Maths AA terms in Geometry and trigonometry

The situation in which two sides and a non-included acute angle of a triangle fit two different triangles, so the sine rule gives two possible values (acute and obtuse) for an angle.

Half the distance between the maximum and minimum values of a sinusoidal function; for $y=a\sin (b(x+c))+d$ it is $\mathrm{\mid }a\mathrm{\mid }$.

The angle between the line and its projection onto the plane (the shadow it casts straight down onto the plane). It always lies between $0\mathrm{°}$ and $90\mathrm{°}$.

The acute angle between two intersecting planes, equal to the acute angle between their normal vectors.

The angle between the horizontal and the line of sight from an observer down to an object below them.

The angle between the horizontal and the line of sight from an observer up to an object above them.

Part of the circumference of a circle between two points on it. Its length is $l=r\theta$ when the angle $\theta$ at the centre is in radians.

The unit vectors $\mathit{i}$, $\mathit{j}$ and $\mathit{k}$ in the positive directions of the $x$-, $y$- and $z$-axes. Every vector in three dimensions is a sum of multiples of them.

The direction of one point from another, given as an angle measured clockwise from north and written with three digits, such as $045\mathrm{°}$ or $210\mathrm{°}$.

The equations $\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}$ of the line through $(x_0,y_0,z_0)$ with direction vector $\left(\begin{array}{c}l\\ m\\ n\end{array}\right)$, found by eliminating the parameter.

A straight line segment joining two points on a curve. On a circle the longest chord is a diameter; on the graph of a function, the gradient of a chord is the average rate of change between its two endpoints.

One of the functions $\sin x$, $\cos x$ and $\tan x$ of a real number $x$, defined from the coordinates of the point at angle $x$ radians on the unit circle.

Two lines that are the same line: their direction vectors are parallel and they share a point, so every point of one lies on the other.

Lying on the same straight line. Points $A$, $B$ and $C$ are collinear when $\overrightarrow{AC}=k\overrightarrow{AB}$ for some scalar $k$.

The line segment that joins two skew lines and is perpendicular to both. Its length is the shortest distance between the lines.

Two angles whose sum is a right angle, $\frac{\pi }{2}$ (or ${90}^{\circ }$); the angles $\theta$ and $\frac{\pi }{2}-\theta$.

One of the numbers $v_1$, $v_2$, $v_3$ in $\mathit{v}=v_1\mathit{i}+v_2\mathit{j}+v_3\mathit{k}$, giving how far the vector goes in each axis direction. Written as a column vector, $\mathit{v}=\left(\begin{array}{c}{v}_1\\ v_2\\ v_3\end{array}\right)$.

An identity that writes a trigonometric function of a sum or difference of two angles, such as $\sin (A+B)$, in terms of the sines, cosines or tangents of $A$ and $B$ separately.

Lying in the same plane. Any three points are coplanar; four points or two lines may or may not be.

In any triangle $ABC$, $c^2=a^2+b^2-2ab\cos C$. It links all three sides with one angle and extends Pythagoras's theorem to triangles without a right angle.

One complete repetition of the pattern of a periodic graph, covering an interval of $x$ equal to one period.

A non-zero vector parallel to a line, written $\mathit{b}$ in $\mathit{r}=\mathit{a}+\lambda \mathit{b}$. Any non-zero multiple of it is also a direction vector of the same line.

The vector from one point to another, giving the change in position: $\overrightarrow{AB}=\mathit{b}-\mathit{a}$, where $\mathit{a}$ and $\mathit{b}$ are the position vectors of $A$ and $B$.

An identity expressing a trigonometric ratio of $2\theta$ in terms of ratios of $\theta$, such as $\sin 2\theta =2\sin \theta \cos \theta$ and $\cos 2\theta =1-2{\sin }^2\theta$.

A value written without rounding, using fractions, surds or $\pi$, such as $\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}$ rather than $0.866$.

The point on a line or plane closest to a given point, where the segment from the given point meets the line or plane at a right angle.

The single-wave form $R\cos (\theta -\alpha )$ or $R\sin (\theta +\alpha )$, with $R>0$, of an expression $a\cos \theta +b\sin \theta$; here $R=\sqrt{a^2+b^2}$ is its amplitude.

