyperbolic Cotangent Form"
] |
proofwiki-9911 | Primitive of x by Inverse Hyperbolic Secant of x over a | :$\ds \int x \arsech \frac x a \rd x = \dfrac {x^2} 2 \arsech \dfrac x a - \dfrac {a \sqrt {a^2 - x^2} } 2 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arsech \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x \sqrt {a^2 - x^2} }
| ... | :$\ds \int x \arsech \frac x a \rd x = \dfrac {x^2} 2 \arsech \dfrac x a - \dfrac {a \sqrt {a^2 - x^2} } 2 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arsech \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x \s... | Primitive of x by Inverse Hyperbolic Secant of x over a | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Secant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Secant_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Secant Function",
"Primitive of x by Inverse Hyperbolic Secant of x over a"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Secant of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of x over Root of a squared minus x squared"
] |
proofwiki-9912 | Primitive of x by Inverse Hyperbolic Cosecant of x over a | :<nowiki>$\ds \int x \arcsch \frac x a \rd x = \begin {cases}
\dfrac {x^2} 2 \arcsch \dfrac x a + \dfrac {a \sqrt {x^2 + a^2} } 2 + C & : x > 0 \\ \\
\dfrac {x^2} 2 \arcsch \dfrac x a - \dfrac {a \sqrt {x^2 + a^2} } 2 + C & : x < 0
\end {cases}$</nowiki> | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsch \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \dfrac {-a} {\size x \sqrt {a^2 + x^2} }
... | :<nowiki>$\ds \int x \arcsch \frac x a \rd x = \begin {cases}
\dfrac {x^2} 2 \arcsch \dfrac x a + \dfrac {a \sqrt {x^2 + a^2} } 2 + C & : x > 0 \\ \\
\dfrac {x^2} 2 \arcsch \dfrac x a - \dfrac {a \sqrt {x^2 + a^2} } 2 + C & : x < 0
\end {cases}$</nowiki> | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsch \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \dfrac {-a} {\si... | Primitive of x by Inverse Hyperbolic Cosecant of x over a | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Cosecant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_by_Inverse_Hyperbolic_Cosecant_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Cosecant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Cosecant of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of x over Root of x squared plus a squared"
] |
proofwiki-9913 | Primitive of x squared by Inverse Hyperbolic Sine of x over a | :$\ds \int x^2 \arsinh \frac x a \rd x = \frac {x^3} 3 \arsinh \frac x a + \frac {\paren {2 a^2 - x^2} \sqrt {x^2 + a^2} } 9 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arsinh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {x^2 + a^2} }
| c = D... | :$\ds \int x^2 \arsinh \frac x a \rd x = \frac {x^3} 3 \arsinh \frac x a + \frac {\paren {2 a^2 - x^2} \sqrt {x^2 + a^2} } 9 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arsinh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {... | Primitive of x squared by Inverse Hyperbolic Sine of x over a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Inverse_Hyperbolic_Sine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Inverse_Hyperbolic_Sine_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Sine of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of x cubed over Root of x squared plus a squared"
] |
proofwiki-9914 | Primitive of x squared by Inverse Hyperbolic Cosine of x over a | :$\ds \int x^2 \arcosh \frac x a \rd x = \dfrac {x^3} 3 \arcosh \dfrac x a - \dfrac {\paren {x^2 + 2 a^2} \sqrt {x^2 - a^2} } 9 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcosh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {x^2 - a^2} }
| c = D... | :$\ds \int x^2 \arcosh \frac x a \rd x = \dfrac {x^3} 3 \arcosh \dfrac x a - \dfrac {\paren {x^2 + 2 a^2} \sqrt {x^2 - a^2} } 9 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcosh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {... | Primitive of x squared by Inverse Hyperbolic Cosine of x over a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Inverse_Hyperbolic_Cosine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Inverse_Hyperbolic_Cosine_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Cosine Function",
"Primitive of x squared by Inverse Hyperbolic Cosine of x over a"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Cosine of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of x cubed over Root of x squared minus a squared"
] |
proofwiki-9915 | Primitive of x squared by Inverse Hyperbolic Tangent of x over a | :$\ds \int x^2 \artanh \frac x a \rd x = \frac {a x^2} 6 + \frac {x^3} 3 \artanh \frac x a + \frac {a^3} 6 \map \ln {a^2 - x^2} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \artanh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {a^2 - x^2}
| c = Derivative... | :$\ds \int x^2 \artanh \frac x a \rd x = \frac {a x^2} 6 + \frac {x^3} 3 \artanh \frac x a + \frac {a^3} 6 \map \ln {a^2 - x^2} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \artanh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {a^2 - x... | Primitive of x squared by Inverse Hyperbolic Tangent of x over a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Inverse_Hyperbolic_Tangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Inverse_Hyperbolic_Tangent_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Tangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Tangent of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of x cubed over a squared minus x squared"
] |
proofwiki-9916 | Primitive of x squared by Inverse Hyperbolic Cotangent of x over a | :$\ds \int x^2 \arcoth \frac x a \rd x = \frac {a x^2} 6 + \frac {x^3} 3 \arcoth \frac x a + \frac {a^3} 6 \map \ln {x^2 - a^2} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcoth \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 - a^2}
| c = Derivat... | :$\ds \int x^2 \arcoth \frac x a \rd x = \frac {a x^2} 6 + \frac {x^3} 3 \arcoth \frac x a + \frac {a^3} 6 \map \ln {x^2 - a^2} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcoth \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 ... | Primitive of x squared by Inverse Hyperbolic Cotangent of x over a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Inverse_Hyperbolic_Cotangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Inverse_Hyperbolic_Cotangent_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Cotangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Cotangent of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of x cubed over x squared minus a squared"
] |
proofwiki-9917 | Primitive of Inverse Hyperbolic Cosine of x over a over x | {{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \arcosh \dfrac x a \rd x
| r = \dfrac 1 2 \ln^2 \paren {\dfrac {2 x} a} + \sum_{n \mathop \ge 1} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac a x}^{2 n} + C
| c =
}}
{{eqn | r = \dfrac 1 2 \ln^2 \paren {\dfrac {2 x} a} + \dfrac 1 {2 \t... | For $\arcosh \dfrac x a > 0$:
{{begin-eqn}}
{{eqn | l = \arcosh \dfrac x a
| r = \ln \frac {2 x} a - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } \paren {\frac a x}^{2 n} }
| c = Power Series Expansion for Real Area Hyperbolic Cosine
}}
{{eqn | ll= \leadsto... | {{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \arcosh \dfrac x a \rd x
| r = \dfrac 1 2 \ln^2 \paren {\dfrac {2 x} a} + \sum_{n \mathop \ge 1} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac a x}^{2 n} + C
| c =
}}
{{eqn | r = \dfrac 1 2 \ln^2 \paren {\dfrac {2 x} a} + \dfrac 1 {2 \t... | For $\arcosh \dfrac x a > 0$:
{{begin-eqn}}
{{eqn | l = \arcosh \dfrac x a
| r = \ln \frac {2 x} a - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } \paren {\frac a x}^{2 n} }
| c = [[Power Series Expansion for Real Area Hyperbolic Cosine]]
}}
{{eqn | ll= \le... | Primitive of Inverse Hyperbolic Cosine of x over a over x | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Cosine_of_x_over_a_over_x | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Cosine_of_x_over_a_over_x | [
"Primitives involving Inverse Hyperbolic Cosine Function",
"Primitive of Inverse Hyperbolic Cosine of x over a over x"
] | [] | [
"Power Series Expansion for Real Area Hyperbolic Cosine",
"Fubini's Theorem",
"Primitive of Logarithm of a x over x",
"Primitive of Power"
] |
proofwiki-9918 | Primitive of Inverse Hyperbolic Secant of x over a over x | {{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \arsech \dfrac x a \rd x
| r = -\frac 1 2 \map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} - \sum_{n \mathop \ge 1} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C
| c =
}}
{{eqn | r = -\dfrac 1 2 \map \ln {\dfrac a x} \map ... | {{begin-eqn}}
{{eqn | l = \arsech \dfrac x a
| r = \ln \frac {2 a} x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } \paren {\frac x a}^{2 n} }
| c = Power Series Expansion for Real Area Hyperbolic Secant
}}
{{eqn | ll= \leadsto
| l = \frac 1 x \arsech... | {{begin-eqn}}
{{eqn | l = \int \dfrac 1 x \arsech \dfrac x a \rd x
| r = -\frac 1 2 \map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} - \sum_{n \mathop \ge 1} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac x a}^{2 n} + C
| c =
}}
{{eqn | r = -\dfrac 1 2 \map \ln {\dfrac a x} \map ... | {{begin-eqn}}
{{eqn | l = \arsech \dfrac x a
| r = \ln \frac {2 a} x - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } \paren {\frac x a}^{2 n} }
| c = [[Power Series Expansion for Real Area Hyperbolic Secant]]
}}
{{eqn | ll= \leadsto
| l = \frac 1 x \ar... | Primitive of Inverse Hyperbolic Secant of x over a over x | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Secant_of_x_over_a_over_x | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Secant_of_x_over_a_over_x | [
"Primitives involving Inverse Hyperbolic Secant Function",
"Primitive of Inverse Hyperbolic Secant of x over a over x"
] | [] | [
"Power Series Expansion for Real Area Hyperbolic Secant",
"Fubini's Theorem",
"Primitive of Logarithm of a x over x",
"Primitive of Power",
"Definition:Arbitrary Constant"
] |
proofwiki-9919 | Primitive of Inverse Hyperbolic Sine of x over a over x squared | :$\ds \int \frac 1 {x^2} \arsinh \dfrac x a \rd x = -\frac 1 x \arsinh \dfrac x a - \frac 1 a \map \ln {\frac a x + \frac {\sqrt {a^2 + x^2} } {\size x} }$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arsinh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {x^2 + a^2} }
| c = D... | :$\ds \int \frac 1 {x^2} \arsinh \dfrac x a \rd x = -\frac 1 x \arsinh \dfrac x a - \frac 1 a \map \ln {\frac a x + \frac {\sqrt {a^2 + x^2} } {\size x} }$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arsinh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {... | Primitive of Inverse Hyperbolic Sine of x over a over x squared | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Sine_of_x_over_a_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Sine_of_x_over_a_over_x_squared | [
"Primitives involving Inverse Hyperbolic Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Sine of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form"
] |
proofwiki-9920 | Primitive of Inverse Hyperbolic Cosine of x over a over x squared | :$\ds \int \frac 1 {x^2} \arcosh \dfrac x a \rd x = -\frac 1 x \arcosh \dfrac x a + \frac 1 a \arcsec \size {\frac x a} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcosh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {x^2 - a^2} }
| c = D... | :$\ds \int \frac 1 {x^2} \arcosh \dfrac x a \rd x = -\frac 1 x \arcosh \dfrac x a + \frac 1 a \arcsec \size {\frac x a} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcosh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {... | Primitive of Inverse Hyperbolic Cosine of x over a over x squared | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Cosine_of_x_over_a_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Cosine_of_x_over_a_over_x_squared | [
"Primitives involving Inverse Hyperbolic Cosine Function",
"Primitive of Inverse Hyperbolic Cosine of x over a over x squared"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Cosine of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of Reciprocal of x by Root of x squared minus a squared"
] |
proofwiki-9921 | Primitive of Inverse Hyperbolic Tangent of x over a over x squared | :$\ds \int \frac 1 {x^2} \artanh \dfrac x a \rd x = -\frac 1 x \artanh \dfrac x a + \frac 1 {2 a} \map \ln {\frac {x^2} {a^2 - x^2} } + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \artanh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {a^2 - x^2}
| c = Derivative... | :$\ds \int \frac 1 {x^2} \artanh \dfrac x a \rd x = -\frac 1 x \artanh \dfrac x a + \frac 1 {2 a} \map \ln {\frac {x^2} {a^2 - x^2} } + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \artanh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {a^2 - x... | Primitive of Inverse Hyperbolic Tangent of x over a over x squared | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Tangent_of_x_over_a_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Tangent_of_x_over_a_over_x_squared | [
"Primitives involving Inverse Hyperbolic Tangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Tangent of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of Reciprocal of x by a squared minus x squared"
] |
proofwiki-9922 | Primitive of Inverse Hyperbolic Cotangent of x over a over x squared | :$\ds \int \frac 1 {x^2} \arcoth \dfrac x a \rd x = -\frac 1 x \arcoth \dfrac x a + \frac 1 {2 a} \map \ln {\frac {x^2} {x^2 - a^2} } + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcoth \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 - a^2}
| c = Derivat... | :$\ds \int \frac 1 {x^2} \arcoth \dfrac x a \rd x = -\frac 1 x \arcoth \dfrac x a + \frac 1 {2 a} \map \ln {\frac {x^2} {x^2 - a^2} } + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcoth \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 ... | Primitive of Inverse Hyperbolic Cotangent of x over a over x squared | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Cotangent_of_x_over_a_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Inverse_Hyperbolic_Cotangent_of_x_over_a_over_x_squared | [
"Primitives involving Inverse Hyperbolic Cotangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Cotangent of x over a",
"Primitive of Power",
"Integration by Parts",
"Primitive of Reciprocal of x by x squared minus a squared",
"Logarithm of Reciprocal"
] |
proofwiki-9923 | Primitive of Power of x by Inverse Hyperbolic Sine of x over a | :$\ds \int x^m \sinh^{-1} \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \sinh^{-1} \frac x a - \frac 1 {m + 1} \int \frac {x^{m + 1} } {\sqrt {x^2 + a^2} } \rd x + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sinh^{-1} \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {x^2 + a^2} }
| c ... | :$\ds \int x^m \sinh^{-1} \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \sinh^{-1} \frac x a - \frac 1 {m + 1} \int \frac {x^{m + 1} } {\sqrt {x^2 + a^2} } \rd x + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sinh^{-1} \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqr... | Primitive of Power of x by Inverse Hyperbolic Sine of x over a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Sine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Sine_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Sine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Sine of x over a",
"Primitive of Power",
"Integration by Parts"
] |
proofwiki-9924 | Primitive of Power of x by Inverse Hyperbolic Cosine of x over a | :$\ds \int x^m \arcosh \frac x a \rd x = \dfrac {x^{m + 1} } {m + 1} \arcosh \dfrac x a - \dfrac 1 {m + 1} \int \dfrac {x^{m + 1} } {\sqrt {x^2 - a^2} } \rd x + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcosh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {x^2 - a^2} }
| c = D... | :$\ds \int x^m \arcosh \frac x a \rd x = \dfrac {x^{m + 1} } {m + 1} \arcosh \dfrac x a - \dfrac 1 {m + 1} \int \dfrac {x^{m + 1} } {\sqrt {x^2 - a^2} } \rd x + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcosh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {\sqrt {... | Primitive of Power of x by Inverse Hyperbolic Cosine of x over a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Cosine_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Cosine_of_x_over_a | [
