hcju/proofwikips huggingface dataset:
Area of Parallelogram/Parallelogram
Let $ABCD$ be a parallelogram whose adjacent sides are of length $a$ and $b$ enclosing an angle $\theta$. The area of $ABCD$ equals the product of one of its bases and the associated altitude: {{begin-eqn}} {{eqn | l = \map \Area {ABCD}       | r = b h       | c =  }} {{eqn | r = a b \sin \theta       | c =  }} {{end-eqn}} where: :$b$ is the side of $ABCD$ which has been chosen to be the base :$h$ is the altitude of $ABCD$ from $b$.
Perimeter of Rectangle
Let $ABCD$ be a rectangle whose side lengths are $a$ and $b$. The perimeter of $ABCD$ is $2 a + 2 b$.
Perimeter of Parallelogram
Let $ABCD$ be a parallelogram whose side lengths are $a$ and $b$. The perimeter of $ABCD$ is $2 a + 2 b$.
Perimeter of Triangle
Let $ABC$ be a triangle. Then the perimeter $P$ of $ABC$ is given by: :$P = a + b + c$ where $a, b, c$ are the lengths of the sides of $ABC$.
Area of Trapezoid
:410px Let $ABCD$ be a trapezoid: :whose parallel sides are of lengths $a$ and $b$ and :whose height is $h$. Then the area of $ABCD$ is given by: :$\Box ABCD = \dfrac {h \paren {a + b} } 2$
Perimeter of Trapezoid
:400px Let $ABCD$ be a trapezoid: :whose parallel sides are of lengths $a$ and $b$ :whose height is $h$. and :whose non-parallel sides are at angles $\theta$ and $\phi$ with the parallels. The perimeter $P$ of $ABCD$ is given by: :$P = a + b + h \paren {\csc \theta + \csc \phi}$ where $\csc$ denotes cosecant.
Perimeter of Regular Polygon
Let $P$ be a regular $n$-sided polygon whose side length is $b$. Then the perimeter $L$ of $P$ is given by: :$L = n b$
Commensurability is Transitive
Let $a$, $b$, $c$ be three real numbers. Let $a$ and $b$ be commensurable, and $b$ and $c$ be commensurable. Then $a$ and $c$ are commensurable.
Arc Length of Sector
Let $\mathcal C = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$. Let $BAC$ be the sector of $\mathcal C$ whose angle between $AB$ and $AC$ is $\theta$. :300px Then the length $s$ of arc $BC$ is given by: :$\mathcal s = r \theta$ where: :$r = AB$ is the length of the radius of the circle :$\theta$ is measured in radians.
Length of Inradius of Triangle
Let $\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$. Then the length of the inradius $r$ of $\triangle ABC$ is given by: :$r = \dfrac {\sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} } } s$ where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.
Length of Circumradius of Triangle
Let $\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$. Then the length of the circumradius $R$ of $\triangle ABC$ is given by: :$R = \dfrac {abc} {4 \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} } }$ where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.
Area of Triangle in Terms of Inradius
Let $\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$. Then the area $\mathcal A$ of $\triangle ABC$ is given by: :$\mathcal A = r s$ where: :$r$ is the inradius of $\triangle ABC$ :$s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.
Area of Regular Polygon by Circumradius
Let $P$ be a regular $n$-gon. Let $C$ be a circumcircle of $P$. Let the radius of $C$ be $r$. Then the area $\mathcal A$ of $P$ is given by: :$\mathcal A = \dfrac 1 2 n r^2 \sin \dfrac {2 \pi} n$
Perimeter of Regular Polygon by Inradius
Let $P$ be a regular $n$-gon. Let $C$ be an incircle of $P$. Let the radius of $C$ be $r$. Then the perimeter $\mathcal P$ of $P$ is given by: :$\mathcal P = 2 n r \tan \dfrac \pi n$
Image of Interval by Continuous Function is Interval/Proof 1
Let $I$ be a real interval. Let $f: I \to \R$ be a continuous real function. Then the image of $f$ is a real interval.