Half of a sphere, cut off by a plane through its centre. A solid hemisphere of radius $r$ has volume $\frac{2}{3}\pi r^3$ and total surface area $3\pi r^2$.

The longest side of a right-angled triangle, opposite the right angle.

The angle between two given sides of a triangle, at the vertex where those two sides meet.

The functions $\arcsin x$, $\arccos x$ and $\arctan x$, which return the angle with a given sine, cosine or tangent. Each is the inverse of sine, cosine or tangent restricted to a domain on which it is one-to-one.

The line common to two non-parallel planes. Its direction vector is perpendicular to both normals, for example ${\mathit{n}}_1\times {\mathit{n}}_2$.

The length of a vector, written $\mathrm{\mid }\mathit{v}\mathrm{\mid }$. For $\mathit{v}=v_1\mathit{i}+v_2\mathit{j}+v_3\mathit{k}$ it is $\mathrm{\mid }\mathit{v}\mathrm{\mid }=\sqrt{v_1^2+v_2^2+v_3^2}$, which is never negative.

The point exactly halfway along a line segment. Its coordinates are the means of the endpoints' coordinates: $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2})$ in three dimensions.

A flat shape that folds up to make the surface of a solid. The area of the net equals the surface area of the solid.

A non-zero vector perpendicular to a plane, and so perpendicular to every line lying in that plane.

Non-zero vectors that are scalar multiples of each other: $\mathit{w}=k\mathit{v}$ with $k\mathrm{\ne }0$. They point in the same direction if $k>0$ and in opposite directions if $k<0$.

A variable such as $\lambda$ or $t$ whose value picks out one point of a line or curve. Each real value of the parameter gives exactly one point.

A function whose values repeat at regular intervals: $f(x+p)=f(x)$ for all $x$ in its domain. The smallest such positive $p$ is the period.

A flat surface extending without limit in every direction. In three dimensions a plane has a Cartesian equation of the form $ax+by+cz=d$.

A graph has point symmetry about a point $M$ if a rotation of ${180}^{\circ }$ about $M$ maps the graph onto itself; $M$ is the centre of symmetry.

The vector from the origin $O$ to a point $A$, written $\overrightarrow{OA}=\mathit{a}$. Its components are the coordinates of $A$.

The horizontal line midway between the maximum and minimum values of a sinusoidal graph; for $y=a\sin (b(x+c))+d$ it is $y=d$.

The one angle an inverse trigonometric function returns: in $[-\frac{\pi }{2},\frac{\pi }{2}]$ for $\arcsin$, in $[0,\pi ]$ for $\arccos$ and strictly between $-\frac{\pi }{2}$ and $\frac{\pi }{2}$ for $\arctan$.

A solid with two identical, parallel end faces joined by flat sides, so every slice parallel to the ends is the same shape, its cross-section. Volume = cross-sectional area × length.

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: $c^2=a^2+b^2$.

The identity ${\cos }^2\theta +{\sin }^2\theta =1$, true for every angle $\theta$. It follows from Pythagoras's theorem applied to a point on the unit circle.

One of the four regions into which the coordinate axes divide the plane, numbered anticlockwise from the first quadrant, where $x>0$ and $y>0$.

The angle at the centre of a circle subtended by an arc whose length equals the radius. A full turn is $2\pi$ radians, so $\pi \text{ radians}=180\mathrm{°}$.

The functions $\sec \theta =\frac{1}{\cos \theta }$, $\mathrm{cosec}\theta =\frac{1}{\sin \theta }$ and $\cot \theta =\frac{\cos \theta }{\sin \theta }$, each defined for every angle at which its denominator is not zero.

The acute angle between the radius to a point on the unit circle and the $x$-axis. Angles with the same related angle have trigonometric ratios equal in size, differing at most in sign.

A cone whose apex lies directly above the centre of its circular base. Its slant height $l$, from the apex to the rim, satisfies $l^2=r^2+h^2$.

A pyramid whose apex lies directly above the centre of its base, so the line from the apex to that centre is perpendicular to the base. Its volume is $V=\frac{1}{3}Ah$.

A way to find the direction of $\mathit{v}\times \mathit{w}$: with the fingers of the right hand curling from $\mathit{v}$ towards $\mathit{w}$, the thumb points along $\mathit{v}\times \mathit{w}$.