"Primitive of Power of x by Inverse Hyperbolic Cosine of x over a",
"Primitives involving Inverse Hyperbolic Cosine Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Cosine of x over a",
"Primitive of Power",
"Integration by Parts"
] |
proofwiki-9925 | Primitive of Power of x by Inverse Hyperbolic Tangent of x over a | :$\ds \int x^m \artanh \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \artanh \frac x a - \frac a {m + 1} \int \frac {x^{m + 1} } {a^2 - x^2} \rd x + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \artanh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {a^2 - x^2}
| c = Derivative... | :$\ds \int x^m \artanh \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \artanh \frac x a - \frac a {m + 1} \int \frac {x^{m + 1} } {a^2 - x^2} \rd x + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \artanh \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {a^2 - x... | Primitive of Power of x by Inverse Hyperbolic Tangent of x over a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Tangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Tangent_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Tangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Tangent of x over a",
"Primitive of Power",
"Integration by Parts"
] |
proofwiki-9926 | Primitive of Power of x by Inverse Hyperbolic Cotangent of x over a | :$\ds \int x^m \arcoth \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arcoth \frac x a - \frac a {m + 1} \int \frac {x^{m + 1} } {a^2 - x^2} \rd x + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcoth \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 - a^2}
| c = Derivat... | :$\ds \int x^m \arcoth \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arcoth \frac x a - \frac a {m + 1} \int \frac {x^{m + 1} } {a^2 - x^2} \rd x + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcoth \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 ... | Primitive of Power of x by Inverse Hyperbolic Cotangent of x over a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Cotangent_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Cotangent_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Cotangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Cotangent of x over a",
"Primitive of Power",
"Integration by Parts"
] |
proofwiki-9927 | Primitive of Power of x by Inverse Hyperbolic Secant of x over a | :$\ds \int x^m \arsech \frac x a \rd x = \dfrac {x^{m + 1} } {m + 1} \arsech \dfrac x a + \dfrac a {m + 1} \int \dfrac {x^m} {\sqrt {a^2 - x^2} } \rd x + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arsech \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x \sqrt {a^2 - x^2} }
| ... | :$\ds \int x^m \arsech \frac x a \rd x = \dfrac {x^{m + 1} } {m + 1} \arsech \dfrac x a + \dfrac a {m + 1} \int \dfrac {x^m} {\sqrt {a^2 - x^2} } \rd x + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arsech \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x \s... | Primitive of Power of x by Inverse Hyperbolic Secant of x over a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Secant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Secant_of_x_over_a | [
"Primitive of Power of x by Inverse Hyperbolic Secant of x over a",
"Primitives involving Inverse Hyperbolic Secant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Secant of x over a",
"Primitive of Power",
"Integration by Parts"
] |
proofwiki-9928 | Primitive of Power of x by Inverse Hyperbolic Cosecant of x over a | :<nowiki>$\ds \int x^m \arcsch \frac x a \rd x = \begin{cases}
\ds \frac {x^{m + 1} } {m + 1} \arcsch \frac x a + \frac a {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x > 0 \\ \\
\ds \frac {x^{m + 1} } {m + 1} \arcsch \frac x a - \frac a {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x < 0
\... | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\rd v} {\rd x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsch \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\rd u} {\rd x}
| r = \frac {-a} {\size x \sqrt{a^2 + x^2}... | :<nowiki>$\ds \int x^m \arcsch \frac x a \rd x = \begin{cases}
\ds \frac {x^{m + 1} } {m + 1} \arcsch \frac x a + \frac a {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x > 0 \\ \\
\ds \frac {x^{m + 1} } {m + 1} \arcsch \frac x a - \frac a {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x < 0
\... | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\rd v} {\rd x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsch \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\rd u} {\rd x}
| r = \frac {-a}... | Primitive of Power of x by Inverse Hyperbolic Cosecant of x over a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Cosecant_of_x_over_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Cosecant_of_x_over_a | [
"Primitives involving Inverse Hyperbolic Cosecant Function"
] | [] | [
"Definition:Primitive",
"Derivative of Real Area Hyperbolic Cosecant of x over a",
"Primitive of Power",
"Integration by Parts"
] |
proofwiki-9929 | Hyperbolic Tangent Half-Angle Substitution | :$\ds \int \map F {\sinh x, \cosh x} \rd x = 2 \int \map F {\frac {2 u} {1 - u^2}, \frac {1 + u^2} {1 - u^2} } \frac {\d u} {1 - u^2}$
where $u = \tanh \dfrac x 2$. | {{begin-eqn}}
{{eqn | l = u
| r = \tanh \dfrac x 2
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = 2 \tanh^{-1} u
| c = {{Defof|Inverse Hyperbolic Tangent|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d x} {\d u}
| r = \dfrac 2 {1 - u^2}
| c = Derivative o... | :$\ds \int \map F {\sinh x, \cosh x} \rd x = 2 \int \map F {\frac {2 u} {1 - u^2}, \frac {1 + u^2} {1 - u^2} } \frac {\d u} {1 - u^2}$
where $u = \tanh \dfrac x 2$. | {{begin-eqn}}
{{eqn | l = u
| r = \tanh \dfrac x 2
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = 2 \tanh^{-1} u
| c = {{Defof|Inverse Hyperbolic Tangent|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d x} {\d u}
| r = \dfrac 2 {1 - u^2}
| c = [[Derivative... | Hyperbolic Tangent Half-Angle Substitution | https://proofwiki.org/wiki/Hyperbolic_Tangent_Half-Angle_Substitution | https://proofwiki.org/wiki/Hyperbolic_Tangent_Half-Angle_Substitution | [
"Integral Substitutions",
"Hyperbolic Tangent Function",
"Primitives involving Hyperbolic Sine Function",
"Primitives involving Hyperbolic Cosine Function",
"Hyperbolic Tangent Half-Angle Substitutions"
] | [] | [
"Derivative of Inverse Hyperbolic Tangent",
"Derivative of Constant Multiple",
"Hyperbolic Tangent Half-Angle Substitution for Sine",
"Hyperbolic Tangent Half-Angle Substitution for Cosine",
"Integration by Substitution"
] |
proofwiki-9930 | Reciprocal of One Plus Cosine | :$\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$ | {{begin-eqn}}
{{eqn | l = 1 + \cos x
| r = \cos 0 + \cos x
| c = Cosine of Zero is One
}}
{{eqn | r = 2 \map \cos {\dfrac {0 + x} 2} \map \cos {\dfrac {0 - x} 2}
| c = Cosine plus Cosine
}}
{{eqn | r = 2 \map \cos {\dfrac x 2} \map \cos {\dfrac {-x} 2}
| c = simplifying
}}
{{eqn | r = 2 \map \co... | :$\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$ | {{begin-eqn}}
{{eqn | l = 1 + \cos x
| r = \cos 0 + \cos x
| c = [[Cosine of Zero is One]]
}}
{{eqn | r = 2 \map \cos {\dfrac {0 + x} 2} \map \cos {\dfrac {0 - x} 2}
| c = [[Cosine plus Cosine]]
}}
{{eqn | r = 2 \map \cos {\dfrac x 2} \map \cos {\dfrac {-x} 2}
| c = simplifying
}}
{{eqn | r = 2 ... | Reciprocal of One Plus Cosine/Proof 1 | https://proofwiki.org/wiki/Reciprocal_of_One_Plus_Cosine | https://proofwiki.org/wiki/Reciprocal_of_One_Plus_Cosine/Proof_1 | [
"Reciprocal of One Plus Cosine",
"Trigonometric Identities",
"Cosine Function"
] | [] | [
"Cosine of Zero is One",
"Prosthaphaeresis Formulas/Cosine plus Cosine",
"Cosine Function is Even"
] |
proofwiki-9931 | Reciprocal of One Plus Cosine | :$\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$ | {{begin-eqn}}
{{eqn | l = \cos x
| r = 2 \cos^2 \frac x 2 - 1
| c = {{Corollary|Double Angle Formula for Cosine|1}}
}}
{{eqn | ll= \leadstoandfrom
| l = 1 + \cos x
| r = 2 \cos^2 \frac x 2
| c = adding $1$ to both sides
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 {1 + \cos x}
... | :$\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$ | {{begin-eqn}}
{{eqn | l = \cos x
| r = 2 \cos^2 \frac x 2 - 1
| c = {{Corollary|Double Angle Formula for Cosine|1}}
}}
{{eqn | ll= \leadstoandfrom
| l = 1 + \cos x
| r = 2 \cos^2 \frac x 2
| c = adding $1$ to both sides
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 {1 + \cos x}
... | Reciprocal of One Plus Cosine/Proof 2 | https://proofwiki.org/wiki/Reciprocal_of_One_Plus_Cosine | https://proofwiki.org/wiki/Reciprocal_of_One_Plus_Cosine/Proof_2 | [
"Reciprocal of One Plus Cosine",
"Trigonometric Identities",
"Cosine Function"
] | [] | [
"Definition:Reciprocal"
] |
proofwiki-9932 | Reciprocal of One Plus Cosine | :$\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$ | {{begin-eqn}}
{{eqn | l = \frac 1 {1 + \cos x}
| r = \frac 1 {1 + \frac {1- \tan^2 \frac x 2} {1 + \tan^2 \frac x 2} }
| c = Tangent Half-Angle Substitution for Cosine
}}
{{eqn | r = \frac {1 + \tan^2 \frac x 2} 2
| c = multiplying through $\frac {1 + \tan^2 \frac x 2} {1 + \tan^2 \frac x 2}$
}}
{{eqn | r = \frac 1... | :$\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$ | {{begin-eqn}}
{{eqn | l = \frac 1 {1 + \cos x}
| r = \frac 1 {1 + \frac {1- \tan^2 \frac x 2} {1 + \tan^2 \frac x 2} }
| c = [[Tangent Half-Angle Substitution for Cosine]]
}}
{{eqn | r = \frac {1 + \tan^2 \frac x 2} 2
| c = multiplying through $\frac {1 + \tan^2 \frac x 2} {1 + \tan^2 \frac x 2}$
}}
{{eqn | r = \fr... | Reciprocal of One Plus Cosine/Proof 3 | https://proofwiki.org/wiki/Reciprocal_of_One_Plus_Cosine | https://proofwiki.org/wiki/Reciprocal_of_One_Plus_Cosine/Proof_3 | [
"Reciprocal of One Plus Cosine",
"Trigonometric Identities",
"Cosine Function"
] | [] | [
"Tangent Half-Angle Substitution for Cosine",
"Sum of Squares of Sine and Cosine/Corollary 1"
] |
proofwiki-9933 | Reciprocal of One Minus Cosine | :$\dfrac 1 {1 - \cos x} = \dfrac 1 2 \map {\csc^2} {\dfrac x 2}$ | {{begin-eqn}}
{{eqn | l = 1 - \cos x
| r = \cos 0 - \cos x
| c = Cosine of Zero is One
}}
{{eqn | r = -2 \map \sin {\dfrac {0 + x} 2} \map \sin {\dfrac {0 - x} 2}
| c = Cosine minus Cosine
}}
{{eqn | r = -2 \map \sin {\dfrac x 2} \map \sin {\dfrac {-x} 2}
| c = simplifying
}}
{{eqn | r = 2 \map ... | :$\dfrac 1 {1 - \cos x} = \dfrac 1 2 \map {\csc^2} {\dfrac x 2}$ | {{begin-eqn}}
{{eqn | l = 1 - \cos x
| r = \cos 0 - \cos x
| c = [[Cosine of Zero is One]]
}}
{{eqn | r = -2 \map \sin {\dfrac {0 + x} 2} \map \sin {\dfrac {0 - x} 2}
| c = [[Cosine minus Cosine]]
}}
{{eqn | r = -2 \map \sin {\dfrac x 2} \map \sin {\dfrac {-x} 2}
| c = simplifying
}}
{{eqn | r =... | Reciprocal of One Minus Cosine/Proof 1 | https://proofwiki.org/wiki/Reciprocal_of_One_Minus_Cosine | https://proofwiki.org/wiki/Reciprocal_of_One_Minus_Cosine/Proof_1 | [
"Reciprocal of One Minus Cosine",
"Trigonometric Identities",
"Cosine Function"
] | [] | [
"Cosine of Zero is One",
"Prosthaphaeresis Formulas/Cosine minus Cosine",
"Sine Function is Odd"
] |
proofwiki-9934 | Reciprocal of One Minus Cosine | :$\dfrac 1 {1 - \cos x} = \dfrac 1 2 \map {\csc^2} {\dfrac x 2}$ | {{begin-eqn}}
{{eqn | l = \cos x
| r = 1 - 2 \sin^2 \frac x 2
| c = {{Corollary|Double Angle Formula for Cosine|2}}
}}
{{eqn | ll= \leadstoandfrom
| l = 1 - \cos x
| r = 2 \sin^2 \frac x 2
| c = rearranging
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 {1 - \cos x}
| r = \frac 1... | :$\dfrac 1 {1 - \cos x} = \dfrac 1 2 \map {\csc^2} {\dfrac x 2}$ | {{begin-eqn}}
{{eqn | l = \cos x
| r = 1 - 2 \sin^2 \frac x 2
| c = {{Corollary|Double Angle Formula for Cosine|2}}
}}
{{eqn | ll= \leadstoandfrom
| l = 1 - \cos x
| r = 2 \sin^2 \frac x 2
| c = rearranging
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 {1 - \cos x}
| r = \frac 1... | Reciprocal of One Minus Cosine/Proof 2 | https://proofwiki.org/wiki/Reciprocal_of_One_Minus_Cosine | https://proofwiki.org/wiki/Reciprocal_of_One_Minus_Cosine/Proof_2 | [
"Reciprocal of One Minus Cosine",
"Trigonometric Identities",
"Cosine Function"
] | [] | [
"Definition:Reciprocal"
] |
proofwiki-9935 | Reciprocal of One Minus Cosine | :$\dfrac 1 {1 - \cos x} = \dfrac 1 2 \map {\csc^2} {\dfrac x 2}$ | {{begin-eqn}}
{{eqn | l = \frac 1 {1 - \cos x}
| r = \frac 1 {1 - \frac {1 - \map {\tan^2} {\frac x 2} } {1 + \map {\tan^2} {\frac x 2} } }
| c = Tangent Half-Angle Substitution for Cosine
}}
{{eqn | r = \frac {1 + \map {\tan^2} {\frac x 2} } {1 + \map {\tan^2} {\frac x 2} - 1 + \map {\tan^2} {\frac x 2} }
... | :$\dfrac 1 {1 - \cos x} = \dfrac 1 2 \map {\csc^2} {\dfrac x 2}$ | {{begin-eqn}}
{{eqn | l = \frac 1 {1 - \cos x}
| r = \frac 1 {1 - \frac {1 - \map {\tan^2} {\frac x 2} } {1 + \map {\tan^2} {\frac x 2} } }
| c = [[Tangent Half-Angle Substitution for Cosine]]
}}
{{eqn | r = \frac {1 + \map {\tan^2} {\frac x 2} } {1 + \map {\tan^2} {\frac x 2} - 1 + \map {\tan^2} {\frac x 2... | Reciprocal of One Minus Cosine/Proof 3 | https://proofwiki.org/wiki/Reciprocal_of_One_Minus_Cosine | https://proofwiki.org/wiki/Reciprocal_of_One_Minus_Cosine/Proof_3 | [
"Reciprocal of One Minus Cosine",
"Trigonometric Identities",
"Cosine Function"
] | [] | [
"Tangent Half-Angle Substitution for Cosine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Squares of Sine and Cosine/Corollary 1"
] |
proofwiki-9936 | Irrationality of Logarithm | Let $a, b \in \N_{>0}$ such that both $\nexists m, n \in \N_{>0}: a^m = b^n$.
Then $\log_b a$ is irrational. | {{AimForCont}} $\log_b a$ is rational.
Then:
:$\exists p, q \in \N_{>0} : \log_b a = \dfrac p q$
where $p \perp q$.
Then:
{{begin-eqn}}
{{eqn | l = \log_b a
| r = \dfrac p q
| c =
}}
{{eqn | ll= \leadsto
| l = b^{\frac p q}
| r = a
| c = {{Defof|Real General Logarithm}}
}}
{{eqn | ll= \le... | Let $a, b \in \N_{>0}$ such that both $\nexists m, n \in \N_{>0}: a^m = b^n$.
Then $\log_b a$ is [[Definition:Irrational Number|irrational]]. | {{AimForCont}} $\log_b a$ is [[Definition:Rational Number|rational]].
Then:
:$\exists p, q \in \N_{>0} : \log_b a = \dfrac p q$
where $p \perp q$.
Then:
{{begin-eqn}}
{{eqn | l = \log_b a
| r = \dfrac p q
| c =
}}
{{eqn | ll= \leadsto
| l = b^{\frac p q}
| r = a
| c = {{Defof|Real G... | Irrationality of Logarithm | https://proofwiki.org/wiki/Irrationality_of_Logarithm | https://proofwiki.org/wiki/Irrationality_of_Logarithm | [
"Number Theory",
"Irrationality Proofs",
"Logarithms"
] | [
"Definition:Irrational Number"
] | [
"Definition:Rational Number",
"Definition:Contradiction",
"Proof by Contradiction",
"Category:Number Theory",
"Category:Irrationality Proofs",
"Category:Logarithms"
] |
proofwiki-9937 | Tangent of 22.5 Degrees | :$\tan 22.5 \degrees = \tan \dfrac \pi 8 = \sqrt 2 - 1$ | {{begin-eqn}}
{{eqn | l = \tan 22.5 \degrees
| r = \tan \dfrac {45 \degrees} 2
| c =
}}
{{eqn | r = \dfrac {1 - \cos 45\degrees} {\sin 45\degrees}
| c = {{Corollary|Half Angle Formula for Tangent|2}}
}}
{{eqn | r = \dfrac {1 - \frac {\sqrt 2} 2} {\frac {\sqrt 2} 2}
| c = {{cos|45}} and {{sin|45... | :$\tan 22.5 \degrees = \tan \dfrac \pi 8 = \sqrt 2 - 1$ | {{begin-eqn}}
{{eqn | l = \tan 22.5 \degrees
| r = \tan \dfrac {45 \degrees} 2
| c =
}}
{{eqn | r = \dfrac {1 - \cos 45\degrees} {\sin 45\degrees}
| c = {{Corollary|Half Angle Formula for Tangent|2}}
}}
{{eqn | r = \dfrac {1 - \frac {\sqrt 2} 2} {\frac {\sqrt 2} 2}
| c = {{cos|45}} and {{sin|45... | Tangent of 22.5 Degrees/Proof 1 | https://proofwiki.org/wiki/Tangent_of_22.5_Degrees | https://proofwiki.org/wiki/Tangent_of_22.5_Degrees/Proof_1 | [
"Tangent of 22.5 Degrees",
"Tangent Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-9938 | Tangent of 22.5 Degrees | :$\tan 22.5 \degrees = \tan \dfrac \pi 8 = \sqrt 2 - 1$ | {{begin-eqn}}
{{eqn | l = \tan 22.5 \degrees
| r = \dfrac {\sin 22.5 \degrees} {\cos 22.5 \degrees}
| c =
}}
{{eqn | r = \dfrac {\dfrac 1 2 \sqrt {2 - \sqrt 2} } {\dfrac 1 2 \sqrt {2 + \sqrt 2} }
| c = {{sin|22.5}}, {{cos|22.5}}
}}
{{eqn | r = \dfrac {\sqrt {2 - \sqrt 2} \sqrt {2 - \sqrt 2} } {\sqrt... | :$\tan 22.5 \degrees = \tan \dfrac \pi 8 = \sqrt 2 - 1$ | {{begin-eqn}}
{{eqn | l = \tan 22.5 \degrees
| r = \dfrac {\sin 22.5 \degrees} {\cos 22.5 \degrees}