Image of Interval by Continuous Function is Interval/Proof 2
Let $I$ be a real interval. Let $f: I \to \R$ be a continuous real function. Then the image of $f$ is a real interval.
Area of Isosceles Triangle
Let $\triangle ABC$ be an isosceles triangle whose apex is $A$. Let $\theta$ be the angle of the apex $A$. Let $r$ be the length of a leg of $\triangle ABC$. Then the area $\mathcal A$ of $\triangle ABC$ is given by: :$\mathcal A = \dfrac 1 2 r^2 \sin \theta$
Area of Segment of Circle
Let $C$ be a circle of radius $r$. Let $S$ be a segment of $C$ such that its base subtends an angle of $\theta$ at the center of the circle. Then the area $\AA$ of $S$ is given by: :$\AA = \dfrac 1 2 r^2 \paren {\theta - \sin \theta}$ where $\theta$ is measured in radians.
Mills' Theorem
There exists a real number $A$ such that $\left\lfloor{A^{3^n} }\right\rfloor$ is a prime number for all $n \in \N_{>0}$, where: :$\left\lfloor{x}\right\rfloor$ denotes the floor function of $x$ :$\N$ denotes the set of all natural numbers.
Equation of Ellipse in Reduced Form
Let $K$ be an ellipse aligned in a cartesian plane in reduced form. Let: :the major axis of $K$ have length $2 a$ :the minor axis of $K$ have length $2 b$.
Area of Ellipse
Let $K$ be an ellipse whose major axis is of length $2 a$ and whose minor axis is of length $2 b$. The area $\mathcal A$ of $K$ is given by: : $\mathcal A = \pi a b$
Equation of Ellipse in Reduced Form/Cartesian Frame
The equation of $K$ is: :$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
Equation of Ellipse in Reduced Form/Cartesian Frame/Parametric Form
The equation of $K$ in parametric form is: :$x = a \cos \theta, y = b \sin \theta$
Perimeter of Ellipse
Let $K$ be an ellipse whose major axis is of length $2 a$ and whose minor axis is of length $2 b$. The perimeter $\PP$ of $K$ is given by: :$\ds \PP = 4 a \int_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \theta} \rd \theta$ where: :$k = \dfrac {\sqrt {a^2 - b^2} } a$ The definite integral: :$\displaystyle \PP = \int_0^{\pi / 2} \sqrt{1 - k^2 \sin^2 \theta} \rd \theta$ is the complete elliptic integral of the second kind.
Distance of Point from Origin in Cartesian Coordinates
Let $P = \tuple {x, y}$ be a point in the cartesian plane. Then $P$ is at a distance of $\sqrt {x^2 + y^2}$ from the origin.
Composite of Monomorphisms is Monomorphism
Let: :$\struct {S_1, \circ_1, \circ_2, \ldots, \circ_n}$ :$\struct {S_2, *_1, *_2, \ldots, *_n}$ :$\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$ be algebraic structures. Let: :$\phi: \struct {S_1, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$ :$\psi: \struct {S_2, *_1, *_2, \ldots, *_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$ be monomorphisms. Then the composite of $\phi$ and $\psi$ is also a monomorphism.
Composite of Automorphisms is Automorphism
Let $\struct {S, \circ_1, \circ_2, \ldots, \circ_n}$ be an algebraic structure. Let: :$\phi: \struct {S, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S, \circ_1, \circ_2, \ldots, \circ_n}$ :$\psi: \struct {S, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S, \circ_1, \circ_2, \ldots, \circ_n}$ be automorphisms. Then the composite of $\phi$ and $\psi$ is also an automorphism.
Composite of Endomorphisms is Endomorphism
Let $\left({S, \circ_1, \circ_2, \ldots, \circ_n}\right)$ be an algebraic structure. Let: : $\phi: \left({S, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S, \circ_1, \circ_2, \ldots, \circ_n}\right)$ : $\psi: \left({S, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S, \circ_1, \circ_2, \ldots, \circ_n}\right)$ be endomorphisms. Then the composite of $\phi$ and $\psi$ is also an endomorphism.