A quantity with size only and no direction, such as a mass, a time or a temperature; a real number. Multiplying a vector by a scalar changes its length and, if the scalar is negative, reverses its direction.

The number $\mathit{v}\cdot \mathit{w}=v_1{w}_1+v_2{w}_2+v_3{w}_3=\mathrm{\mid }\mathit{v}\mathrm{\mid }\mathrm{\mid }\mathit{w}\mathrm{\mid }\cos \theta$, where $\theta$ is the angle between $\mathit{v}$ and $\mathit{w}$. It is zero exactly when non-zero vectors are perpendicular.

The signed length of the shadow of $\mathit{v}$ on the direction of $\mathit{w}$: $\frac{\mathit{v}\cdot \mathit{w}}{\mathrm{\mid }\mathit{w}\mathrm{\mid }}=\mathrm{\mid }\mathit{v}\mathrm{\mid }\cos \theta$. It is negative when the angle between them is obtuse.

The region of a circle bounded by two radii and the arc between them. Its area is $A=\frac{1}{2}{r}^2\theta$ when $\theta$ is in radians.

The region of a circle bounded by a chord and the arc it cuts off. Its area is the sector area minus the triangle area: $\frac{1}{2}{r}^2(\theta -\sin \theta )$.

Three or more planes that all contain the same line, like the pages of a partly open book.

In any triangle $ABC$, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$, where side $a$ is opposite angle $A$, and so on. It is used when a side and the angle opposite it are both known.

A function of the form $y=a\sin (b(x+c))+d$ (or the same with cosine): a sine curve stretched and translated, with a constant amplitude and period.

Two lines in three dimensions that are not parallel and do not intersect. Skew lines cannot lie in a common plane.

The set of all points in space at a fixed distance $r$, the radius, from a fixed point, the centre. A solid sphere has volume $\frac{4}{3}\pi r^3$ and surface area $4\pi r^2$.

The total area of all the faces and curved surfaces that enclose a solid, measured in square units.

Finding a distance or position that cannot be measured directly by measuring angles from the two ends of a baseline of known length and solving the triangle formed.

An equation involving circular functions of an unknown angle, such as $2\sin x=1$. It usually has infinitely many solutions, so it is solved on a stated interval.

For an acute angle $\theta$ in a right-angled triangle, one of $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$. Each depends only on $\theta$, not on the size of the triangle.

The circle of radius 1 centred at the origin. A point on it at angle $\theta$ anticlockwise from the positive $x$-axis has coordinates $(\cos \theta ,\sin \theta )$.

A vector of magnitude 1. The unit vector in the direction of a non-zero vector $\mathit{v}$ is $\frac{\mathit{v}}{\mathrm{\mid }\mathit{v}\mathrm{\mid }}$.

A quantity with both a size (magnitude) and a direction, such as a displacement, a velocity or a force. It is drawn as a directed line segment and written in bold, $\mathit{v}$, or as $\overrightarrow{AB}$.

An equation $\mathit{r}=\mathit{a}+\lambda \mathit{b}$ giving the position vector $\mathit{r}$ of every point on a line, where $\mathit{a}$ is the position vector of a known point, $\mathit{b}$ a direction vector and $\lambda$ a real parameter.

The equation $\mathit{r}=\mathit{a}+\lambda \mathit{b}+\mu \mathit{c}$, giving every point of a plane through the point with position vector $\mathit{a}$, where $\mathit{b}$ and $\mathit{c}$ are non-parallel vectors in the plane.

The vector $\mathit{v}\times \mathit{w}$ perpendicular to both $\mathit{v}$ and $\mathit{w}$, with magnitude $\mathrm{\mid }\mathit{v}\mathrm{\mid }\mathrm{\mid }\mathit{w}\mathrm{\mid }\sin \theta$ and direction given by the right-hand rule.

The vector along $\mathit{w}$ whose length is the scalar projection of $\mathit{v}$ on $\mathit{w}$: $\frac{\mathit{v}\cdot \mathit{w}}{\mathrm{\mid }\mathit{w}{\mathrm{\mid }}^2}\mathit{w}$.

The vector $\mathbf{0}$ with every component $0$ and magnitude $0$. It has no direction, and $\mathit{v}+\mathbf{0}=\mathit{v}$ for every vector $\mathit{v}$.