| c =
}}
{{eqn | r = \dfrac {\dfrac 1 2 \sqrt {2 - \sqrt 2} } {\dfrac 1 2 \sqrt {2 + \sqrt 2} }
| c = {{sin|22.5}}, {{cos|22.5}}
}}
{{eqn | r = \dfrac {\sqrt {2 - \sqrt 2} \sqrt {2 - \sqrt 2} } {\sqrt... | Tangent of 22.5 Degrees/Proof 2 | https://proofwiki.org/wiki/Tangent_of_22.5_Degrees | https://proofwiki.org/wiki/Tangent_of_22.5_Degrees/Proof_2 | [
"Tangent of 22.5 Degrees",
"Tangent Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares"
] |
proofwiki-9939 | Tangent of 67.5 Degrees | :$\tan 67.5 \degrees = \tan \dfrac {3 \pi} 8 = \sqrt 2 + 1$ | {{begin-eqn}}
{{eqn | l = \tan 67.5 \degrees
| r = \map \tan {45 \degrees + 22.5 \degrees}
| c =
}}
{{eqn | r = \frac {\tan 45 \degrees + \tan 22.5 \degrees} {1 - \tan 45 \degrees \tan 22.5 \degrees}
| c = Tangent of Sum
}}
{{eqn | r = \frac {1 + \paren {\sqrt 2 - 1} } {1 - 1 \times \paren {\sqrt 2 -... | :$\tan 67.5 \degrees = \tan \dfrac {3 \pi} 8 = \sqrt 2 + 1$ | {{begin-eqn}}
{{eqn | l = \tan 67.5 \degrees
| r = \map \tan {45 \degrees + 22.5 \degrees}
| c =
}}
{{eqn | r = \frac {\tan 45 \degrees + \tan 22.5 \degrees} {1 - \tan 45 \degrees \tan 22.5 \degrees}
| c = [[Tangent of Sum]]
}}
{{eqn | r = \frac {1 + \paren {\sqrt 2 - 1} } {1 - 1 \times \paren {\sqrt... | Tangent of 67.5 Degrees | https://proofwiki.org/wiki/Tangent_of_67.5_Degrees | https://proofwiki.org/wiki/Tangent_of_67.5_Degrees | [
"Tangent Function"
] | [] | [
"Tangent of Sum",
"Tangent of 45 Degrees",
"Tangent of 22.5 Degrees",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares",
"Category:Tangent Function"
] |
proofwiki-9940 | Sine of x plus Cosine of x/Cosine Form | :$\sin x + \cos x = \sqrt 2 \, \map \cos {x - \dfrac \pi 4}$ | {{begin-eqn}}
{{eqn | l = \sin x + \cos x
| r = \sin x + \map \sin {\frac \pi 2 - x}
| c = Sine of Complement equals Cosine
}}
{{eqn | r = 2 \, \map \sin {\frac {x + \paren {\frac \pi 2 - x} } 2} \map \cos {\frac {x - \paren {\frac \pi 2 - x} } 2}
| c = Sine plus Sine
}}
{{eqn | r = 2 \sin \frac \pi 4... | :$\sin x + \cos x = \sqrt 2 \, \map \cos {x - \dfrac \pi 4}$ | {{begin-eqn}}
{{eqn | l = \sin x + \cos x
| r = \sin x + \map \sin {\frac \pi 2 - x}
| c = [[Sine of Complement equals Cosine]]
}}
{{eqn | r = 2 \, \map \sin {\frac {x + \paren {\frac \pi 2 - x} } 2} \map \cos {\frac {x - \paren {\frac \pi 2 - x} } 2}
| c = [[Sine plus Sine]]
}}
{{eqn | r = 2 \sin \fr... | Sine of x plus Cosine of x/Cosine Form | https://proofwiki.org/wiki/Sine_of_x_plus_Cosine_of_x/Cosine_Form | https://proofwiki.org/wiki/Sine_of_x_plus_Cosine_of_x/Cosine_Form | [
"Sine Function",
"Cosine Function"
] | [] | [
"Sine of Complement equals Cosine",
"Prosthaphaeresis Formulas/Sine plus Sine",
"Sine of 45 Degrees",
"Category:Sine Function",
"Category:Cosine Function"
] |
proofwiki-9941 | Sine of x minus Cosine of x/Sine Form | :$\sin x - \cos x = \sqrt 2 \map \sin {x - \dfrac \pi 4}$ | {{begin-eqn}}
{{eqn | l = \sin x - \cos x
| r = \sin x - \map \sin {\frac \pi 2 - x}
| c = Sine of Complement equals Cosine
}}
{{eqn | r = 2 \map \cos {\frac {x + \paren {\frac \pi 2 - x} } 2} \map \sin {\frac {x - \paren {\frac \pi 2 - x} } 2}
| c = Sine minus Sine
}}
{{eqn | r = 2 \cos \frac \pi 4 \... | :$\sin x - \cos x = \sqrt 2 \map \sin {x - \dfrac \pi 4}$ | {{begin-eqn}}
{{eqn | l = \sin x - \cos x
| r = \sin x - \map \sin {\frac \pi 2 - x}
| c = [[Sine of Complement equals Cosine]]
}}
{{eqn | r = 2 \map \cos {\frac {x + \paren {\frac \pi 2 - x} } 2} \map \sin {\frac {x - \paren {\frac \pi 2 - x} } 2}
| c = [[Sine minus Sine]]
}}
{{eqn | r = 2 \cos \frac... | Sine of x minus Cosine of x/Sine Form | https://proofwiki.org/wiki/Sine_of_x_minus_Cosine_of_x/Sine_Form | https://proofwiki.org/wiki/Sine_of_x_minus_Cosine_of_x/Sine_Form | [
"Sine Function",
"Cosine Function"
] | [] | [
"Sine of Complement equals Cosine",
"Prosthaphaeresis Formulas/Sine minus Sine",
"Cosine of 45 Degrees",
"Category:Sine Function",
"Category:Cosine Function"
] |
proofwiki-9942 | Sine of x minus Cosine of x/Cosine Form | :$\sin x - \cos x = \sqrt 2 \, \map \cos {x - \dfrac {3 \pi} 4}$ | {{begin-eqn}}
{{eqn | l = \sin x - \cos x
| r = \sqrt 2 \, \map \sin {x - \dfrac \pi 4}
| c = Sine of x minus Cosine of x: Sine Form
}}
{{eqn | r = \sqrt 2 \, \map \cos {\frac \pi 2 - \paren {x - \dfrac \pi 4} }
| c = Cosine of Complement equals Sine
}}
{{eqn | r = \sqrt 2 \, \map \cos {\frac \pi 2 - ... | :$\sin x - \cos x = \sqrt 2 \, \map \cos {x - \dfrac {3 \pi} 4}$ | {{begin-eqn}}
{{eqn | l = \sin x - \cos x
| r = \sqrt 2 \, \map \sin {x - \dfrac \pi 4}
| c = [[Sine of x minus Cosine of x/Sine Form|Sine of x minus Cosine of x: Sine Form]]
}}
{{eqn | r = \sqrt 2 \, \map \cos {\frac \pi 2 - \paren {x - \dfrac \pi 4} }
| c = [[Cosine of Complement equals Sine]]
}}
{{... | Sine of x minus Cosine of x/Cosine Form | https://proofwiki.org/wiki/Sine_of_x_minus_Cosine_of_x/Cosine_Form | https://proofwiki.org/wiki/Sine_of_x_minus_Cosine_of_x/Cosine_Form | [
"Sine Function",
"Cosine Function"
] | [] | [
"Sine of x minus Cosine of x/Sine Form",
"Cosine of Complement equals Sine",
"Cosine Function is Even",
"Category:Sine Function",
"Category:Cosine Function"
] |
proofwiki-9943 | Multiple of Sine plus Multiple of Cosine/Cosine Form | :$p \sin x + q \cos x = \sqrt {p^2 + q^2} \map \cos {x + \arctan \dfrac {-p} q}$ | Let it be assumed that $p \sin x + q \cos x$ can be expressed in the form $M \map \cos {x + \phi}$.
Then:
{{begin-eqn}}
{{eqn | l = p \sin x + q \cos x
| r = M \map \cos {x + \phi}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac p M \sin x + \frac q M \cos x
| r = \map \cos {x + \phi}
| c =
}}... | :$p \sin x + q \cos x = \sqrt {p^2 + q^2} \map \cos {x + \arctan \dfrac {-p} q}$ | Let it be assumed that $p \sin x + q \cos x$ can be expressed in the form $M \map \cos {x + \phi}$.
Then:
{{begin-eqn}}
{{eqn | l = p \sin x + q \cos x
| r = M \map \cos {x + \phi}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac p M \sin x + \frac q M \cos x
| r = \map \cos {x + \phi}
| c = ... | Multiple of Sine plus Multiple of Cosine/Cosine Form | https://proofwiki.org/wiki/Multiple_of_Sine_plus_Multiple_of_Cosine/Cosine_Form | https://proofwiki.org/wiki/Multiple_of_Sine_plus_Multiple_of_Cosine/Cosine_Form | [
"Multiple of Sine plus Multiple of Cosine"
] | [] | [
"Cosine of Sum",
"Tangent is Sine divided by Cosine",
"Category:Multiple of Sine plus Multiple of Cosine"
] |
proofwiki-9944 | Multiple of Sine plus Multiple of Cosine/Sine Form | :$p \sin x + q \cos x = \sqrt {p^2 + q^2} \map \sin {x + \arctan \dfrac q p}$ | Let it be assumed that $p \sin x + q \cos x$ can be expressed in the form $M \map \sin {x + \phi}$.
Then:
{{begin-eqn}}
{{eqn | l = p \sin x + q \cos x
| r = M \map \sin {x + \phi}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac p M \sin x + \frac q M \cos x
| r = \map \sin {x + \phi}
| c =
}}... | :$p \sin x + q \cos x = \sqrt {p^2 + q^2} \map \sin {x + \arctan \dfrac q p}$ | Let it be assumed that $p \sin x + q \cos x$ can be expressed in the form $M \map \sin {x + \phi}$.
Then:
{{begin-eqn}}
{{eqn | l = p \sin x + q \cos x
| r = M \map \sin {x + \phi}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac p M \sin x + \frac q M \cos x
| r = \map \sin {x + \phi}
| c = ... | Multiple of Sine plus Multiple of Cosine/Sine Form | https://proofwiki.org/wiki/Multiple_of_Sine_plus_Multiple_of_Cosine/Sine_Form | https://proofwiki.org/wiki/Multiple_of_Sine_plus_Multiple_of_Cosine/Sine_Form | [
"Multiple of Sine plus Multiple of Cosine"
] | [] | [
"Sine of Sum",
"Tangent is Sine divided by Cosine",
"Category:Multiple of Sine plus Multiple of Cosine"
] |
proofwiki-9945 | Primitive of Cosine of a x over Sine of a x plus phi | :$\ds \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} } = \frac {\ln \size {\map \sin {a x + \phi} } } {a \cos \phi} + \tan \phi \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} } + C$ | First note that:
{{begin-eqn}}
{{eqn | n = 1
| l = \map {\frac \d {\d x} } {\map \sin {a x + \phi} }
| r = a \map \cos {a x + \phi}
| c = Derivative of $\sin a x$ etc.
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} }
| r = \int \frac {\cos a x ... | :$\ds \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} } = \frac {\ln \size {\map \sin {a x + \phi} } } {a \cos \phi} + \tan \phi \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} } + C$ | First note that:
{{begin-eqn}}
{{eqn | n = 1
| l = \map {\frac \d {\d x} } {\map \sin {a x + \phi} }
| r = a \map \cos {a x + \phi}
| c = [[Derivative of Sine of a x|Derivative of $\sin a x$]] etc.
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} }
... | Primitive of Cosine of a x over Sine of a x plus phi | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_Sine_of_a_x_plus_phi | https://proofwiki.org/wiki/Primitive_of_Cosine_of_a_x_over_Sine_of_a_x_plus_phi | [
"Primitives involving Sine Function",
"Primitives involving Cosine Function"
] | [] | [
"Derivative of Sine Function/Corollary",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Cosine of Sum",
"Linear Combination of Integrals/Indefinite",
"Primitive of Function under its Derivative",
"Tangent is Sine divided by Cosine",
"Category:Primitives involving Sine Function",
... |
proofwiki-9946 | Primitive of Sine of a x over Sine of a x plus phi | :$\ds \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} } = \frac x {\cos \phi} - \tan \phi \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} }
| r = \frac 1 {\cos \phi} \int \frac {\sin a x \cos \phi \rd x} {\map \sin {a x + \phi} }
| c = multiplying top and bottom by $\cos \phi$
}}
{{eqn | r = \frac 1 {\cos \phi} \int \frac {\paren {\sin a x \cos \phi + \cos a x \sin ... | :$\ds \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} } = \frac x {\cos \phi} - \tan \phi \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} }
| r = \frac 1 {\cos \phi} \int \frac {\sin a x \cos \phi \rd x} {\map \sin {a x + \phi} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $\cos \phi$
}}
{{eqn | r = \frac 1 {\cos \phi} \in... | Primitive of Sine of a x over Sine of a x plus phi | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_Sine_of_a_x_plus_phi | https://proofwiki.org/wiki/Primitive_of_Sine_of_a_x_over_Sine_of_a_x_plus_phi | [
"Primitives involving Sine Function",
"Primitives involving Cosine Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sine of Sum",
"Sine of Sum",
"Linear Combination of Integrals/Indefinite",
"Tangent is Sine divided by Cosine",
"Category:Primitives involving Sine Function",
"Category:Primitives involving Cosine Function"
] |
proofwiki-9947 | Reciprocal of Hyperbolic Cosine Plus One | :$\dfrac 1 {\cosh x + 1} = \dfrac 1 2 \sech^2 \dfrac x 2$ | {{begin-eqn}}
{{eqn | l = \cosh x
| r = 2 \cosh^2 \frac x 2 - 1
| c = {{Corollary|Double Angle Formula for Hyperbolic Cosine|1}}
}}
{{eqn | ll= \leadstoandfrom
| l = \cosh x + 1
| r = 2 \cosh^2 \frac x 2
| c = adding $1$ to both sides
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 {\co... | :$\dfrac 1 {\cosh x + 1} = \dfrac 1 2 \sech^2 \dfrac x 2$ | {{begin-eqn}}
{{eqn | l = \cosh x
| r = 2 \cosh^2 \frac x 2 - 1
| c = {{Corollary|Double Angle Formula for Hyperbolic Cosine|1}}
}}
{{eqn | ll= \leadstoandfrom
| l = \cosh x + 1
| r = 2 \cosh^2 \frac x 2
| c = adding $1$ to both sides
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 {\co... | Reciprocal of Hyperbolic Cosine Plus One | https://proofwiki.org/wiki/Reciprocal_of_Hyperbolic_Cosine_Plus_One | https://proofwiki.org/wiki/Reciprocal_of_Hyperbolic_Cosine_Plus_One | [
"Hyperbolic Cosine Function"
] | [] | [
"Definition:Reciprocal"
] |
proofwiki-9948 | Reciprocal of Hyperbolic Cosine Minus One | :$\dfrac 1 {\cosh x - 1} = \dfrac 1 2 \csch^2 \dfrac x 2$ | {{begin-eqn}}
{{eqn | l = \cosh x
| r = 1 + 2 \sinh^2 \frac x 2
| c = {{Corollary|Double Angle Formula for Hyperbolic Cosine|2}}
}}
{{eqn | ll= \leadstoandfrom
| l = \cosh x - 1
| r = 2 \sinh^2 \frac x 2
| c = subtracting $1$ from both sides
}}
{{eqn | ll= \leadstoandfrom
| l = \frac... | :$\dfrac 1 {\cosh x - 1} = \dfrac 1 2 \csch^2 \dfrac x 2$ | {{begin-eqn}}
{{eqn | l = \cosh x
| r = 1 + 2 \sinh^2 \frac x 2
| c = {{Corollary|Double Angle Formula for Hyperbolic Cosine|2}}
}}
{{eqn | ll= \leadstoandfrom
| l = \cosh x - 1
| r = 2 \sinh^2 \frac x 2
| c = subtracting $1$ from both sides
}}
{{eqn | ll= \leadstoandfrom
| l = \frac... | Reciprocal of Hyperbolic Cosine Minus One | https://proofwiki.org/wiki/Reciprocal_of_Hyperbolic_Cosine_Minus_One | https://proofwiki.org/wiki/Reciprocal_of_Hyperbolic_Cosine_Minus_One | [
"Hyperbolic Cosine Function"
] | [] | [
"Definition:Reciprocal"
] |
proofwiki-9949 | Steiner-Lehmus Theorem | Let $ABC$ be a triangle.
Denote the lengths of the angle bisectors through the vertices $A$ and $B$ by $\omega_\alpha$ and $\omega_\beta$.
Let $\omega_\alpha = \omega_\beta$.
Then $ABC$ is an isosceles triangle.
:250px | Let $a$, $b$, and $c$ be the sides opposite $A$, $B$ and $C$ respectively.
By Length of Angle Bisector, $\omega_\alpha, \omega_\beta$ are given by:
:$\omega_\alpha^2 = \dfrac {b c} {\paren {b + c}^2} \paren {\paren {b + c}^2 - a^2}$
:$\omega_\beta^2 = \dfrac {a c} {\paren {a + c}^2} \paren {\paren {a + c}^2 - b^2}$
Equ... | Let $ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Denote the lengths of the [[Definition:Angle Bisector|angle bisectors]] through the [[Definition:Vertex of Polygon|vertices]] $A$ and $B$ by $\omega_\alpha$ and $\omega_\beta$.
Let $\omega_\alpha = \omega_\beta$.
Then $ABC$ is an [[Definition:Isosceles Tri... | Let $a$, $b$, and $c$ be the [[Definition:Opposite (in Triangle)|sides opposite]] $A$, $B$ and $C$ respectively.
By [[Length of Angle Bisector]], $\omega_\alpha, \omega_\beta$ are given by:
:$\omega_\alpha^2 = \dfrac {b c} {\paren {b + c}^2} \paren {\paren {b + c}^2 - a^2}$
:$\omega_\beta^2 = \dfrac {a c} {\paren {a... | Steiner-Lehmus Theorem/Proof 1 | https://proofwiki.org/wiki/Steiner-Lehmus_Theorem | https://proofwiki.org/wiki/Steiner-Lehmus_Theorem/Proof_1 | [
"Triangles",
"Steiner-Lehmus Theorem"
] | [
"Definition:Triangle (Geometry)",
"Definition:Angle Bisector",
"Definition:Polygon/Vertex",
"Definition:Triangle (Geometry)/Isosceles",
"File:Steiner-Lehmus.png"
] | [
"Definition:Triangle (Geometry)/Opposite",
"Length of Angle Bisector",
"Definition:Triangle (Geometry)/Isosceles"
] |
proofwiki-9950 | Steiner-Lehmus Theorem | Let $ABC$ be a triangle.
Denote the lengths of the angle bisectors through the vertices $A$ and $B$ by $\omega_\alpha$ and $\omega_\beta$.
Let $\omega_\alpha = \omega_\beta$.
Then $ABC$ is an isosceles triangle.
:250px | Let $a$, $b$, and $c$ be the sides opposite $A$, $B$ and $C$ respectively.
By Length of Angle Bisector, $\omega_\alpha, \omega_\beta$ are given by:
{{begin-eqn}}
{{eqn | l = \omega_\alpha^2
| r = b c \paren {1 - \dfrac {a^2} {\paren {b + c}^2} }
}}
{{eqn | l = \omega_\beta^2
| r = a c \paren {1 - \dfrac {b^... | Let $ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Denote the lengths of the [[Definition:Angle Bisector|angle bisectors]] through the [[Definition:Vertex of Polygon|vertices]] $A$ and $B$ by $\omega_\alpha$ and $\omega_\beta$.
Let $\omega_\alpha = \omega_\beta$.
Then $ABC$ is an [[Definition:Isosceles Tri... | Let $a$, $b$, and $c$ be the [[Definition:Opposite (in Triangle)|sides opposite]] $A$, $B$ and $C$ respectively.
By [[Length of Angle Bisector]], $\omega_\alpha, \omega_\beta$ are given by:
{{begin-eqn}}
{{eqn | l = \omega_\alpha^2
| r = b c \paren {1 - \dfrac {a^2} {\paren {b + c}^2} }
}}
{{eqn | l = \omega_\b... | Steiner-Lehmus Theorem/Proof 2 | https://proofwiki.org/wiki/Steiner-Lehmus_Theorem | https://proofwiki.org/wiki/Steiner-Lehmus_Theorem/Proof_2 | [
"Triangles",
"Steiner-Lehmus Theorem"
] | [
"Definition:Triangle (Geometry)",
"Definition:Angle Bisector",
"Definition:Polygon/Vertex",
"Definition:Triangle (Geometry)/Isosceles",
"File:Steiner-Lehmus.png"
] | [
"Definition:Triangle (Geometry)/Opposite",
"Length of Angle Bisector",
"Definition:Positive/Real Number",
"Definition:Strictly Positive",
"Definition:Real Number",
"Real Number Ordering is Compatible with Multiplication",
"Transitive Law",
"Definition:Negative",
"Definition:Parenthesis",
"Definiti... |
proofwiki-9951 | Steiner-Lehmus Theorem | Let $ABC$ be a triangle.