Sine of Angle in Cartesian Plane
Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\sin \theta = \dfrac y r$ where $\sin$ denotes the sine of $\theta$.
Equivalence of Definitions of Sine of Angle
Let $\theta$ be an angle. {{TFAE|def = Sine of Angle|view = sine of $\theta$}}
Cardinality of Subset Relation on Power Set of Finite Set
Let $S$ be a set such that: :$\card S = n$ where $\card S$ denotes the cardinality of $S$. From Subset Relation on Power Set is Partial Ordering we have that $\struct {\powerset S, \subseteq}$ is an ordered set. The cardinality of $\subseteq$ as a relation is $3^n$.
Quotient Structure of Semigroup is Semigroup
Let $\RR$ be a congruence relation on a semigroup $\struct {S, \circ}$. Then the quotient structure $\struct {S / \RR, \circ_\RR}$ is a semigroup.
Quotient Structure of Monoid is Monoid
Let $\RR$ be a congruence relation on a monoid $\struct {S, \circ}$ with an identity $e$. Then the quotient structure $\struct {S / \RR, \circ_\RR}$ is a monoid.
Quotient Structure of Group is Group
Let $\RR$ be a congruence relation on a group $\struct {G, \circ}$. Then the quotient structure $\struct {G / \RR, \circ_\RR}$ is a group.
Quotient Structure of Abelian Group is Abelian Group
Let $\RR$ be a congruence relation on an abelian group $\struct {G, \circ}$. Then the quotient structure $\struct {G / \RR, \circ_\RR}$ is an abelian group.
Quotient Structure is Similar to Structure
Let $\RR$ be a congruence relation on a algebraic structure $\struct {G, \circ}$. {{MissingLinks|"similar" structure?}} Then the quotient structure $\struct {G / \RR, \circ_\RR}$ is a similar structure to $\struct {G, \circ}$.
Shape of Secant Function
The nature of the secant function on the set of real numbers $\R$ is as follows: :$(1): \quad \sec x$ is continuous and strictly increasing on the intervals $\hointr 0 {\dfrac \pi 2}$ and $\hointl {\dfrac \pi 2} \pi$ :$(2): \quad \sec x$ is continuous and strictly decreasing on the intervals $\hointr {-\pi} {-\dfrac \pi 2}$ and $\hointl {-\dfrac \pi 2} 0$ :$(3): \quad \sec x \to + \infty$ as $x \to -\dfrac \pi 2^+$ :$(4): \quad \sec x \to + \infty$ as $x \to \dfrac \pi 2^-$ :$(5): \quad \sec x \to - \infty$ as $x \to \dfrac \pi 2^+$ :$(6): \quad \sec x \to - \infty$ as $x \to \dfrac {3 \pi} 2^-$
Shape of Cosecant Function
The nature of the cosecant function on the set of real numbers $\R$ is as follows: :$(1): \quad$ strictly decreasing on the intervals $\hointr {-\dfrac \pi 2} 0$ and $\hointl 0 {\dfrac \pi 2}$ :$(2): \quad$ strictly increasing on the intervals $\hointr {\dfrac \pi 2} \pi$ and $\hointl \pi {\dfrac {3 \pi} 2}$ :$(3): \quad$ $\csc x \to +\infty$ as $x \to 0^+$ :$(4): \quad$ $\csc x \to +\infty$ as $x \to \pi^-$ :$(5): \quad$ $\csc x \to -\infty$ as $x \to \pi^+$ :$(6): \quad$ $\csc x \to -\infty$ as $x \to 2 \pi^-$
Integer Multiples under Multiplication form Semigroup
Let $n \Z$ be the set of integer multiples of $n$. Then $\struct {n \Z, \times}$ is a semigroup. If $\size n > 1$ then $\struct {n \Z, \times}$ has no identity.
Order of External Direct Product
Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be algebraic structures. Then the order of $\struct {S \times T, \circ}$ is $\card S \times \card T$.
Congruence (Number Theory) is Congruence Relation
Congruence modulo $m$ is a congruence relation on $\left({\Z, +}\right)$.