Denote the lengths of the angle bisectors through the vertices $A$ and $B$ by $\omega_\alpha$ and $\omega_\beta$.
Let $\omega_\alpha = \omega_\beta$.
Then $ABC$ is an isosceles triangle.
:250px | :300px
Draw $DF \parallel BE$ and $EF \parallel BD$.
By Quadrilateral is Parallelogram iff Both Pairs of Opposite Sides are Equal or Parallel:
:$\Box BEFD$ is a parallelogram.
:$\leadsto \angle DFE = \beta$
Draw $FA$.
Let $\angle EFA = \gamma$.
Let $\angle EAF = \delta$.
{{AimForCont}} $\alpha > \beta$.
Compare $\trian... | Let $ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Denote the lengths of the [[Definition:Angle Bisector|angle bisectors]] through the [[Definition:Vertex of Polygon|vertices]] $A$ and $B$ by $\omega_\alpha$ and $\omega_\beta$.
Let $\omega_\alpha = \omega_\beta$.
Then $ABC$ is an [[Definition:Isosceles Tri... | :[[File:Steiner-Lehmus Theorem.png|300px]]
Draw $DF \parallel BE$ and $EF \parallel BD$.
By [[Quadrilateral is Parallelogram iff Both Pairs of Opposite Sides are Equal or Parallel]]:
:$\Box BEFD$ is a [[Definition:Parallelogram|parallelogram]].
:$\leadsto \angle DFE = \beta$
Draw $FA$.
Let $\angle EFA = \gamma$.
L... | Steiner-Lehmus Theorem/Proof 3 | https://proofwiki.org/wiki/Steiner-Lehmus_Theorem | https://proofwiki.org/wiki/Steiner-Lehmus_Theorem/Proof_3 | [
"Triangles",
"Steiner-Lehmus Theorem"
] | [
"Definition:Triangle (Geometry)",
"Definition:Angle Bisector",
"Definition:Polygon/Vertex",
"Definition:Triangle (Geometry)/Isosceles",
"File:Steiner-Lehmus.png"
] | [
"File:Steiner-Lehmus Theorem.png",
"Quadrilateral is Parallelogram iff Both Pairs of Opposite Sides are Equal or Parallel",
"Definition:Quadrilateral/Parallelogram",
"Definition:Triangle (Geometry)/Base",
"Greater Angle of Triangle Subtended by Greater Side",
"Greater Side of Triangle Subtends Greater Ang... |
proofwiki-9952 | Steiner-Lehmus Theorem | Let $ABC$ be a triangle.
Denote the lengths of the angle bisectors through the vertices $A$ and $B$ by $\omega_\alpha$ and $\omega_\beta$.
Let $\omega_\alpha = \omega_\beta$.
Then $ABC$ is an isosceles triangle.
:250px | === {{Lemma|Steiner-Lehmus Theorem|1}} ===
{{:Steiner-Lehmus Theorem/Lemma 1}}{{qed|lemma}}
=== {{Lemma|Steiner-Lehmus Theorem|2}} ===
{{:Steiner-Lehmus Theorem/Lemma 2}}{{qed|lemma}}
400px
Let $\triangle ABC$ be a triangle.
Let $\angle ABC$ be bisected by $BM$
Let $\angle ACB$ be bisected by $CN$.
Let $BM = CN$.
Suppo... | Let $ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Denote the lengths of the [[Definition:Angle Bisector|angle bisectors]] through the [[Definition:Vertex of Polygon|vertices]] $A$ and $B$ by $\omega_\alpha$ and $\omega_\beta$.
Let $\omega_\alpha = \omega_\beta$.
Then $ABC$ is an [[Definition:Isosceles Tri... | === {{Lemma|Steiner-Lehmus Theorem|1}} ===
{{:Steiner-Lehmus Theorem/Lemma 1}}{{qed|lemma}}
=== {{Lemma|Steiner-Lehmus Theorem|2}} ===
{{:Steiner-Lehmus Theorem/Lemma 2}}{{qed|lemma}}
[[File:Steiner-Lehmus Proof 4.png|400px]]
Let $\triangle ABC$ be a [[Definition:Triangle|triangle]].
Let $\angle ABC$ be [[Definit... | Steiner-Lehmus Theorem/Proof 4 | https://proofwiki.org/wiki/Steiner-Lehmus_Theorem | https://proofwiki.org/wiki/Steiner-Lehmus_Theorem/Proof_4 | [
"Triangles",
"Steiner-Lehmus Theorem"
] | [
"Definition:Triangle (Geometry)",
"Definition:Angle Bisector",
"Definition:Polygon/Vertex",
"Definition:Triangle (Geometry)/Isosceles",
"File:Steiner-Lehmus.png"
] | [
"File:Steiner-Lehmus Proof 4.png",
"Definition:Triangle",
"Definition:Bisection",
"Definition:Bisection",
"Definition:Contradiction",
"Definition:Contradiction",
"Triangle with Two Equal Angles is Isosceles",
"Definition:Triangle (Geometry)/Isosceles"
] |
proofwiki-9953 | Triangle Right-Angle-Hypotenuse-Side Congruence | If two right triangles have:
:their hypotenuses equal
:another of their respective sides equal
they will also have:
:their third sides equal
:the remaining two angles equal to their respective remaining angles. | Let $\triangle ABC$ and $\triangle DEF$ be two triangles having sides $AB = DE$ and $AC = DF$, and with $\angle ABC = \angle DEF = 90^\circ$.
By Pythagoras' Theorem:
:$BC = \sqrt {AB^2 + AC^2}$
and:
:$EF = \sqrt {DE^2 + DF^2}$
:$\therefore BC = \sqrt {AB^2 + AC^2} = \sqrt {DE^2 + DF^2} = EF$
The part that the remaining... | If two [[Definition:Right Triangle|right triangles]] have:
:their [[Definition:Hypotenuse|hypotenuses]] equal
:another of their respective [[Definition:Side of Polygon|sides]] equal
they will also have:
:their third [[Definition:Side of Polygon|sides]] equal
:the remaining two [[Definition:Internal Angle|angles]] equa... | Let $\triangle ABC$ and $\triangle DEF$ be two [[Definition:Triangle (Geometry)|triangles]] having sides $AB = DE$ and $AC = DF$, and with $\angle ABC = \angle DEF = 90^\circ$.
By [[Pythagoras' Theorem]]:
:$BC = \sqrt {AB^2 + AC^2}$
and:
:$EF = \sqrt {DE^2 + DF^2}$
:$\therefore BC = \sqrt {AB^2 + AC^2} = \sqrt {DE^2 ... | Triangle Right-Angle-Hypotenuse-Side Congruence/Proof 1 | https://proofwiki.org/wiki/Triangle_Right-Angle-Hypotenuse-Side_Congruence | https://proofwiki.org/wiki/Triangle_Right-Angle-Hypotenuse-Side_Congruence/Proof_1 | [
"Triangle Right-Angle-Hypotenuse-Side Congruence",
"Right Triangles",
"Congruence (Geometry)"
] | [
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Polygon/Internal Angle",
"Definition:Polygon/Internal Angle"
] | [
"Definition:Triangle (Geometry)",
"Pythagoras's Theorem",
"Definition:Polygon/Internal Angle",
"Definition:Polygon/Internal Angle",
"Triangle Side-Side-Side Congruence"
] |
proofwiki-9954 | Triangle Right-Angle-Hypotenuse-Side Congruence | If two right triangles have:
:their hypotenuses equal
:another of their respective sides equal
they will also have:
:their third sides equal
:the remaining two angles equal to their respective remaining angles. | 350px
Let $\triangle ADB$ and $\triangle ADC$ both be right triangles.
Let them have equal hypotenuse and one leg ($AD$) equal.
{{hypothesis}}:
:$AB = AC$
:$\angle ADB = \angle ADC = \ $ one right angle
:$AD$ is shared
So the two triangles can be drawn as shown with $BD$ and $DC$ joined at $D$.
By addition:
:$\angle BD... | If two [[Definition:Right Triangle|right triangles]] have:
:their [[Definition:Hypotenuse|hypotenuses]] equal
:another of their respective [[Definition:Side of Polygon|sides]] equal
they will also have:
:their third [[Definition:Side of Polygon|sides]] equal
:the remaining two [[Definition:Internal Angle|angles]] equa... | [[File:HL.png|350px]]
Let $\triangle ADB$ and $\triangle ADC$ both be [[Definition:Right Triangle|right triangles]].
Let them have equal [[Definition:Hypotenuse|hypotenuse]] and one [[Definition:Leg of Right Triangle|leg]] ($AD$) equal.
{{hypothesis}}:
:$AB = AC$
:$\angle ADB = \angle ADC = \ $ one [[Definition:Rig... | Triangle Right-Angle-Hypotenuse-Side Congruence/Proof 2 | https://proofwiki.org/wiki/Triangle_Right-Angle-Hypotenuse-Side_Congruence | https://proofwiki.org/wiki/Triangle_Right-Angle-Hypotenuse-Side_Congruence/Proof_2 | [
"Triangle Right-Angle-Hypotenuse-Side Congruence",
"Right Triangles",
"Congruence (Geometry)"
] | [
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Polygon/Internal Angle",
"Definition:Polygon/Internal Angle"
] | [
"File:HL.png",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Triangle (Geometry)/Right-Angled/Legs",
"Definition:Right Angle",
"Definition:Triangle (Geometry)",
"Definition:Right Angle",
"Two Angles making Two Right Angles make Str... |
proofwiki-9955 | Internal Angles of Regular Polygon | The size $A$ of each internal angle of a regular $n$-gon is given by:
:$A = \dfrac {\paren {n - 2} 180 \degrees} n$ | From Sum of Internal Angles of Polygon, we have that the sum $S$ of all internal angles of a $n$-gon is:
:$S = \paren {n - 2} 180 \degrees$
From the definition of a regular polygon, all the internal angles of a regular polygon are equal.
Therefore, the size $A$ of each internal angle of a regular polygon with $n$ sides... | The size $A$ of each [[Definition:Internal Angle|internal angle]] of a [[Definition:Regular Polygon|regular $n$-gon]] is given by:
:$A = \dfrac {\paren {n - 2} 180 \degrees} n$ | From [[Sum of Internal Angles of Polygon]], we have that the sum $S$ of all [[Definition:Internal Angle|internal angles]] of a [[Definition:Polygon|$n$-gon]] is:
:$S = \paren {n - 2} 180 \degrees$
From the definition of a [[Definition:Regular Polygon|regular polygon]], all the [[Definition:Internal Angle|internal angl... | Internal Angles of Regular Polygon | https://proofwiki.org/wiki/Internal_Angles_of_Regular_Polygon | https://proofwiki.org/wiki/Internal_Angles_of_Regular_Polygon | [
"Regular Polygons"
] | [
"Definition:Polygon/Internal Angle",
"Definition:Polygon/Regular"
] | [
"Sum of Internal Angles of Polygon",
"Definition:Polygon/Internal Angle",
"Definition:Polygon",
"Definition:Polygon/Regular",
"Definition:Polygon/Internal Angle",
"Definition:Polygon/Regular",
"Definition:Polygon/Internal Angle",
"Definition:Polygon/Regular",
"Definition:Polygon/Side"
] |
proofwiki-9956 | Sum of Internal Angles of Polygon | The sum $S$ of all internal angles of a polygon with $n$ sides is given by the formula $S = \paren {n - 2} 180 \degrees$. | The Polygon Triangulation Theorem shows that there exists a triangulation of the polygon that consists of $n - 2$ triangles.
The sides of these triangles are sides and chords of the polygon, where the chords lie completely in the interior of $P$.
Hence the vertices of the triangles are vertices of the polygon.
Sum of A... | The sum $S$ of all [[Definition:Internal Angle|internal angles]] of a [[Definition:Polygon|polygon]] with $n$ sides is given by the formula $S = \paren {n - 2} 180 \degrees$. | The [[Polygon Triangulation Theorem]] shows that there exists a [[Definition:Triangulation of Polygon|triangulation]] of the [[Definition:Polygon|polygon]] that consists of $n - 2$ [[Definition:Triangle (Geometry)|triangles]].
The [[Definition:Side of Polygon|sides]] of these [[Definition:Triangle (Geometry)|triangles... | Sum of Internal Angles of Polygon/Proof 1 | https://proofwiki.org/wiki/Sum_of_Internal_Angles_of_Polygon | https://proofwiki.org/wiki/Sum_of_Internal_Angles_of_Polygon/Proof_1 | [
"Sum of Internal Angles of Polygon",
"Polygons"
] | [
"Definition:Polygon/Internal Angle",
"Definition:Polygon"
] | [
"Polygon Triangulation Theorem",
"Definition:Triangulation of Polygon",
"Definition:Polygon",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Polygon/Chord",
"Definition:Polygon",
"Definition:Polygon/Chord",
"D... |
proofwiki-9957 | Sum of Internal Angles of Polygon | The sum $S$ of all internal angles of a polygon with $n$ sides is given by the formula $S = \paren {n - 2} 180 \degrees$. | This proof assumes that the polygon is convex.
Name a vertex as $A_1$, go clockwise and name the vertices as $A_2, A_3, \ldots, A_n$.
By joining $A_1$ to every vertex except $A_2$ and $A_n$, one can form $\paren {n - 2}$ triangles in a fan triangulation of the convex polygon.
From Sum of Angles of Triangle equals Two R... | The sum $S$ of all [[Definition:Internal Angle|internal angles]] of a [[Definition:Polygon|polygon]] with $n$ sides is given by the formula $S = \paren {n - 2} 180 \degrees$. | This proof assumes that the [[Definition:Polygon|polygon]] is [[Definition:Convex Polygon|convex]].
Name a [[Definition:Vertex of Polygon|vertex]] as $A_1$, go [[Definition:Clockwise|clockwise]] and name the [[Definition:Vertex of Polygon|vertices]] as $A_2, A_3, \ldots, A_n$.
By joining $A_1$ to every [[Definition:V... | Sum of Internal Angles of Polygon/Proof 2 | https://proofwiki.org/wiki/Sum_of_Internal_Angles_of_Polygon | https://proofwiki.org/wiki/Sum_of_Internal_Angles_of_Polygon/Proof_2 | [
"Sum of Internal Angles of Polygon",
"Polygons"
] | [
"Definition:Polygon/Internal Angle",
"Definition:Polygon"
] | [
"Definition:Polygon",
"Definition:Convex Polygon",
"Definition:Polygon/Vertex",
"Definition:Clockwise",
"Definition:Polygon/Vertex",
"Definition:Polygon/Vertex",
"Definition:Triangle (Geometry)",
"Definition:Triangulation of Polygon/Fan",
"Definition:Convex Polygon",
"Sum of Angles of Triangle equ... |
proofwiki-9958 | Five Platonic Solids | There exist exactly five platonic solids:
{{begin-itemize}}
{{item|(1):|the regular tetrahedron}}
{{item|(2):|the cube}}
{{item|(3):|the regular octahedron}}
{{item|(4):|the regular dodecahedron}}
{{item|(5):|the regular icosahedron.}}
{{end-itemize}}
{{:Euclid:Proposition/XIII/18/Endnote}} | A solid angle cannot be constructed from only two planes.
Therefore at least three faces need to come together to form a vertex.
Let $P$ be a platonic solid.
Let the polygon which forms each face of $P$ be a equilateral triangles.
We have that:
:each vertex of a regular tetrahedron is composed of $3$ equilateral triang... | There exist exactly five [[Definition:Platonic Solid|platonic solids]]:
{{begin-itemize}}
{{item|(1):|the [[Definition:Regular Tetrahedron|regular tetrahedron]]}}
{{item|(2):|the [[Definition:Cube (Geometry)|cube]]}}
{{item|(3):|the [[Definition:Regular Octahedron|regular octahedron]]}}
{{item|(4):|the [[Definition:Reg... | A [[Definition:Solid Angle|solid angle]] cannot be constructed from only two [[Definition:Plane|planes]].
Therefore at least three [[Definition:Face of Polyhedron|faces]] need to come together to form a [[Definition:Vertex of Polyhedron|vertex]].
Let $P$ be a [[Definition:Platonic Solid|platonic solid]].
Let the [[D... | Five Platonic Solids/Proof 1 | https://proofwiki.org/wiki/Five_Platonic_Solids | https://proofwiki.org/wiki/Five_Platonic_Solids/Proof_1 | [
"Five Platonic Solids",
"Platonic Solids",
"5"
] | [
"Definition:Platonic Solid",
"Definition:Tetrahedron/Regular",
"Definition:Cube/Geometry",
"Definition:Octahedron/Regular",
"Definition:Dodecahedron/Regular",
"Definition:Icosahedron/Regular"
] | [
"Definition:Solid Angle",
"Definition:Plane Surface",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Vertex",
"Definition:Platonic Solid",
"Definition:Polygon",
"Definition:Polyhedron/Face",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Polyhedron/Vertex",
"Definition:Tetrahedron... |
proofwiki-9959 | Five Platonic Solids | There exist exactly five platonic solids:
{{begin-itemize}}
{{item|(1):|the regular tetrahedron}}
{{item|(2):|the cube}}
{{item|(3):|the regular octahedron}}
{{item|(4):|the regular dodecahedron}}
{{item|(5):|the regular icosahedron.}}
{{end-itemize}}
{{:Euclid:Proposition/XIII/18/Endnote}} | Consider a convex regular polyhedron $P$.
Let $m$ be the number of sides of each of the regular polygons that form the faces of $P$.
Let $n$ be the number of those polygons which meet at each vertex of $P$.
From Internal Angles of Regular Polygon, the internal angles of each face of $P$ measure $180^\circ - \dfrac {360... | There exist exactly five [[Definition:Platonic Solid|platonic solids]]:
{{begin-itemize}}
{{item|(1):|the [[Definition:Regular Tetrahedron|regular tetrahedron]]}}
{{item|(2):|the [[Definition:Cube (Geometry)|cube]]}}
{{item|(3):|the [[Definition:Regular Octahedron|regular octahedron]]}}
{{item|(4):|the [[Definition:Reg... | Consider a [[Definition:Convex Polyhedron|convex]] [[Definition:Regular Polyhedron|regular polyhedron]] $P$.
Let $m$ be the number of [[Definition:Side of Polygon|sides]] of each of the [[Definition:Regular Polygon|regular polygons]] that form the [[Definition:Face of Polyhedron|faces]] of $P$.
Let $n$ be the number ... | Five Platonic Solids/Proof 2 | https://proofwiki.org/wiki/Five_Platonic_Solids | https://proofwiki.org/wiki/Five_Platonic_Solids/Proof_2 | [
"Five Platonic Solids",
"Platonic Solids",
"5"
] | [
"Definition:Platonic Solid",
"Definition:Tetrahedron/Regular",
"Definition:Cube/Geometry",
"Definition:Octahedron/Regular",
"Definition:Dodecahedron/Regular",
"Definition:Icosahedron/Regular"
] | [
"Definition:Convex Polyhedron",
"Definition:Regular Polyhedron",
"Definition:Polygon/Side",
"Definition:Polygon/Regular",
"Definition:Polyhedron/Face",
"Definition:Polygon/Regular",
"Definition:Polygon/Vertex",
"Internal Angles of Regular Polygon",
"Definition:Polygon/Internal Angle",
"Definition:... |
proofwiki-9960 | Space in which All Convergent Sequences have Unique Limit not necessarily Hausdorff | Let $T = \struct{S, \tau}$ be a topological space.
Let $T$ be such that all convergent sequences have a unique limit.
Then it is not necessarily the case that $T$ is a Hausdorff space. | Let $T = \struct{\R, \tau}$ be the set of real numbers $\R$ with the countable complement topology.
From Countable Complement Space is not $T_2$, $T$ is not a Hausdorff space.
Suppose $\sequence{x_n}$ is a sequence in $\R$ which converges to $x$.
Then $C = \set{x_n: x_n \ne x}$ is closed in $T$ because it is countable.... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T$ be such that all [[Definition:Convergent Sequence (Topology)|convergent sequences]] have a unique [[Definition:Limit of Sequence (Topology)|limit]].
Then it is not necessarily the case that $T$ is a [[Definition:Hausdorff Spa... | Let $T = \struct{\R, \tau}$ be the [[Definition:Real Number|set of real numbers]] $\R$ with the [[Definition:Countable Complement Topology|countable complement topology]].
From [[Countable Complement Space is not T2|Countable Complement Space is not $T_2$]], $T$ is not a [[Definition:Hausdorff Space|Hausdorff space]].... | Space in which All Convergent Sequences have Unique Limit not necessarily Hausdorff | https://proofwiki.org/wiki/Space_in_which_All_Convergent_Sequences_have_Unique_Limit_not_necessarily_Hausdorff | https://proofwiki.org/wiki/Space_in_which_All_Convergent_Sequences_have_Unique_Limit_not_necessarily_Hausdorff | [
"Countable Complement Topologies",
"Hausdorff Spaces"
] | [
"Definition:Topological Space",
"Definition:Convergent Sequence/Topology",
"Definition:Limit of Sequence/Topological Space",
"Definition:T2 Space"
] | [
"Definition:Real Number",
"Definition:Countable Complement Topology",
"Countable Complement Space is not T2",
"Definition:T2 Space",
"Definition:Sequence",
"Definition:Closed Set/Topology",
"Definition:Countable Set",
"Definition:Neighborhood (Topology)/Point",
"Definition:Limit of Sequence/Topologi... |
proofwiki-9961 | Geometric Sequence with Coprime Extremes is in Lowest Terms | Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be a geometric sequence of integers.
Let:
:$a_0 \perp a_n$
where $\perp$ denotes coprimality.
Then $G_n$ is in its lowest terms. | Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be natural numbers in geometric sequence such that $a_0 \perp a_n$.
{{AimForCont}} there were to exist another set of natural numbers in geometric sequence:
:$G\,'_n = \sequence {b_0, b_1, \cdots, b_n}$
with the same common ratio where:
:$\forall k \in \N_{\le n}: a_k > b_k... | Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
Let:
:$a_0 \perp a_n$
where $\perp$ denotes [[Definition:Coprime Integers|coprimality]].
Then $G_n$ is [[Definition:Geometric Sequence of Integers in Lowest Terms|in its lowest terms]]. | Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be [[Definition:Natural Number|natural numbers]] in [[Definition:Geometric Sequence|geometric sequence]] such that $a_0 \perp a_n$.
{{AimForCont}} there were to exist another set of [[Definition:Natural Number|natural numbers]] in [[Definition:Geometric Sequence|geometric ... | Geometric Sequence with Coprime Extremes is in Lowest Terms/Proof 1 | https://proofwiki.org/wiki/Geometric_Sequence_with_Coprime_Extremes_is_in_Lowest_Terms | https://proofwiki.org/wiki/Geometric_Sequence_with_Coprime_Extremes_is_in_Lowest_Terms/Proof_1 | [
"Geometric Sequences of Integers",
"Geometric Sequence with Coprime Extremes is in Lowest Terms"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Coprime/Integers",
"Definition:Geometric Sequence of Integers in Lowest Terms"
] | [
"Definition:Natural Numbers",
"Definition:Geometric Sequence",
"Definition:Natural Numbers",
"Definition:Geometric Sequence",
"Definition:Geometric Sequence/Common Ratio",
"Definition:By Hypothesis",
"Definition:Geometric Sequence/Common Ratio"
] |
proofwiki-9962 | Geometric Sequence with Coprime Extremes is in Lowest Terms | Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be a geometric sequence of integers.
Let:
:$a_0 \perp a_n$
where $\perp$ denotes coprimality.
Then $G_n$ is in its lowest terms. | Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be natural numbers in geometric sequence such that $a_0 \perp a_n$.
{{AimForCont}} $G\,'_n = \sequence {b_0, b_1, \cdots, b_n}$ be another set of natural numbers in geometric sequence with the same common ratio where:
:$\forall k \in \N_{\le n}: a_k > b_k$
By definition of ... | Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
Let:
:$a_0 \perp a_n$
where $\perp$ denotes [[Definition:Coprime Integers|coprimality]].
Then $G_n$ is [[Definition:Geometric Sequence of Integers in Lowest Terms|in its lowest terms]]. | Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be [[Definition:Natural Number|natural numbers]] in [[Definition:Geometric Sequence|geometric sequence]] such that $a_0 \perp a_n$.
{{AimForCont}} $G\,'_n = \sequence {b_0, b_1, \cdots, b_n}$ be another set of [[Definition:Natural Number|natural numbers]] in [[Definition:G... | Geometric Sequence with Coprime Extremes is in Lowest Terms/Proof 2 | https://proofwiki.org/wiki/Geometric_Sequence_with_Coprime_Extremes_is_in_Lowest_Terms | https://proofwiki.org/wiki/Geometric_Sequence_with_Coprime_Extremes_is_in_Lowest_Terms/Proof_2 | [
"Geometric Sequences of Integers",
"Geometric Sequence with Coprime Extremes is in Lowest Terms"
] | [
"Definition:Geometric Sequence/Integers",
"Definition:Coprime/Integers",
"Definition:Geometric Sequence of Integers in Lowest Terms"
] | [
"Definition:Natural Numbers",
"Definition:Geometric Sequence",
"Definition:Natural Numbers",
"Definition:Geometric Sequence",
"Definition:Geometric Sequence/Common Ratio",
"Definition:Geometric Sequence",
"Definition:By Hypothesis",
"Definition:Rational Number/Canonical Form",
"Canonical Form of Rat... |
proofwiki-9963 | Measurements of Common Angles/Straight Angle | The measurement of a straight angle is $180 \degrees$ or $\pi$ radians. | From Measurement of Full Angle, a full rotation is defined to be $360 \degrees$ or $2 \pi$ radians.
Since lines are straight, it therefore follows that from any point on a line, the angle between one side of the line and the other is one half of a full rotation.
Therefore, the measurement of a straight angle is:
:$\dfr... | The measurement of a [[Definition:Straight Angle|straight angle]] is $180 \degrees$ or $\pi$ [[Definition:Radian|radians]]. | From [[Measurement of Full Angle]], a full rotation is defined to be $360 \degrees$ or $2 \pi$ [[Definition:Radian|radians]].
Since [[Definition:Straight Line|lines]] are straight, it therefore follows that from any [[Definition:Point|point]] on a line, the [[Definition:Angle|angle]] between one side of the line and t... | Measurements of Common Angles/Straight Angle | https://proofwiki.org/wiki/Measurements_of_Common_Angles/Straight_Angle | https://proofwiki.org/wiki/Measurements_of_Common_Angles/Straight_Angle | [
"Straight Angles",
"Measurements of Common Angles"
] | [
"Definition:Straight Angle",
"Definition:Angular Measure/Radian"
] | [
"Measurements of Common Angles/Full Angle",
"Definition:Angular Measure/Radian",
"Definition:Line/Straight Line",
"Definition:Point",
"Definition:Angle",
"Definition:Straight Angle"
] |
proofwiki-9964 | Measurements of Common Angles/Right Angle | The measurement of a '''right angle''' is $\dfrac {180 \degrees} 2 = 90 \degrees$ or $\dfrac \pi 2$. | A right angle is equal to one half of a straight angle.
From Measurement of Straight Angle it follows that the measurement of a right angle is $\dfrac {180 \degrees} 2 = 90 \degrees$ or $\dfrac \pi 2$.
{{qed}} | The measurement of a '''[[Definition:Right Angle|right angle]]''' is $\dfrac {180 \degrees} 2 = 90 \degrees$ or $\dfrac \pi 2$. | A [[Definition:Right Angle|right angle]] is equal to one half of a [[Definition:Straight Angle|straight angle]].
From [[Measurement of Straight Angle]] it follows that the measurement of a [[Definition:Right Angle|right angle]] is $\dfrac {180 \degrees} 2 = 90 \degrees$ or $\dfrac \pi 2$.
{{qed}} | Measurements of Common Angles/Right Angle | https://proofwiki.org/wiki/Measurements_of_Common_Angles/Right_Angle | https://proofwiki.org/wiki/Measurements_of_Common_Angles/Right_Angle | [
"Right Angles",
"Measurements of Common Angles"
] | [
"Definition:Right Angle"
] | [
"Definition:Right Angle",
"Definition:Straight Angle",
"Measurements of Common Angles/Straight Angle",
"Definition:Right Angle"
] |
proofwiki-9965 | Measurements of Common Angles/Full Angle | A full angle is equal to $360 \degrees$ or $2 \pi$ radians. | By definition, $1$ '''radian''' is the angle which sweeps out an arc on a circle whose length is the radius $r$ of the circle.
From Perimeter of Circle, the length of the circumference of a circle of radius $r$ is equal to $2 \pi r$.
Therefore, $1$ radian sweeps out $\dfrac 1 {2 \pi}$ of a circle.
It follows that $2 \p... | A [[Definition:Full Angle|full angle]] is equal to $360 \degrees$ or $2 \pi$ [[Definition:Radian|radians]]. | By definition, $1$ '''[[Definition:Radian|radian]]''' is the [[Definition:Angle|angle]] which sweeps out an [[Definition:Arc of Circle|arc]] on a [[Definition:Circle|circle]] whose [[Definition:Length (Linear Measure)|length]] is the [[Definition:Radius of Circle|radius]] $r$ of the [[Definition:Circle|circle]].
From ... | Measurements of Common Angles/Full Angle | https://proofwiki.org/wiki/Measurements_of_Common_Angles/Full_Angle | https://proofwiki.org/wiki/Measurements_of_Common_Angles/Full_Angle | [
"Full Angles",
"Measurements of Common Angles"
] | [
"Definition:Full Angle",
"Definition:Angular Measure/Radian"
] | [
"Definition:Angular Measure/Radian",
"Definition:Angle",
"Definition:Circle/Arc",
"Definition:Circle",
"Definition:Linear Measure/Length",
"Definition:Circle/Radius",
"Definition:Circle",
"Perimeter of Circle",
"Definition:Linear Measure/Length",
"Definition:Circle/Circumference",
"Definition:Ci... |
proofwiki-9966 | Measurements of Common Angles/Acute Angle | An acute angle measures $\theta$, where:
:$0 \degrees < \theta < 90 \degrees$
or:
:$0 < \theta < \dfrac \pi 2$ | An acute angle is defined to be an angle whose measure is between that of a zero angle and a right angle.
A zero angle measures $0$ by definition, and a right angle measures $90 \degrees$ or $\dfrac \pi 2$.
Hence the result.
{{qed}} | An [[Definition:Acute Angle|acute angle]] measures $\theta$, where:
:$0 \degrees < \theta < 90 \degrees$
or:
:$0 < \theta < \dfrac \pi 2$ | An [[Definition:Acute Angle|acute angle]] is defined to be an [[Definition:Angle|angle]] whose measure is between that of a [[Definition:Zero Angle|zero angle]] and a [[Definition:Right Angle|right angle]].
A [[Definition:Zero Angle|zero angle]] measures $0$ by definition, and a [[Measurement of Right Angle|right angl... | Measurements of Common Angles/Acute Angle | https://proofwiki.org/wiki/Measurements_of_Common_Angles/Acute_Angle | https://proofwiki.org/wiki/Measurements_of_Common_Angles/Acute_Angle | [
"Acute Angles",
"Measurements of Common Angles"
] | [
"Definition:Acute Angle"
] | [
"Definition:Acute Angle",
"Definition:Angle",
"Definition:Zero Angle",
"Definition:Right Angle",
"Definition:Zero Angle",
"Measurements of Common Angles/Right Angle"
] |
proofwiki-9967 | Measurements of Common Angles/Obtuse Angle | An '''obtuse angle''' measures $\theta$, where:
:$90 \degrees < \theta < 180 \degrees$
or:
:$\dfrac \pi 2 < \theta < \pi$ | An obtuse angle is defined to be an angle whose measure is between that of a right angle and a straight angle.
A right angle measures $90 \degrees$ or $\dfrac \pi 2$ and a straight angle measures $180 \degrees$ or $\pi$.
Hence the result.
{{qed}} | An '''[[Definition:Obtuse Angle|obtuse angle]]''' measures $\theta$, where:
:$90 \degrees < \theta < 180 \degrees$
or:
:$\dfrac \pi 2 < \theta < \pi$ | An [[Definition:Obtuse Angle|obtuse angle]] is defined to be an [[Definition:Angle|angle]] whose measure is between that of a [[Definition:Right Angle|right angle]] and a [[Definition:Straight Angle|straight angle]].
A [[Measurement of Right Angle|right angle measures $90 \degrees$ or $\dfrac \pi 2$]] and a [[Measurem... | Measurements of Common Angles/Obtuse Angle | https://proofwiki.org/wiki/Measurements_of_Common_Angles/Obtuse_Angle | https://proofwiki.org/wiki/Measurements_of_Common_Angles/Obtuse_Angle | [
"Obtuse Angles",
"Measurements of Common Angles"
] | [
"Definition:Obtuse Angle"
] | [
"Definition:Obtuse Angle",
"Definition:Angle",
"Definition:Right Angle",
"Definition:Straight Angle",
"Measurements of Common Angles/Right Angle",
"Measurements of Common Angles/Straight Angle"
] |
proofwiki-9968 | Measurements of Common Angles/Reflex Angle | A reflex angle measures $\theta$, where:
:$180 \degrees < \theta < 360 \degrees$
or:
:$\pi < \theta < 2 \pi$ | A reflex angle is defined to be an angle whose measure is between that of a straight angle and a full angle.
A straight angle measures $180 \degrees$ or $\pi$ and a full angle measures $360 \degrees$ or $2 \pi$.
Hence the result.
{{qed}} | A [[Definition:Reflex Angle|reflex angle]] measures $\theta$, where:
:$180 \degrees < \theta < 360 \degrees$
or:
:$\pi < \theta < 2 \pi$ | A [[Definition:Reflex Angle|reflex angle]] is defined to be an [[Definition:Angle|angle]] whose measure is between that of a [[Definition:Straight Angle|straight angle]] and a [[Definition:Full Angle|full angle]].
A [[Measurement of Straight Angle|straight angle measures $180 \degrees$ or $\pi$]] and a [[Measurement o... | Measurements of Common Angles/Reflex Angle | https://proofwiki.org/wiki/Measurements_of_Common_Angles/Reflex_Angle | https://proofwiki.org/wiki/Measurements_of_Common_Angles/Reflex_Angle | [
"Reflex Angles",
"Measurements of Common Angles"
] | [
"Definition:Reflex Angle"
] | [
"Definition:Reflex Angle",
"Definition:Angle",
"Definition:Straight Angle",
"Definition:Full Angle",
"Measurements of Common Angles/Straight Angle",
"Measurements of Common Angles/Full Angle"
] |
proofwiki-9969 | Supplementary Interior Angles implies Parallel Lines | Given two infinite straight lines which are cut by a transversal, if the interior angles on the same side of the transversal are supplementary, then the lines are parallel. | :200px
Let $AB$ and $CD$ be infinite straight lines.
Let $EF$ be a transversal that cuts them.
Let at least one pair of interior angles on the same side of the transversal be supplementary.
{{WLOG}}, let those interior angles be $\angle BGH$ and $\angle DHG$.
So, by definition, $\angle DHG + \angle BGH$ equals two righ... | Given two [[Definition:Infinite Straight Line|infinite straight lines]] which are cut by a [[Definition:Transversal (Geometry)|transversal]], if the [[Definition:Interior Angle of Transversal|interior angles]] on the same side of the transversal are [[Definition:Supplementary Angles|supplementary]], then the lines are ... | :[[File:Parallel Cut by Transversal.png|200px]]
Let $AB$ and $CD$ be [[Definition:Infinite Straight Line|infinite straight lines]].
Let $EF$ be a [[Definition:Transversal (Geometry)|transversal]] that cuts them.