Equivalence of Definitions of Cotangent of Angle
Let $\theta$ be an angle. {{TFAE|def = Cotangent of Angle|cotangent}}
Equivalence of Definitions of Cosecant of Angle
Let $\theta$ be an angle. {{TFAE|def = Cosecant of Angle|view = cosecant}}
Cosine of Angle in Cartesian Plane
Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\cos \theta = \dfrac x r$ where $\cos$ denotes the cosine of $\theta$.
Tangent of Angle in Cartesian Plane
Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\tan \theta = \dfrac y x$ where $\tan$ denotes the tangent of $\theta$.
Cotangent of Angle in Cartesian Plane
Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\cot \theta = \dfrac x y$ where $\cot$ denotes the cotangent of $\theta$.
Secant of Angle in Cartesian Plane
Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\sec \theta = \dfrac r x$ where $\sec$ denotes the secant of $\theta$.
Cosecant of Angle in Cartesian Plane
Let $P = \tuple {x, y}$ be a point in the cartesian plane whose origin is at $O$. Let $\theta$ be the angle between the $x$-axis and the line $OP$. Let $r$ be the length of $OP$. Then: :$\csc \theta = \dfrac r x$ where $\csc$ denotes the secant of $\theta$.
Tangent is Sine divided by Cosine
Let $\theta$ be an angle such that $\cos \theta \ne 0$. Then: :$\tan \theta = \dfrac {\sin \theta} {\cos \theta}$ where $\tan$, $\sin$ and $\cos$ mean tangent, sine and cosine respectively.
Cotangent is Cosine divided by Sine
Let $\theta$ be an angle such that $\sin \theta \ne 0$. Then: :$\cot \theta = \dfrac {\cos \theta} {\sin \theta}$ where $\cot$, $\sin$ and $\cos$ mean cotangent, sine and cosine respectively.
Cotangent is Reciprocal of Tangent
:$\cot \theta = \dfrac 1 {\tan \theta}$
Secant is Reciprocal of Cosine
:$\sec \theta = \dfrac 1 {\cos \theta}$
Cosecant is Reciprocal of Sine
:$\csc \theta = \dfrac 1 {\sin \theta}$
Particular Values of Sine Function
The following values of the sine function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Sine of Zero === {{:Sine of Zero is Zero}} === Sine of 15 Degrees === {{:Sine of 15 Degrees}} === Sine of 30 Degrees === {{:Sine of 30 Degrees}} === Sine of 45 Degrees === {{:Sine of 45 Degrees}} === Sine of 60 Degrees === {{:Sine of 60 Degrees}} === Sine of 75 Degrees === {{:Sine of 75 Degrees}} === Sine of Right Angle === {{:Sine of Right Angle}} === Sine of 105 Degrees === {{:Sine of 105 Degrees}} === Sine of 120 Degrees === {{:Sine of 120 Degrees}} === Sine of 135 Degrees === {{:Sine of 135 Degrees}} === Sine of 150 Degrees === {{:Sine of 150 Degrees}} === Sine of 165 Degrees === {{:Sine of 165 Degrees}} === Sine of Straight Angle === {{:Sine of Straight Angle}} === Sine of 195 Degrees === {{:Sine of 195 Degrees}} === Sine of 210 Degrees === {{:Sine of 210 Degrees}} === Sine of 225 Degrees === {{:Sine of 225 Degrees}} === Sine of 240 Degrees === {{:Sine of 240 Degrees}} === Sine of 255 Degrees === {{:Sine of 255 Degrees}} === Sine of Three Right Angles === {{:Sine of Three Right Angles}} === Sine of 285 Degrees === {{:Sine of 285 Degrees}} === Sine of 300 Degrees === {{:Sine of 300 Degrees}} === Sine of 315 Degrees === {{:Sine of 315 Degrees}} === Sine of 330 Degrees === {{:Sine of 330 Degrees}} === Sine of 345 Degrees === {{:Sine of 345 Degrees}} === Sine of Full Angle === {{:Sine of Full Angle}}
Positive-Term Generalized Sum Converges iff Supremum
Let $\struct {G, \circ, \le}$ be an abelian totally ordered group, considered under the order topology. Let $\set {x_i: i \in I}$ be an indexed set of positive elements of $G$. {{explain|What actually does "positive" mean in this context?}} Then: :the generalized sum $\displaystyle \sum \set {x_i: i \in I}$ converges to a point $x \in G$  :$x$ is the supremum of: ::$P := \displaystyle \set {\sum_{i \mathop \in F} x_i: \text{$F \subseteq I$ and $F$ is finite} }$