Let at least one pair of [[Definition:Interior Angle of Transversal|interior angles]] on the same side of... | Supplementary Interior Angles implies Parallel Lines | https://proofwiki.org/wiki/Supplementary_Interior_Angles_implies_Parallel_Lines | https://proofwiki.org/wiki/Supplementary_Interior_Angles_implies_Parallel_Lines | [
"Transversals (Geometry)",
"Parallel Lines",
"Supplementary Angles"
] | [
"Definition:Line/Infinite Straight Line",
"Definition:Transversal (Geometry)",
"Definition:Transversal (Geometry)/Interior Angle",
"Definition:Supplementary Angles",
"Definition:Parallel (Geometry)/Lines"
] | [
"File:Parallel Cut by Transversal.png",
"Definition:Line/Infinite Straight Line",
"Definition:Transversal (Geometry)",
"Definition:Transversal (Geometry)/Interior Angle",
"Definition:Transversal (Geometry)",
"Definition:Supplementary Angles",
"Definition:Transversal (Geometry)/Interior Angle",
"Defini... |
proofwiki-9970 | Equal Corresponding Angles implies Parallel Lines | Given two infinite straight lines which are cut by a transversal, if the corresponding angles are equal, then the lines are parallel. | :200px
Let $AB$ and $CD$ be infinite straight lines.
Let $EF$ be a transversal that cuts them.
Let at least one pair of corresponding angles be equal.
{{WLOG}}, let $\angle EGB = \angle GHD$.
By the Vertical Angle Theorem:
:$\angle GHD = \angle EGB = \angle AGH$
Thus by Equal Alternate Angles implies Parallel Lines:
:$... | Given two [[Definition:Infinite Straight Line|infinite straight lines]] which are cut by a [[Definition:Transversal (Geometry)|transversal]], if the [[Definition:Corresponding Angles of Transversal|corresponding angles]] are equal, then the lines are [[Definition:Parallel Lines|parallel]]. | :[[File:Parallel Cut by Transversal.png|200px]]
Let $AB$ and $CD$ be [[Definition:Infinite Straight Line|infinite straight lines]].
Let $EF$ be a [[Definition:Transversal (Geometry)|transversal]] that cuts them.
Let at least one pair of [[Definition:Corresponding Angles of Transversal|corresponding angles]] be equal... | Equal Corresponding Angles implies Parallel Lines | https://proofwiki.org/wiki/Equal_Corresponding_Angles_implies_Parallel_Lines | https://proofwiki.org/wiki/Equal_Corresponding_Angles_implies_Parallel_Lines | [
"Angles",
"Parallel Lines"
] | [
"Definition:Line/Infinite Straight Line",
"Definition:Transversal (Geometry)",
"Definition:Transversal (Geometry)/Corresponding Angles",
"Definition:Parallel (Geometry)/Lines"
] | [
"File:Parallel Cut by Transversal.png",
"Definition:Line/Infinite Straight Line",
"Definition:Transversal (Geometry)",
"Definition:Transversal (Geometry)/Corresponding Angles",
"Two Straight Lines make Equal Opposite Angles",
"Equal Alternate Angles implies Parallel Lines"
] |
proofwiki-9971 | Equivalence of Definitions of Tangent Vector | Let $M$ be a smooth manifold.
Let $m \in M$ be a point.
Let $V$ be an open neighborhood of $m$.
Let $\map {C^\infty} {V, \R}$ be defined as the set of all smooth mappings $f: V \to \R$.
{{TFAE|def = Tangent Vector}} | === Definition 2 implies Definition 1 ===
Let $\lambda \in \R$ and $f, g \in \map {C^\infty} {V, \R}$.
{{begin-eqn}}
{{eqn | l = \map {X_m} {f + \lambda g}
| r = \map {\frac \d {\d \tau} {\restriction_0} } {\map {\paren {f + \lambda g} \circ \gamma} \tau}
| c = Definition 2
}}
{{eqn | r = \map {\frac \d {\... | Let $M$ be a [[Definition:Smooth Manifold|smooth manifold]].
Let $m \in M$ be a [[Definition:Point|point]].
Let $V$ be an [[Definition:Open Neighborhood of Point|open neighborhood]] of $m$.
Let $\map {C^\infty} {V, \R}$ be defined as the [[Definition:Set|set]] of all [[Definition:Smooth Mapping|smooth mappings]] $... | === Definition 2 implies Definition 1 ===
Let $\lambda \in \R$ and $f, g \in \map {C^\infty} {V, \R}$.
{{begin-eqn}}
{{eqn | l = \map {X_m} {f + \lambda g}
| r = \map {\frac \d {\d \tau} {\restriction_0} } {\map {\paren {f + \lambda g} \circ \gamma} \tau}
| c = [[Definition:Tangent Vector/Definition 2|De... | Equivalence of Definitions of Tangent Vector | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Tangent_Vector | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Tangent_Vector | [
"Tangent Vectors"
] | [
"Definition:Topological Manifold/Smooth Manifold",
"Definition:Point",
"Definition:Open Neighborhood/Point",
"Definition:Set",
"Definition:Smooth Mapping"
] | [
"Definition:Tangent Vector/Definition 2",
"Definition:Linear Transformation",
"Product Rule for Derivatives",
"Definition:Tangent Vector",
"Definition:Linear Transformation",
"Definition:Tangent Vector",
"Definition:Linear Transformation",
"Definition:Tangent Vector"
] |
proofwiki-9972 | Parallelism implies Equal Alternate Angles | Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the alternate angles are equal. | :200px
Let $AB$ and $CD$ be parallel infinite straight lines.
Let $EF$ be a transversal that cuts them.
{{AimForCont}} the alternate angles are not equal.
Then one of the pair $\angle AGH$ and $\angle GHD$ must be greater.
{{WLOG}}, let $\angle AGH$ be greater.
From Two Angles on Straight Line make Two Right Angles, $\... | Given two [[Definition:Infinite Straight Line|infinite straight lines]] which are cut by a [[Definition:Transversal (Geometry)|transversal]], if the lines are [[Definition:Parallel Lines|parallel]], then the [[Definition:Alternate Angles of Transversal|alternate angles]] are equal. | :[[File:Parallel Cut by Transversal.png|200px]]
Let $AB$ and $CD$ be [[Definition:Parallel Lines|parallel]] [[Definition:Infinite Straight Line|infinite straight lines]].
Let $EF$ be a [[Definition:Transversal (Geometry)|transversal]] that cuts them.
{{AimForCont}} the [[Definition:Alternate Angles of Transversal|al... | Parallelism implies Equal Alternate Angles | https://proofwiki.org/wiki/Parallelism_implies_Equal_Alternate_Angles | https://proofwiki.org/wiki/Parallelism_implies_Equal_Alternate_Angles | [
"Transversals (Geometry)",
"Parallel Lines"
] | [
"Definition:Line/Infinite Straight Line",
"Definition:Transversal (Geometry)",
"Definition:Parallel (Geometry)/Lines",
"Definition:Transversal (Geometry)/Alternate Angles"
] | [
"File:Parallel Cut by Transversal.png",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Infinite Straight Line",
"Definition:Transversal (Geometry)",
"Definition:Transversal (Geometry)/Alternate Angles",
"Two Angles on Straight Line make Two Right Angles",
"Definition:Right Angle",
"Definitio... |
proofwiki-9973 | Parallelism implies Equal Corresponding Angles | Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the corresponding angles are equal. | :200px
Let $AB$ and $CD$ be parallel infinite straight lines.
Let $EF$ be a transversal that cuts them.
From Parallelism implies Equal Alternate Angles:
:$\angle AGH = \angle DHG$
By the Vertical Angle Theorem:
:$\angle EGB = \angle AGH = \angle DHG$
{{qed}}
{{Euclid Note|29|I|{{EuclidNoteConverse|prop = 28|title = Equ... | Given two [[Definition:Infinite Straight Line|infinite straight lines]] which are cut by a [[Definition:Transversal (Geometry)|transversal]], if the lines are [[Definition:Parallel Lines|parallel]], then the [[Definition:Corresponding Angles of Transversal|corresponding angles]] are equal. | :[[File:Parallel Cut by Transversal.png|200px]]
Let $AB$ and $CD$ be [[Definition:Parallel Lines|parallel]] [[Definition:Infinite Straight Line|infinite straight lines]].
Let $EF$ be a [[Definition:Transversal (Geometry)|transversal]] that cuts them.
From [[Parallelism implies Equal Alternate Angles]]:
:$\angle AGH ... | Parallelism implies Equal Corresponding Angles | https://proofwiki.org/wiki/Parallelism_implies_Equal_Corresponding_Angles | https://proofwiki.org/wiki/Parallelism_implies_Equal_Corresponding_Angles | [
"Transversals (Geometry)",
"Parallel Lines"
] | [
"Definition:Line/Infinite Straight Line",
"Definition:Transversal (Geometry)",
"Definition:Parallel (Geometry)/Lines",
"Definition:Transversal (Geometry)/Corresponding Angles"
] | [
"File:Parallel Cut by Transversal.png",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Infinite Straight Line",
"Definition:Transversal (Geometry)",
"Parallelism implies Equal Alternate Angles",
"Two Straight Lines make Equal Opposite Angles"
] |
proofwiki-9974 | Proportion is Reflexive | Proportion is a reflexive relation.
That is, every real variable is proportional to itself:
:$\forall x \in \R: x \propto x$ | Let $x$ be arbitrary.
Then $x = 1 \times x$.
The result follows from the definition of reflexivity and proportion.
{{qed}}
Category:Proportion
Category:Examples of Reflexive Relations
5hd7asghtbu63z4rden9azmj4kwse4i | [[Definition:Proportion|Proportion]] is a [[Definition:Reflexive Relation|reflexive relation]].
That is, every [[Definition:Real Number|real]] [[Definition:Variable|variable]] is [[Definition:Proportional|proportional]] to itself:
:$\forall x \in \R: x \propto x$ | Let $x$ be arbitrary.
Then $x = 1 \times x$.
The result follows from the definition of [[Definition:Reflexive Relation|reflexivity]] and [[Definition:Proportion|proportion]].
{{qed}}
[[Category:Proportion]]
[[Category:Examples of Reflexive Relations]]
5hd7asghtbu63z4rden9azmj4kwse4i | Proportion is Reflexive | https://proofwiki.org/wiki/Proportion_is_Reflexive | https://proofwiki.org/wiki/Proportion_is_Reflexive | [
"Proportion",
"Examples of Reflexive Relations"
] | [
"Definition:Proportion",
"Definition:Reflexive Relation",
"Definition:Real Number",
"Definition:Variable",
"Definition:Proportion"
] | [
"Definition:Reflexive Relation",
"Definition:Proportion",
"Category:Proportion",
"Category:Examples of Reflexive Relations"
] |
proofwiki-9975 | Proportion is Symmetric | Proportion is a symmetric relation.
That is:
:$\forall x, y \in \R: x \propto y \implies y \propto x$ | Let $x, y$ be arbitrary.
Let $x$ be proportional to $y$:
:$x \propto y$
Then by definition:
{{begin-eqn}}
{{eqn | q = \exists k \ne 0
| l = x
| r = k \times y
| c =
}}
{{eqn | ll= \leadsto
| l = y
| r = k^{-1} \times x
| c =
}}
{{end-eqn}}
The result follows from the definition of ... | [[Definition:Proportion|Proportion]] is a [[Definition:Symmetric Relation|symmetric relation]].
That is:
:$\forall x, y \in \R: x \propto y \implies y \propto x$ | Let $x, y$ be arbitrary.
Let $x$ be [[Definition:Proportional|proportional]] to $y$:
:$x \propto y$
Then by definition:
{{begin-eqn}}
{{eqn | q = \exists k \ne 0
| l = x
| r = k \times y
| c =
}}
{{eqn | ll= \leadsto
| l = y
| r = k^{-1} \times x
| c =
}}
{{end-eqn}}
The resul... | Proportion is Symmetric | https://proofwiki.org/wiki/Proportion_is_Symmetric | https://proofwiki.org/wiki/Proportion_is_Symmetric | [
"Proportion",
"Examples of Symmetric Relations"
] | [
"Definition:Proportion",
"Definition:Symmetric Relation"
] | [
"Definition:Proportion",
"Definition:Symmetric Relation",
"Definition:Proportion",
"Category:Proportion",
"Category:Examples of Symmetric Relations"
] |
proofwiki-9976 | Proportion is Transitive | Proportion is a transitive relation.
That is:
:$\forall x, y, z \in \R: x \propto y \land y \propto z \implies x \propto z$ | Let $x, y, z$ be arbitrary.
Let $x$ be proportional to $y$ and $y$ to $z$:
:$x \propto y \land y \propto z$
Then by definition:
:$\exists j, k \ne 0: x = j \times y \land y = k \times z$
Substituting $k \times z$ for $y$:
:$x = \paren {j \times k} \times z$
so $j \times k$ is the desired constant of proportion.
The res... | [[Definition:Proportion|Proportion]] is a [[Definition:Transitive Relation|transitive relation]].
That is:
:$\forall x, y, z \in \R: x \propto y \land y \propto z \implies x \propto z$ | Let $x, y, z$ be arbitrary.
Let $x$ be [[Definition:Proportional|proportional]] to $y$ and $y$ to $z$:
:$x \propto y \land y \propto z$
Then by definition:
:$\exists j, k \ne 0: x = j \times y \land y = k \times z$
Substituting $k \times z$ for $y$:
:$x = \paren {j \times k} \times z$
so $j \times k$ is the desi... | Proportion is Transitive | https://proofwiki.org/wiki/Proportion_is_Transitive | https://proofwiki.org/wiki/Proportion_is_Transitive | [
"Proportion",
"Examples of Transitive Relations"
] | [
"Definition:Proportion",
"Definition:Transitive Relation"
] | [
"Definition:Proportion",
"Definition:Proportion/Constant of Proportion",
"Definition:Transitive Relation",
"Definition:Proportion",
"Category:Proportion",
"Category:Examples of Transitive Relations"
] |
proofwiki-9977 | Proportion is Equivalence Relation | Proportion is an equivalence relation. | :Proportion is Reflexive: $\forall x \in \R: x \propto x$
:Proportion is Symmetric: $\forall x, y \in \R: x \propto y \implies y \propto x$
:Proportion is Transitive: $\forall x, y, z \in \R: x \propto y \land y \propto z \implies x \propto z$
The result follows from the definition of an equivalence relation.
{{qed}}
C... | [[Definition:Proportion|Proportion]] is an [[Definition:Equivalence Relation|equivalence relation]]. | :[[Proportion is Reflexive]]: $\forall x \in \R: x \propto x$
:[[Proportion is Symmetric]]: $\forall x, y \in \R: x \propto y \implies y \propto x$
:[[Proportion is Transitive]]: $\forall x, y, z \in \R: x \propto y \land y \propto z \implies x \propto z$
The result follows from the definition of an [[Definition:Equiv... | Proportion is Equivalence Relation | https://proofwiki.org/wiki/Proportion_is_Equivalence_Relation | https://proofwiki.org/wiki/Proportion_is_Equivalence_Relation | [
"Proportion",
"Examples of Equivalence Relations"
] | [
"Definition:Proportion",
"Definition:Equivalence Relation"
] | [
"Proportion is Reflexive",
"Proportion is Symmetric",
"Proportion is Transitive",
"Definition:Equivalence Relation",
"Category:Proportion",
"Category:Examples of Equivalence Relations"
] |
proofwiki-9978 | Parallelism implies Supplementary Interior Angles | Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the interior angles on the same side of the transversal are supplementary. | :200px
Let $AB$ and $CD$ be parallel infinite straight lines.
Let $EF$ be a transversal that cuts them.
From Parallelism implies Equal Corresponding Angles and Euclid's second common notion:
:$\angle EGB + \angle BGH = \angle DHG + \angle BGH$
From Two Angles on Straight Line make Two Right Angles, $\angle EGB + \angle... | Given two [[Definition:Infinite Straight Line|infinite straight lines]] which are cut by a [[Definition:Transversal (Geometry)|transversal]], if the lines are [[Definition:Parallel Lines|parallel]], then the [[Definition:Interior Angle of Transversal|interior angles]] on the same side of the [[Definition:Transversal (G... | :[[File:Parallel Cut by Transversal.png|200px]]
Let $AB$ and $CD$ be [[Definition:Parallel Lines|parallel]] [[Definition:Infinite Straight Line|infinite straight lines]].
Let $EF$ be a [[Definition:Transversal (Geometry)|transversal]] that cuts them.
From [[Parallelism implies Equal Corresponding Angles]] and [[Axio... | Parallelism implies Supplementary Interior Angles | https://proofwiki.org/wiki/Parallelism_implies_Supplementary_Interior_Angles | https://proofwiki.org/wiki/Parallelism_implies_Supplementary_Interior_Angles | [
"Transversals (Geometry)",
"Parallel Lines",
"Supplementary Angles"
] | [
"Definition:Line/Infinite Straight Line",
"Definition:Transversal (Geometry)",
"Definition:Parallel (Geometry)/Lines",
"Definition:Transversal (Geometry)/Interior Angle",
"Definition:Transversal (Geometry)",
"Definition:Supplementary Angles"
] | [
"File:Parallel Cut by Transversal.png",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Infinite Straight Line",
"Definition:Transversal (Geometry)",
"Parallelism implies Equal Corresponding Angles",
"Axiom:Euclid's Common Notions",
"Two Angles on Straight Line make Two Right Angles",
"Defini... |
proofwiki-9979 | Proportion of Power | Let $x$ and $y$ be proportional.
{{explain|Establish what types of object $x$ and $y$ are. As it stands here, they could be anything.}}
Let $n \in \Z$.
Then $x^n \propto y^n$. | Let $x \propto y$.
Then $\exists k \ne 0: x = k \times y$ by the definition of proportion.
Raising both sides of this equation to the $n$th power:
{{begin-eqn}}
{{eqn | l = x^n
| r = \paren {k \times y}^n
}}
{{eqn | r = k^n \times y^n
}}
{{end-eqn}}
so $k^n$ is the desired constant of proportion.
The result follo... | Let $x$ and $y$ be [[Definition:Proportional|proportional]].
{{explain|Establish what types of object $x$ and $y$ are. As it stands here, they could be anything.}}
Let $n \in \Z$.
Then $x^n \propto y^n$. | Let $x \propto y$.
Then $\exists k \ne 0: x = k \times y$ by the definition of [[Definition:Proportion|proportion]].
Raising both sides of this equation to the [[Definition:Integer Power|$n$th power]]:
{{begin-eqn}}
{{eqn | l = x^n
| r = \paren {k \times y}^n
}}
{{eqn | r = k^n \times y^n
}}
{{end-eqn}}
so $k... | Proportion of Power | https://proofwiki.org/wiki/Proportion_of_Power | https://proofwiki.org/wiki/Proportion_of_Power | [
"Proportion"
] | [
"Definition:Proportion"
] | [
"Definition:Proportion",
"Definition:Power (Algebra)/Integer",
"Definition:Proportion/Constant of Proportion",
"Definition:Proportion",
"Category:Proportion"
] |
proofwiki-9980 | Sine of 18 Degrees | :$\sin 18 \degrees = \sin \dfrac \pi {10} = \dfrac 1 {2 \phi} = 2^{-1} \phi^{-1} = \dfrac {\sqrt 5 - 1} 4$
where $\sin$ denotes the sine function.
and $\phi$ denotes the golden mean. | From Sine of $90 \degrees$:
:$\map \sin {5 \times 18 \degrees} = \sin 90 \degrees = 1$.
Consider the equation:
:$\sin 5x = 1$
where $x = 18 \degrees$ is one of the solutions.
From Quintuple Angle Formula for Sine:
:$16 \sin^5 \theta - 20 \sin^3 \theta + 5 \sin \theta = 1$
Let $s = \sin \theta$:
:$16 s^5 - 20 s^3 + 5s -... | :$\sin 18 \degrees = \sin \dfrac \pi {10} = \dfrac 1 {2 \phi} = 2^{-1} \phi^{-1} = \dfrac {\sqrt 5 - 1} 4$
where $\sin$ denotes the [[Definition:Sine Function|sine function]].
and $\phi$ denotes the [[Definition:Golden Mean|golden mean]]. | From [[Sine of 90 Degrees|Sine of $90 \degrees$]]:
:$\map \sin {5 \times 18 \degrees} = \sin 90 \degrees = 1$.
Consider the equation:
:$\sin 5x = 1$
where $x = 18 \degrees$ is one of the solutions.
From [[Quintuple Angle Formula for Sine]]:
:$16 \sin^5 \theta - 20 \sin^3 \theta + 5 \sin \theta = 1$
Let $s = \sin \th... | Sine of 18 Degrees | https://proofwiki.org/wiki/Sine_of_18_Degrees | https://proofwiki.org/wiki/Sine_of_18_Degrees | [
"Sine Function",
"Golden Mean"
] | [
"Definition:Sine",
"Definition:Golden Mean"
] | [
"Sine of Right Angle",
"Quintuple Angle Formulas/Sine",
"Solution to Quadratic Equation",
"Definition:Square Root/Negative",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Category:Sine Function",
"Category:Golden Mean"
] |
proofwiki-9981 | Cosine of 72 Degrees | :$\cos 72 \degrees = \cos \dfrac {2 \pi} 5 = 2^{-1} \phi ^{-1} = 2^{-1} \paren {\phi - 1} = \dfrac {\sqrt 5 - 1} 4$ | {{begin-eqn}}
{{eqn | l = \cos 72 \degrees
| r = \map \cos {90 \degrees - 18 \degrees}
}}
{{eqn | r = \sin 18 \degrees
| c = Cosine of Complement equals Sine
}}
{{eqn | r = \dfrac {\sqrt 5 - 1} 4
| c = {{sin|18}}
}}
{{eqn | r = \frac 1 2 \times \paren {\dfrac {\sqrt 5 + 1} 2 - 1}
| c =
}}
{{eqn... | :$\cos 72 \degrees = \cos \dfrac {2 \pi} 5 = 2^{-1} \phi ^{-1} = 2^{-1} \paren {\phi - 1} = \dfrac {\sqrt 5 - 1} 4$ | {{begin-eqn}}
{{eqn | l = \cos 72 \degrees
| r = \map \cos {90 \degrees - 18 \degrees}
}}
{{eqn | r = \sin 18 \degrees
| c = [[Cosine of Complement equals Sine]]
}}
{{eqn | r = \dfrac {\sqrt 5 - 1} 4
| c = {{sin|18}}
}}
{{eqn | r = \frac 1 2 \times \paren {\dfrac {\sqrt 5 + 1} 2 - 1}
| c =
}}
{... | Cosine of 72 Degrees/Proof 1 | https://proofwiki.org/wiki/Cosine_of_72_Degrees | https://proofwiki.org/wiki/Cosine_of_72_Degrees/Proof_1 | [
"Cosine of 72 Degrees",
"Cosine Function",
"Golden Mean"
] | [] | [
"Cosine of Complement equals Sine"
] |
proofwiki-9982 | Cosine of 72 Degrees | :$\cos 72 \degrees = \cos \dfrac {2 \pi} 5 = 2^{-1} \phi ^{-1} = 2^{-1} \paren {\phi - 1} = \dfrac {\sqrt 5 - 1} 4$ | {{begin-eqn}}
{{eqn | l = \cos 72 \degrees
| r = 2 \paren {\cos 36 \degrees}^2 - 1
| c = {{Corollary|Double Angle Formula for Cosine|1}}
}}
{{eqn | r = 2 \paren {\dfrac \phi 2}^2 - 1
| c = {{cos|36}}
}}
{{eqn | r = \dfrac {\phi^2} 2 - 1
| c =
}}
{{eqn | r = \dfrac {\phi + 1} 2 - 1
| c = S... | :$\cos 72 \degrees = \cos \dfrac {2 \pi} 5 = 2^{-1} \phi ^{-1} = 2^{-1} \paren {\phi - 1} = \dfrac {\sqrt 5 - 1} 4$ | {{begin-eqn}}
{{eqn | l = \cos 72 \degrees
| r = 2 \paren {\cos 36 \degrees}^2 - 1
| c = {{Corollary|Double Angle Formula for Cosine|1}}
}}
{{eqn | r = 2 \paren {\dfrac \phi 2}^2 - 1
| c = {{cos|36}}
}}
{{eqn | r = \dfrac {\phi^2} 2 - 1
| c =
}}
{{eqn | r = \dfrac {\phi + 1} 2 - 1
| c = [... | Cosine of 72 Degrees/Proof 2 | https://proofwiki.org/wiki/Cosine_of_72_Degrees | https://proofwiki.org/wiki/Cosine_of_72_Degrees/Proof_2 | [
"Cosine of 72 Degrees",
"Cosine Function",
"Golden Mean"
] | [] | [
"Square of Golden Mean equals One plus Golden Mean",
"Reciprocal Form of One Minus Golden Mean"
] |
proofwiki-9983 | Sine of 72 Degrees | :$\sin 72 \degrees = \sin \dfrac {2 \pi} 5 = 2^{-1} 5^{\frac 1 4} \phi^{\frac 1 2} = \dfrac {\sqrt{10 + 2 \sqrt 5} } 4$
where $\sin$ denotes the sine function. | {{begin-eqn}}
{{eqn | l = \sin 72 \degrees
| r = \sqrt {1 - \cos^2 72 \degrees}
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | r = \sqrt {1 - \paren {\frac{\sqrt 5 - 1} 4}^2}
| c = Cosine of $72 \degrees$
}}
{{eqn | r = \sqrt {1 - \frac {6 - 2 \sqrt 5} {16} }
}}
{{eqn | r = \frac {\sqrt {10 + 2 \s... | :$\sin 72 \degrees = \sin \dfrac {2 \pi} 5 = 2^{-1} 5^{\frac 1 4} \phi^{\frac 1 2} = \dfrac {\sqrt{10 + 2 \sqrt 5} } 4$
where $\sin$ denotes the [[Definition:Sine Function|sine function]]. | {{begin-eqn}}
{{eqn | l = \sin 72 \degrees
| r = \sqrt {1 - \cos^2 72 \degrees}
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{eqn | r = \sqrt {1 - \paren {\frac{\sqrt 5 - 1} 4}^2}
| c = [[Cosine of 72 Degrees|Cosine of $72 \degrees$]]
}}
{{eqn | r = \sqrt {1 - \frac {6 - 2 \sqrt 5} {16} }
}}
{{eqn ... | Sine of 72 Degrees | https://proofwiki.org/wiki/Sine_of_72_Degrees | https://proofwiki.org/wiki/Sine_of_72_Degrees | [
"Sine Function",
"Golden Mean"
] | [
"Definition:Sine"
] | [
"Sum of Squares of Sine and Cosine",
"Cosine of 72 Degrees"
] |
proofwiki-9984 | Cosine of 18 Degrees | :$\cos 18 \degrees = \cos \dfrac \pi {10} = \dfrac {\sqrt {10 + 2 \sqrt 5} } 4$
where $\cos$ denotes the cosine function. | {{begin-eqn}}
{{eqn | l = \cos 18 \degrees
| r = \map \cos {90 \degrees - 72 \degrees}
}}
{{eqn | r = \sin 72 \degrees
| c = Cosine of Complement equals Sine
}}
{{eqn | r = \frac {\sqrt {10 + 2 \sqrt 5} } 4
| c = Sine of $72 \degrees$
}}
{{end-eqn}}
{{qed}}
Category:Cosine Function
s31hgo49ify7wrsbz4d... | :$\cos 18 \degrees = \cos \dfrac \pi {10} = \dfrac {\sqrt {10 + 2 \sqrt 5} } 4$
where $\cos$ denotes the [[Definition:Cosine Function|cosine function]]. | {{begin-eqn}}
{{eqn | l = \cos 18 \degrees
| r = \map \cos {90 \degrees - 72 \degrees}
}}
{{eqn | r = \sin 72 \degrees
| c = [[Cosine of Complement equals Sine]]
}}
{{eqn | r = \frac {\sqrt {10 + 2 \sqrt 5} } 4
| c = [[Sine of 72 Degrees|Sine of $72 \degrees$]]
}}
{{end-eqn}}
{{qed}}
[[Category:Cosin... | Cosine of 18 Degrees | https://proofwiki.org/wiki/Cosine_of_18_Degrees | https://proofwiki.org/wiki/Cosine_of_18_Degrees | [
"Cosine Function"
] | [
"Definition:Cosine"
] | [
"Cosine of Complement equals Sine",
"Sine of 72 Degrees",
"Category:Cosine Function"
] |
proofwiki-9985 | Sine of 3 Degrees | :$\sin 3^\circ = \sin \dfrac \pi {60} = \dfrac {\sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16}$
where $\sin$ denotes the sine function. | {{begin-eqn}}
{{eqn | l = \sin 3^\circ
| r = \sin \left({75^\circ - 72^\circ}\right)
}}
{{eqn | r = \sin 75^\circ \cos 72^\circ - \cos 75^\circ \sin 72^\circ
| c = Sine of Difference
}}
{{eqn | r = \dfrac {\sqrt 6 + \sqrt 2} 4 \times \dfrac {\sqrt 5 - 1} 4 - \cos 75^\circ \sin 72^\circ
| c = Sine of $... | :$\sin 3^\circ = \sin \dfrac \pi {60} = \dfrac {\sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16}$
where $\sin$ denotes the [[Definition:Sine Function|sine function]]. | {{begin-eqn}}
{{eqn | l = \sin 3^\circ
| r = \sin \left({75^\circ - 72^\circ}\right)
}}
{{eqn | r = \sin 75^\circ \cos 72^\circ - \cos 75^\circ \sin 72^\circ
| c = [[Sine of Difference]]
}}
{{eqn | r = \dfrac {\sqrt 6 + \sqrt 2} 4 \times \dfrac {\sqrt 5 - 1} 4 - \cos 75^\circ \sin 72^\circ
| c = [[Sin... | Sine of 3 Degrees | https://proofwiki.org/wiki/Sine_of_3_Degrees | https://proofwiki.org/wiki/Sine_of_3_Degrees | [
"Sine Function"
] | [
"Definition:Sine"
] | [
"Sine of Difference",
"Sine of 75 Degrees",
"Cosine of 72 Degrees",
"Cosine of 75 Degrees",
"Sine of 72 Degrees",
"Category:Sine Function"
] |
proofwiki-9986 | Cosine of 3 Degrees | :$\cos 3 \degrees = \cos \dfrac {\pi} {60} = \dfrac {\sqrt {30} - \sqrt {10} - \sqrt 6 + \sqrt 2 + 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16}$
where $\cos$ denotes the cosine function. | {{begin-eqn}}
{{eqn | l = \cos 3 \degrees
| r = \map \cos {75 \degrees - 72 \degrees}
}}
{{eqn | r = \cos 75 \degrees \cos 72 \degrees + \sin 75 \degrees \sin 72 \degrees
| c = Cosine of Difference
}}
{{eqn | r = \dfrac {\sqrt 6 - \sqrt 2} 4 \times \dfrac {\sqrt 5 - 1} 4 + \dfrac {\sqrt 6 + \sqrt 2} 4 \time... | :$\cos 3 \degrees = \cos \dfrac {\pi} {60} = \dfrac {\sqrt {30} - \sqrt {10} - \sqrt 6 + \sqrt 2 + 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16}$
where $\cos$ denotes the [[Definition:Cosine Function|cosine function]]. | {{begin-eqn}}
{{eqn | l = \cos 3 \degrees
| r = \map \cos {75 \degrees - 72 \degrees}
}}
{{eqn | r = \cos 75 \degrees \cos 72 \degrees + \sin 75 \degrees \sin 72 \degrees
| c = [[Cosine of Difference]]
}}
{{eqn | r = \dfrac {\sqrt 6 - \sqrt 2} 4 \times \dfrac {\sqrt 5 - 1} 4 + \dfrac {\sqrt 6 + \sqrt 2} 4 \... | Cosine of 3 Degrees | https://proofwiki.org/wiki/Cosine_of_3_Degrees | https://proofwiki.org/wiki/Cosine_of_3_Degrees | [
"Cosine Function"
] | [
"Definition:Cosine"
] | [
"Cosine of Difference",
"Cosine of 75 Degrees",
"Cosine of 72 Degrees",
"Sine of 75 Degrees",
"Sine of 72 Degrees",
"Category:Cosine Function"
] |
proofwiki-9987 | Square Root of Sum as Sum of Square Roots | Let $a, b \in \R, a \ge b$.
Then:
:$\sqrt {a + b} = \sqrt {\dfrac a 2 + \dfrac {\sqrt {a^2 - b^2}} 2} + \sqrt {\dfrac a 2 - \dfrac {\sqrt {a^2 - b^2}} 2}$ | Let $\sqrt {a + b}$ be expressed in the form $\sqrt c + \sqrt d$.
From Square of Sum:
: $a + b = c + d + 2 \sqrt {c d}$
We now need to solve the simultaneous equations:
: $a = c + d$
: $b = 2 \sqrt {c d}$
First:
{{begin-eqn}}
{{eqn | l = a
| r = c + d
}}
{{eqn | n = 1
| ll= \leadsto
| l = d
| r ... | Let $a, b \in \R, a \ge b$.
Then:
:$\sqrt {a + b} = \sqrt {\dfrac a 2 + \dfrac {\sqrt {a^2 - b^2}} 2} + \sqrt {\dfrac a 2 - \dfrac {\sqrt {a^2 - b^2}} 2}$ | Let $\sqrt {a + b}$ be expressed in the form $\sqrt c + \sqrt d$.
From [[Square of Sum]]:
: $a + b = c + d + 2 \sqrt {c d}$
We now need to solve the [[Definition:Simultaneous Equations|simultaneous equations]]:
: $a = c + d$
: $b = 2 \sqrt {c d}$
First:
{{begin-eqn}}
{{eqn | l = a
| r = c + d
}}
{{eqn | n = 1... | Square Root of Sum as Sum of Square Roots/Proof 1 | https://proofwiki.org/wiki/Square_Root_of_Sum_as_Sum_of_Square_Roots | https://proofwiki.org/wiki/Square_Root_of_Sum_as_Sum_of_Square_Roots/Proof_1 | [
"Algebra",
"Square Root of Sum as Sum of Square Roots"
] | [] | [
"Square of Sum",
"Definition:Simultaneous Equations",
"Definition:Square/Function",
"Real Multiplication Distributes over Addition",
"Solution to Quadratic Equation",
"Real Addition is Commutative",
"Definition:Square Root"
] |
proofwiki-9988 | Square Root of Sum as Sum of Square Roots | Let $a, b \in \R, a \ge b$.
Then:
:$\sqrt {a + b} = \sqrt {\dfrac a 2 + \dfrac {\sqrt {a^2 - b^2}} 2} + \sqrt {\dfrac a 2 - \dfrac {\sqrt {a^2 - b^2}} 2}$ | From Sum of Square Roots as Square Root of Sum:
:$\sqrt p + \sqrt q = \sqrt {p + q + \sqrt {4pq}}$
Let
:$p = \dfrac a 2 + \dfrac {\sqrt {a^2 - b^2}} 2$,
:$q = \dfrac a 2 - \dfrac {\sqrt {a^2 - b^2}} 2$.
Then
{{begin-eqn}}
{{eqn | l = p + q
| r = \frac a 2 + \frac {\sqrt {a^2 - b^2} } 2 + \frac a 2 - \frac {\sqrt ... | Let $a, b \in \R, a \ge b$.
Then:
:$\sqrt {a + b} = \sqrt {\dfrac a 2 + \dfrac {\sqrt {a^2 - b^2}} 2} + \sqrt {\dfrac a 2 - \dfrac {\sqrt {a^2 - b^2}} 2}$ | From [[Sum of Square Roots as Square Root of Sum]]:
:$\sqrt p + \sqrt q = \sqrt {p + q + \sqrt {4pq}}$
Let
:$p = \dfrac a 2 + \dfrac {\sqrt {a^2 - b^2}} 2$,
:$q = \dfrac a 2 - \dfrac {\sqrt {a^2 - b^2}} 2$.