Half Angle Formulas/Sine
{{begin-eqn}} {{eqn | l = \sin \frac \theta 2       | r = +\sqrt {\frac {1 - \cos \theta} 2}       | c = for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {II}$ }} {{eqn | l = \sin \frac \theta 2       | r = -\sqrt {\dfrac {1 - \cos \theta} 2}       | c = for $\dfrac \theta 2$ in quadrant $\text {III}$ or quadrant $\text {IV}$ }} {{end-eqn}}
Half Angle Formulas/Cosine
{{begin-eqn}} {{eqn | l = \cos \frac \theta 2       | r = +\sqrt {\frac {1 + \cos \theta} 2}       | c = for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {IV}$ }} {{eqn | l = \cos \frac \theta 2       | r = -\sqrt {\frac {1 + \cos \theta} 2}       | c = for $\dfrac \theta 2$ in quadrant $\text {II}$ or quadrant $\text {III}$ }} {{end-eqn}}
Half Angle Formulas/Tangent
{{begin-eqn}} {{eqn | l = \tan \frac \theta 2       | r = +\sqrt {\dfrac {1 - \cos \theta} {1 + \cos \theta} }       | c = for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {III}$ }} {{eqn | l = \tan \frac \theta 2       | r = -\sqrt {\dfrac {1 - \cos \theta} {1 + \cos \theta} }       | c = for $\dfrac \theta 2$ in quadrant $\text {II}$ or quadrant $\text {IV}$ }} {{end-eqn}} where $\tan$ denotes tangent and $\cos$ denotes cosine. When $\theta = \paren {2 k + 1} \pi$, $\tan \dfrac \theta 2$ is undefined.
Kernel of Normal Operator is Kernel of Adjoint
Let $H$ be a Hilbert space. Let $A \in \map B H$ be a normal operator. Then: :$\ker A = \ker A^*$ where: :$\ker$ denotes kernel :$A^*$ denotes the adjoint of $A$.
Half Angle Formulas/Tangent/Corollary 1
:$\tan \dfrac \theta 2 = \dfrac {\sin \theta} {1 + \cos \theta}$
Half Angle Formulas/Tangent/Corollary 2
:$\tan \dfrac \theta 2 = \dfrac {1 - \cos \theta} {\sin \theta}$
Half Angle Formulas/Tangent/Corollary 3
:$\tan \dfrac \theta 2 = \csc \theta - \cot \theta$
Bisection of Angle in Cartesian Plane
Let $\theta$ be the angular coordinate of a point $P$ in a polar coordinate plane. Let $QOR$ be a straight line that bisects the angle $\theta$. Then the angular coordinates of $Q$ and $R$ are $\dfrac \theta 2$ and $\pi + \dfrac \theta 2$.
Sine of 15 Degrees
:$\sin 15 \degrees = \sin \dfrac \pi {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$
Sine of 30 Degrees
:$\sin 30 \degrees = \sin \dfrac \pi 6 = \dfrac 1 2$
Sine of 45 Degrees
:$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$
Sine of 60 Degrees
:$\sin 60 \degrees = \sin \dfrac \pi 3 = \dfrac {\sqrt 3} 2$
Sine of 75 Degrees
:$\sin 75 \degrees = \sin \dfrac {5 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$
Sine of Right Angle
:$\sin 90^\circ = \sin \dfrac \pi 2 = 1$
Sine of Angle plus Right Angle
: $\sin \paren {x + \dfrac \pi 2} = \cos x$
Sine of 105 Degrees
:$\sin 105^\circ = \sin \dfrac {7 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$
Sine of 120 Degrees
:$\sin 120 \degrees = \sin \dfrac {2 \pi} 3 = \dfrac {\sqrt 3} 2$
Sine of 135 Degrees
:$\sin 135 \degrees = \sin \dfrac {3 \pi} 4 = \dfrac {\sqrt 2} 2$
Sine of 150 Degrees
:$\sin 150 \degrees = \sin \dfrac {5 \pi} 6 = \dfrac 1 2$
Sine of 165 Degrees
:$\sin 165 \degrees = \sin \dfrac {11 \pi} {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$
Sine of Straight Angle
:$\sin 180 \degrees = \sin \pi = 0$
Sine of Supplementary Angle
:$\sin \paren {\pi - \theta} = \sin \theta$ where $\sin$ denotes sine. That is, the sine of an angle equals its supplement.