Then
{{begin-eqn}}
{{eqn | l = p + q
| r = \frac a 2 + \frac {\sqrt {a^2 - b^2} } 2 + \frac a 2 - \frac {... | Square Root of Sum as Sum of Square Roots/Proof 2 | https://proofwiki.org/wiki/Square_Root_of_Sum_as_Sum_of_Square_Roots | https://proofwiki.org/wiki/Square_Root_of_Sum_as_Sum_of_Square_Roots/Proof_2 | [
"Algebra",
"Square Root of Sum as Sum of Square Roots"
] | [] | [
"Sum of Square Roots as Square Root of Sum",
"Real Multiplication Distributes over Addition",
"Difference of Two Squares"
] |
proofwiki-9989 | Sine of 1 Degree | {{begin-eqn}}
{{eqn | l = \sin 1 \degrees = \sin \dfrac \pi {180}
| r = \paren {\dfrac 1 8 + i \dfrac {\sqrt 3} 8} \paren {\sqrt [3] {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } 2} + i \sqrt {32 - 6 \sqrt 3 + 2 \sqrt {15} + 2 \sqrt {50 + 10 \sqr... | {{begin-eqn}}
{{eqn | l = \map \sin {3 \times 1 \degrees}
| r = 3 \sin 1 \degrees - 4 \sin^3 1 \degrees
| c = Triple Angle Formula for Sine
}}
{{eqn | l = \sin 3 \degrees
| r = 3 \sin 1 \degrees - 4 \sin^3 1 \degrees
}}
{{eqn | l = 4 \sin^3 1 \degrees - 3 \sin 1 \degrees + \sin 3 \degrees
| r = ... | {{begin-eqn}}
{{eqn | l = \sin 1 \degrees = \sin \dfrac \pi {180}
| r = \paren {\dfrac 1 8 + i \dfrac {\sqrt 3} 8} \paren {\sqrt [3] {\paren {\dfrac {\sqrt {30} + \sqrt {10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } 2} + i \sqrt {32 - 6 \sqrt 3 + 2 \sqrt {15} + 2 \sqrt {50 + 10 \sqr... | {{begin-eqn}}
{{eqn | l = \map \sin {3 \times 1 \degrees}
| r = 3 \sin 1 \degrees - 4 \sin^3 1 \degrees
| c = [[Triple Angle Formula for Sine]]
}}
{{eqn | l = \sin 3 \degrees
| r = 3 \sin 1 \degrees - 4 \sin^3 1 \degrees
}}
{{eqn | l = 4 \sin^3 1 \degrees - 3 \sin 1 \degrees + \sin 3 \degrees
| ... | Sine of 1 Degree | https://proofwiki.org/wiki/Sine_of_1_Degree | https://proofwiki.org/wiki/Sine_of_1_Degree | [
"Sine Function"
] | [] | [
"Triple Angle Formulas/Sine",
"Cardano's Formula",
"Sum of Squares of Sine and Cosine",
"Definition:Imaginary Number",
"Sine of Complement equals Cosine",
"Roots of Complex Number/Corollary/Examples/Cube Roots",
"Roots of Complex Number/Corollary/Examples/Cube Roots",
"Sine of Complement equals Cosine... |
proofwiki-9990 | Rectangle is Parallelogram | Let $ABCD$ be a rectangle.
Then $ABCD$ is a parallelogram. | Let $ABCD$ be a rectangle.
{{AimForCont}} $ABCD$ is ''not'' a parallelogram.
{{WLOG}}, let line segments $AB$ and $CD$ ''not'' be parallel.
By {{EuclidPostulateLink|Second}}, let us produce $AB$ and $CD$ into two infinite straight lines.
By the Parallel Postulate, the $AD$ and $BC$ will eventually meet at one side or t... | Let $ABCD$ be a [[Definition:Rectangle|rectangle]].
Then $ABCD$ is a [[Definition:Parallelogram|parallelogram]]. | Let $ABCD$ be a [[Definition:Rectangle|rectangle]].
{{AimForCont}} $ABCD$ is ''not'' a [[Definition:Parallelogram|parallelogram]].
{{WLOG}}, let [[Definition:Line Segment|line segments]] $AB$ and $CD$ ''not'' be [[Definition:Parallel Lines|parallel]].
By {{EuclidPostulateLink|Second}}, let us [[Definition:Productio... | Rectangle is Parallelogram | https://proofwiki.org/wiki/Rectangle_is_Parallelogram | https://proofwiki.org/wiki/Rectangle_is_Parallelogram | [
"Rectangles",
"Parallelograms"
] | [
"Definition:Quadrilateral/Rectangle",
"Definition:Quadrilateral/Parallelogram"
] | [
"Definition:Quadrilateral/Rectangle",
"Definition:Quadrilateral/Parallelogram",
"Definition:Line/Segment",
"Definition:Parallel (Geometry)/Lines",
"Definition:Production",
"Definition:Line/Infinite Straight Line",
"Axiom:Parallel Postulate",
"Definition:Intersection (Geometry)",
"Definition:Triangle... |
proofwiki-9991 | Condition for Denesting of Square Root | Let $a, b \in \Q_{\ge 0}$.
Suppose $\sqrt b \notin \Q$.
Then:
:$\exists p, q \in \Q: \sqrt {a + \sqrt b} = \sqrt p + \sqrt q$
{{iff}}:
:$\exists n \in \Q: a^2 - b = n^2$. | === Lemma ===
{{:Condition for Denesting of Square Root/Lemma}}{{qed|lemma}} | Let $a, b \in \Q_{\ge 0}$.
Suppose $\sqrt b \notin \Q$.
Then:
:$\exists p, q \in \Q: \sqrt {a + \sqrt b} = \sqrt p + \sqrt q$
{{iff}}:
:$\exists n \in \Q: a^2 - b = n^2$. | === [[Condition for Denesting of Square Root/Lemma|Lemma]] ===
{{:Condition for Denesting of Square Root/Lemma}}{{qed|lemma}} | Condition for Denesting of Square Root | https://proofwiki.org/wiki/Condition_for_Denesting_of_Square_Root | https://proofwiki.org/wiki/Condition_for_Denesting_of_Square_Root | [
"Square Roots",
"Condition for Denesting of Square Root"
] | [] | [
"Condition for Denesting of Square Root/Lemma",
"Condition for Denesting of Square Root/Lemma",
"Condition for Denesting of Square Root/Lemma"
] |
proofwiki-9992 | Characteristic Function of Union/Variant 1 | :$\chi_{A \mathop \cup B} = \min \set {\chi_A + \chi_B, 1}$ | By Characteristic Function Determined by 1-Fiber, it suffices to show:
:$\min \set {\map {\chi_A} s + \map {\chi_B} s, 1} = 1 \iff s \in A \cup B$
By the nature of the min operation, this amounts to showing that:
:$\map {\chi_A} s + \map {\chi_B} s \ge 1 \iff s \in A \cup B$
As $\chi_A, \chi_B$ are characteristic funct... | :$\chi_{A \mathop \cup B} = \min \set {\chi_A + \chi_B, 1}$ | By [[Characteristic Function Determined by 1-Fiber]], it suffices to show:
:$\min \set {\map {\chi_A} s + \map {\chi_B} s, 1} = 1 \iff s \in A \cup B$
By the nature of the [[Definition:Min Operation|min operation]], this amounts to showing that:
:$\map {\chi_A} s + \map {\chi_B} s \ge 1 \iff s \in A \cup B$
As $\ch... | Characteristic Function of Union/Variant 1 | https://proofwiki.org/wiki/Characteristic_Function_of_Union/Variant_1 | https://proofwiki.org/wiki/Characteristic_Function_of_Union/Variant_1 | [
"Characteristic Function of Union"
] | [] | [
"Characteristic Function Determined by 1-Fiber",
"Definition:Min Operation",
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Set Union"
] |
proofwiki-9993 | Characteristic Function of Union/Variant 2 | :$\chi_{A \mathop \cup B} = \chi_A + \chi_B - \chi_{A \mathop \cap B}$ | From Subset of Union:
:$A, B \subseteq A \cup B$
From Intersection is Subset of Union:
:$A \cap B \subseteq A \cup B$
Thus from Characteristic Function of Subset:
:$\map {\chi_{A \mathop \cup B} } s = 0 \implies \map {\chi_A} s = \map {\chi_B} s = \map {\chi_{A \mathop \cap B} } s = 0$
Now suppose that $\map {\chi_A} s... | :$\chi_{A \mathop \cup B} = \chi_A + \chi_B - \chi_{A \mathop \cap B}$ | From [[Subset of Union]]:
:$A, B \subseteq A \cup B$
From [[Intersection is Subset of Union]]:
:$A \cap B \subseteq A \cup B$
Thus from [[Characteristic Function of Subset]]:
:$\map {\chi_{A \mathop \cup B} } s = 0 \implies \map {\chi_A} s = \map {\chi_B} s = \map {\chi_{A \mathop \cap B} } s = 0$
Now suppose that... | Characteristic Function of Union/Variant 2 | https://proofwiki.org/wiki/Characteristic_Function_of_Union/Variant_2 | https://proofwiki.org/wiki/Characteristic_Function_of_Union/Variant_2 | [
"Characteristic Function of Union"
] | [] | [
"Set is Subset of Union",
"Intersection is Subset of Union",
"Characteristic Function of Subset",
"Definition:Characteristic Function (Set Theory)/Set",
"Characteristic Function Determined by 0-Fiber"
] |
proofwiki-9994 | Characteristic Function of Union/Variant 3 | :$\chi_{A \mathop \cup B} = \max \set {\chi_A, \chi_B}$ | Suppose $\map {\chi_{A \mathop \cup B} } s = 0$.
Then $s \notin A \cup B$, so $s \notin A$ and $s \notin B$.
Hence:
:$\map {\chi_A} s = \map {\chi_B} s = 0$
and by definition of max operation:
:$\max \set {\map {\chi_A} s, \map {\chi_B} s} = 0$
Conversely, suppose:
:$\max \set {\map {\chi_A} s, \map {\chi_B} s} = 0$
Th... | :$\chi_{A \mathop \cup B} = \max \set {\chi_A, \chi_B}$ | Suppose $\map {\chi_{A \mathop \cup B} } s = 0$.
Then $s \notin A \cup B$, so $s \notin A$ and $s \notin B$.
Hence:
:$\map {\chi_A} s = \map {\chi_B} s = 0$
and by definition of [[Definition:Max Operation|max operation]]:
:$\max \set {\map {\chi_A} s, \map {\chi_B} s} = 0$
Conversely, suppose:
:$\max \set {\map {... | Characteristic Function of Union/Variant 3 | https://proofwiki.org/wiki/Characteristic_Function_of_Union/Variant_3 | https://proofwiki.org/wiki/Characteristic_Function_of_Union/Variant_3 | [
"Characteristic Function of Union"
] | [] | [
"Definition:Max Operation",
"Definition:Characteristic Function (Set Theory)/Set",
"Characteristic Function Determined by 0-Fiber"
] |
proofwiki-9995 | Characteristic Function of Intersection/Variant 1 | :$\chi_{A \mathop \cap B} = \chi_A \chi_B$ | By Characteristic Function Determined by 1-Fiber, it suffices to show that:
:$\map {\chi_A} s \map {\chi_B} s = 1 \iff s \in A \cap B$
Now, both $\chi_A$ and $\chi_B$ are characteristic functions.
It follows that, for any $s \in S$:
:$\map {\chi_A} s \map {\chi_B} s = 1 \iff \map {\chi_A} s = \map {\chi_B} s = 1$
By de... | :$\chi_{A \mathop \cap B} = \chi_A \chi_B$ | By [[Characteristic Function Determined by 1-Fiber]], it suffices to show that:
:$\map {\chi_A} s \map {\chi_B} s = 1 \iff s \in A \cap B$
Now, both $\chi_A$ and $\chi_B$ are [[Definition:Characteristic Function of Set|characteristic functions]].
It follows that, for any $s \in S$:
:$\map {\chi_A} s \map {\chi_B} ... | Characteristic Function of Intersection/Variant 1 | https://proofwiki.org/wiki/Characteristic_Function_of_Intersection/Variant_1 | https://proofwiki.org/wiki/Characteristic_Function_of_Intersection/Variant_1 | [
"Characteristic Function of Intersection"
] | [] | [
"Characteristic Function Determined by 1-Fiber",
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Set Intersection"
] |
proofwiki-9996 | Characteristic Function of Intersection/Variant 2 | :$\chi_{A \mathop \cap B} = \min \set {\chi_A, \chi_B}$ | By Characteristic Function Determined by 1-Fiber, it suffices to show that:
:$\min \set {\map {\chi_A} s, \map {\chi_B} s} = 1 \iff s \in A \cap B$
By definition of characteristic function, we have:
:$\min \set {\map {\chi_A} s, \map {\chi_B} s} = 1$
{{iff}}:
:$\map {\chi_A} s = \map {\chi_B} s = 1$
because $\chi_A, \c... | :$\chi_{A \mathop \cap B} = \min \set {\chi_A, \chi_B}$ | By [[Characteristic Function Determined by 1-Fiber]], it suffices to show that:
:$\min \set {\map {\chi_A} s, \map {\chi_B} s} = 1 \iff s \in A \cap B$
By definition of [[Definition:Characteristic Function of Set|characteristic function]], we have:
:$\min \set {\map {\chi_A} s, \map {\chi_B} s} = 1$
{{iff}}:
:$\map... | Characteristic Function of Intersection/Variant 2 | https://proofwiki.org/wiki/Characteristic_Function_of_Intersection/Variant_2 | https://proofwiki.org/wiki/Characteristic_Function_of_Intersection/Variant_2 | [
"Characteristic Function of Intersection"
] | [] | [
"Characteristic Function Determined by 1-Fiber",
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Set Intersection"
] |
proofwiki-9997 | Tangent Space is Vector Space | Let $M$ be a smooth manifold of dimension $n \in \N$.
Let $m \in M$ be a point.
Let $\struct {U, \kappa}$ be a chart with $m \in U$.
Let $T_m M$ be the tangent space at $m$.
Then $T_m M$ is a real vector space of dimension $n$, spanned by the basis:
:$\set {\valueat {\dfrac \partial {\partial \kappa^i} } m : i \in... | Let $V$ be an open neighborhood of $m$ with $V \subseteq U \subseteq M$.
Let $\map {C^\infty} {V, \R}$ be the set of smooth mappings $f: V \to \R$.
Let $X_m, Y_m \in T_m M$.
Let $\lambda \in \R$.
Then, by definition of tangent vector and Equivalence of Definitions of Tangent Vector:
:$X_m, Y_m$ are linear transformat... | Let $M$ be a [[Definition:Smooth Manifold|smooth manifold]] of [[Definition:Dimension of Topological Manifold|dimension]] $n \in \N$.
Let $m \in M$ be a [[Definition:Point|point]].
Let $\struct {U, \kappa}$ be a [[Definition:Chart|chart]] with $m \in U$.
Let $T_m M$ be the [[Definition:Tangent Space|tangent space... | Let $V$ be an [[Definition:Open Neighborhood of Point|open neighborhood]] of $m$ with $V \subseteq U \subseteq M$.
Let $\map {C^\infty} {V, \R}$ be the [[Definition:Set|set]] of [[Definition:Smooth Mapping|smooth mappings]] $f: V \to \R$.
Let $X_m, Y_m \in T_m M$.
Let $\lambda \in \R$.
Then, by definition of [[D... | Tangent Space is Vector Space | https://proofwiki.org/wiki/Tangent_Space_is_Vector_Space | https://proofwiki.org/wiki/Tangent_Space_is_Vector_Space | [
"Smooth Manifolds",
"Tangent Spaces",
"Vector Spaces"
] | [
"Definition:Topological Manifold/Smooth Manifold",
"Definition:Dimension (Topology)/Topological Manifold",
"Definition:Point",
"Definition:Chart",
"Definition:Tangent Space",
"Definition:Real Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Generated Submodule/Linear Span",
"Defini... | [
"Definition:Open Neighborhood/Point",
"Definition:Set",
"Definition:Smooth Mapping",
"Definition:Tangent Vector",
"Equivalence of Definitions of Tangent Vector",
"Definition:Linear Transformation",
"Definition:Linear Transformation",
"Definition:Leibniz Law",
"Definition:Leibniz Law",
"Definition:... |
proofwiki-9998 | Multiplication of Real Numbers is Right Distributive over Subtraction | {{:Euclid:Proposition/V/6}}
That is, for any number $a$ and for any integers $m, n$:
:$m a - n a = \paren {m - n} a$ | Let two magnitudes $AB, CD$ be equimultiples of two magnitudes $E, F$.
Let $AG, CH$ subtracted from them be equimultiples of the same two $E, F$.
We need to show that the remainders $GB, HD$ are either equal to $E, F$ or are equimultiples of them.
:350px
First let $GB = E$.
Let $CK$ be made equal to $F$.
We have that $... | {{:Euclid:Proposition/V/6}}
That is, for any [[Definition:Number|number]] $a$ and for any [[Definition:Integer|integers]] $m, n$:
:$m a - n a = \paren {m - n} a$ | Let two [[Definition:Strictly Positive Real Number|magnitudes]] $AB, CD$ be [[Definition:Equimultiples|equimultiples]] of two magnitudes $E, F$.
Let $AG, CH$ subtracted from them be [[Definition:Equimultiples|equimultiples]] of the same two $E, F$.
We need to show that the remainders $GB, HD$ are either equal to $E, ... | Multiplication of Real Numbers is Right Distributive over Subtraction | https://proofwiki.org/wiki/Multiplication_of_Real_Numbers_is_Right_Distributive_over_Subtraction | https://proofwiki.org/wiki/Multiplication_of_Real_Numbers_is_Right_Distributive_over_Subtraction | [
"Real Multiplication",
"Real Subtraction",
"Examples of Distributive Operations",
"Multiplication of Real Numbers Distributes over Subtraction"
] | [
"Definition:Number",
"Definition:Integer"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Equimultiples",
"Definition:Equimultiples",
"Definition:Equimultiples",
"File:Euclid-V-6.png",
"Distributive Laws/Arithmetic",
"Definition:Strictly Positive/Real Number"
] |
proofwiki-9999 | Sum of Square Roots as Square Root of Sum | :$\sqrt a + \sqrt b = \sqrt {a + b + \sqrt {4 a b} }$ | {{begin-eqn}}
{{eqn | l = \sqrt a + \sqrt b
| r = \sqrt {\paren {\sqrt a + \sqrt b}^2}
}}
{{eqn | r = \sqrt {\sqrt a^2 + \sqrt b^2 + 2 \sqrt a \sqrt b}
| c = Square of Sum
}}
{{eqn | r = \sqrt {a + b + \sqrt {4 a b} }
| c = Power of Product
}}
{{end-eqn}}
{{qed}}
Category:Algebra
grdxa15a4j79pyvvs7dfy... | :$\sqrt a + \sqrt b = \sqrt {a + b + \sqrt {4 a b} }$ | {{begin-eqn}}
{{eqn | l = \sqrt a + \sqrt b
| r = \sqrt {\paren {\sqrt a + \sqrt b}^2}
}}
{{eqn | r = \sqrt {\sqrt a^2 + \sqrt b^2 + 2 \sqrt a \sqrt b}
| c = [[Square of Sum]]
}}
{{eqn | r = \sqrt {a + b + \sqrt {4 a b} }
| c = [[Power of Product]]
}}
{{end-eqn}}
{{qed}}
[[Category:Algebra]]
grdxa15a... | Sum of Square Roots as Square Root of Sum | https://proofwiki.org/wiki/Sum_of_Square_Roots_as_Square_Root_of_Sum | https://proofwiki.org/wiki/Sum_of_Square_Roots_as_Square_Root_of_Sum | [
"Algebra"
] | [] | [
"Square of Sum",
"Exponent Combination Laws/Power of Product",
"Category:Algebra"
] |
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