Cosine of Supplementary Angle
:$\map \cos {\pi - \theta} = -\cos \theta$ where $\cos$ denotes cosine. That is, the cosine of an angle is the negative of its supplement.
Sine of Conjugate Angle
:$\map \sin {2 \pi - \theta} = -\sin \theta$ where $\sin$ denotes sine. That is, the sine of an angle is the negative of its conjugate.
Cosine of Conjugate Angle
:$\map \cos {2 \pi - \theta} = \cos \theta$ where $\cos$ denotes cosine. That is, the cosine of an angle equals its conjugate.
Sine of 195 Degrees
:$\sin 195 \degrees = \sin \dfrac {13 \pi} {12} = -\dfrac {\sqrt 6 - \sqrt 2} 4$
Sine of 210 Degrees
:$\sin 210 \degrees = \sin \dfrac {7 \pi} 6 = -\dfrac 1 2$
Sine of 225 Degrees
:$\sin 225 \degrees = \sin \dfrac {5 \pi} 4 = -\dfrac {\sqrt 2} 2$
Sine of 240 Degrees
:$\sin 240 \degrees = \sin \dfrac {4 \pi} 3 = -\dfrac {\sqrt 3} 2$
Sine of 255 Degrees
:$\sin 255^\circ = \sin \dfrac {17 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$
Sine of Three Right Angles
:$\sin 270 \degrees = \sin \dfrac {3 \pi} 2 = -1$
Sine of 285 Degrees
:$\sin 285^\circ = \sin \dfrac {19 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$
Sine of 300 Degrees
:$\sin 300 \degrees = \sin \dfrac {5 \pi} 3 = -\dfrac {\sqrt 3} 2$
Sine of 315 Degrees
:$\sin 315 \degrees = \sin \dfrac {7 \pi} 4 = -\dfrac {\sqrt 2} 2$
Sine of 330 Degrees
:$\sin 330 \degrees = \sin \dfrac {11 \pi} 6 = -\dfrac 1 2$
Sine of 345 Degrees
:$\sin 345^\circ = \sin \dfrac {23 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$
Sine of Full Angle
:$\sin 360^\circ = \sin 2 \pi = 0$
Particular Values of Cosine Function
The following values of the cosine function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Cosine of Zero === {{:Cosine of Zero is One}} === Cosine of 15 Degrees === {{:Cosine of 15 Degrees}} === Cosine of 30 Degrees === {{:Cosine of 30 Degrees}} === Cosine of 45 Degrees === {{:Cosine of 45 Degrees}} === Cosine of 60 Degrees === {{:Cosine of 60 Degrees}} === Cosine of 75 Degrees === {{:Cosine of 75 Degrees}} === Cosine of Right Angle === {{:Cosine of Right Angle}} === Cosine of 105 Degrees === {{:Cosine of 105 Degrees}} === Cosine of 120 Degrees === {{:Cosine of 120 Degrees}} === Cosine of 135 Degrees === {{:Cosine of 135 Degrees}} === Cosine of 150 Degrees === {{:Cosine of 150 Degrees}} === Cosine of 165 Degrees === {{:Cosine of 165 Degrees}} === Cosine of Straight Angle === {{:Cosine of Straight Angle}} === Cosine of 195 Degrees === {{:Cosine of 195 Degrees}} === Cosine of 210 Degrees === {{:Cosine of 210 Degrees}} === Cosine of 225 Degrees === {{:Cosine of 225 Degrees}} === Cosine of 240 Degrees === {{:Cosine of 240 Degrees}} === Cosine of 255 Degrees === {{:Cosine of 255 Degrees}} === Cosine of Three Right Angles === {{:Cosine of Three Right Angles}} === Cosine of 285 Degrees === {{:Cosine of 285 Degrees}} === Cosine of 300 Degrees === {{:Cosine of 300 Degrees}} === Cosine of 315 Degrees === {{:Cosine of 315 Degrees}} === Cosine of 330 Degrees === {{:Cosine of 330 Degrees}} === Cosine of 345 Degrees === {{:Cosine of 345 Degrees}} === Cosine of Full Angle === {{:Cosine of Full Angle}}
Cosine of 15 Degrees
:$\cos 15 \degrees = \cos \dfrac \pi {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$
Cosine of 30 Degrees
:$\cos 30 \degrees = \cos \dfrac \pi 6 = \dfrac {\sqrt 3} 2$
Cosine of 45 Degrees
:$\cos 45 \degrees = \cos \dfrac \pi 4 = \dfrac {\sqrt 2} 2$
Cosine of 60 Degrees
:$\cos 60 \degrees = \cos \dfrac \pi 3 = \dfrac 1 2$
Cosine of 75 Degrees
:$\cos 75^\circ = \cos \dfrac {5 \pi}{12} = \dfrac {\sqrt 6 - \sqrt 2} 4$
Cosine of Right Angle
:$\cos 90^\circ = \cos \dfrac \pi 2 = 0$
Cosine of 105 Degrees
:$\cos 105 \degrees = \cos \dfrac {7 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$
Cosine of Angle plus Right Angle
:$\map \cos {x + \dfrac \pi 2} = -\sin x$
Cosine of 120 Degrees
:$\cos 120 \degrees = \cos \dfrac {2 \pi} 3 = -\dfrac 1 2$
Cosine of 135 Degrees
:$\cos 135 \degrees = \cos \dfrac {3 \pi} 4 = -\dfrac {\sqrt 2} 2$
Cosine of 150 Degrees
:$\cos 150 \degrees = \cos \dfrac {5 \pi} 6 = -\dfrac {\sqrt 3} 2$
Cosine of 165 Degrees
:$\cos 165 \degrees = \cos \dfrac {11 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$
Cosine of Straight Angle
:$\cos 180^\circ = \cos \pi = - 1$
Cosine of 195 Degrees
:$\cos 195 \degrees = \cos \dfrac {13 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$
Cosine of 210 Degrees
:$\cos 210 \degrees = \cos \dfrac {7 \pi} 6 = -\dfrac {\sqrt 3} 2$
Cosine of 225 Degrees
:$\cos 225 \degrees = \cos \dfrac {5 \pi} 4 = -\dfrac {\sqrt 2} 2$
Cosine of 240 Degrees
:$\cos 240 \degrees = \cos \dfrac {4 \pi} 3 = -\dfrac 1 2$
Cosine of 255 Degrees
:$\cos 255^\circ = \cos \dfrac {17 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$
Cosine of Three Right Angles
:$\cos 270^\circ = \cos \dfrac {3 \pi} 2 = 0$
Cosine of 285 Degrees
:$\cos 285^\circ = \cos \dfrac {19 \pi} {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$
Cosine of 300 Degrees
:$\cos 300 \degrees = \cos \dfrac {5 \pi} 3 = \dfrac 1 2$
Cosine of 315 Degrees
:$\cos 315 \degrees = \cos \dfrac {7 \pi} 4 = \dfrac {\sqrt 2} 2$
Cosine of 330 Degrees
:$\cos 330 \degrees = \cos \dfrac {11 \pi} 6 = \dfrac {\sqrt 3} 2$
Cosine of 345 Degrees
:$\cos 345 \degrees = \cos \dfrac {23 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$
Cosine of Full Angle
:$\cos 360 \degrees = \cos 2 \pi = 1$