IB Maths AA Topic 3 Definitions

This page contains our IB Maths AA definitions for topic 3. By learning each one of these definitions, you will fully cover the content for IB Maths AA 'Geometry & Trigonometry'.

All Topic 3 Sub-topics 3.1: Fundamentals Of Geometry 3.2: Introduction To Trigonometry 3.3: Applied Trigonometry 3.4: Circles 3.5: Unit Circle 3.6: Trigonometric Identities 3.7: Trigonometric Graphs (HL)3.8: Advanced trigonometry (HL)3.9: Compound angles (HL)3.10: Vectors (HL)3.11: Lines (HL)3.12: Planes

adjacent

The side next to the chosen acute angle in a right-angled triangle, not including the hypotenuse.

ambiguous case

The situation when two sides and a non-included angle are given for a triangle, where using the sine rule can lead to no triangle, one triangle, or two different triangles because

\sin\theta=\sin\left(\pi-\theta\right)

amplitude

The distance from the principal axis to a maximum or minimum of a sinusoidal graph, given by '

\left|a\right|

arc

Part of the circumference of a circle between two points.

arc length

The distance along an arc; for radius

r

and central angle

heta

l=r\theta

when

heta

is in radians, or

l=\frac{\theta}{360}\times2\pi r

when

heta

is in degrees.

arccos

The inverse function of cosine that returns the principal angle '

y

' such that '

\cos y=x

', with domain '

-1\leq x\leq1

' and output values '

0\leq y\leq\pi

arcsin

An inverse trigonometric function whose derivative is

\frac{d}{dx}\left(\arcsin x\right)=\frac{1}{\sqrt{1-x^2}}

, and for an input

f(x)

becomes

\frac{f'(x)}{\sqrt{1-\left(f(x)\right)^2}}

arctan

The inverse function of tangent that returns the principal angle '

y

' such that '

\tan y=x

', with domain all real numbers and output values '

-\frac{\pi}{2}\lt y\lt\frac{\pi}{2}

asymptotes

Lines that a rational function’s graph approaches; can be vertical where the denominator is

0

(and the numerator is not

0

there), horizontal depending on degrees, or oblique when the numerator’s degree is one more than the denominator’s.

bearing

A direction given as an angle measured clockwise from north, written using three figures (for example, east is

090^\circ

Cartesian equation

An equation in

x

y

, and

z

that describes a plane in the form

ax+by+cz=d

, where

(a,b,c)

is perpendicular to the plane.

Cartesian form

A way to write a complex number as

z=a+bi

with

a,b\in\mathbb{R}

, separating its real and imaginary components.

Coincident

Describes two lines that are the same line, meaning their direction vectors are parallel and they share at least one common point.

composite

Describes a function formed by putting one function inside another, for example '

f\left(g\left(x\right)\right)

', which typically requires the chain rule to differentiate.

Compound

Involving a combination of two angles (such as a sum or difference) inside a single trigonometric function, for example $\tan\theta$ where $\theta=A\text{+}B$ or $\theta=A\text{-}B$ .

cone

A solid with a circular base and a curved surface meeting at a single vertex; for a right cone, $V=\frac{1}{3}\pi r^2h$ where $r$ is the base radius and $h$ is the perpendicular height.

cos

A trigonometric function whose derivative is '

-\sin x

cosec

A reciprocal trigonometric ratio defined by '

\cosec x=\frac{1}{\sin x}

', so it is undefined where '

\sin x=0

cosine

A trigonometric function whose derivative is

\frac{d}{dx}(\cos x)=-\sin x

cosine rule

A relationship for any triangle linking two sides and the included angle:

c^2=a^2+b^2-2ab\cos C

cot

A reciprocal trigonometric function whose derivative is

\frac{d}{dx}\left(\cot x\right)=-\cosec^2 x

, and for an input

f(x)

becomes

-\cosec^2\left(f(x)\right)f'(x)

depression

The angle measured downwards from a horizontal line to the line of sight to an object.

directed

Having edges with a specified direction, so moving from one vertex to another may be possible without the reverse move being possible.

direction vector

A non-zero vector parallel to a line that gives its direction; in $r=a+\lambda b$ , $b$ determines how $r$ changes as $\lambda$ changes.

displacement

The vector that represents the change in position from one point to another, for example $\rightarrow{AB}=\textbf{b}-\textbf{a}$ .

distance

The length of the straight line segment joining two points; in three-dimensional space it can be found using

$d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}$ .

domain

The set of input values for which a model is intended to be used, chosen using the context rather than only the algebraic definition.

double angle identities

Formulae that express '

\sin 2x

' and '

\cos 2x

' in terms of '

\sin x

' and '

\cos x

', for example '

\sin 2x=2\sin x\cos x

' and '

\cos 2x=\cos^{2}x-\sin^{2}x=2\cos^{2}x-1=1-2\sin^{2}x

elevation

The angle measured upwards from a horizontal line to the line of sight to an object.

even function

A function with symmetry in the

y

-axis, satisfying

f\left(-x\right)=f\left(x\right)

, for example

\cos\left(-x\right)=\cos x

finite interval

A restricted range of values for the variable, such as '

0\le x\le2\pi

', within which all solutions to an equation must be found.

hemisphere

Half of a sphere cut by a plane through its centre; its volume is $V=\frac{2}{3}\pi r^3$ .

hypotenuse

The longest side in a right-angled triangle, opposite the

90^\circ

angle.

Identity

A square matrix with ones on the main diagonal and zeros elsewhere, with the property that

AI=IA=A

for any compatible matrix

A

Intersecting

Describes two lines that share a common point, found by solving their parametric equations simultaneously for a consistent set of parameter values.

midline

The horizontal line halfway between the maximum and minimum values of a sine or cosine graph, with equation

y=\frac{\text{max}+\text{min}}{2}

; for

y=a\sin\left(b\left(x+c\right)\right)+d

it is

y=d

midpoint

The point halfway along the line segment joining two points; for $A\left(x_1,y_1,z_1\right)$ and $B\left(x_2,y_2,z_2\right)$ it is $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$ .

odd function

A function with rotational symmetry about the origin, satisfying

f\left(-x\right)=-f\left(x\right)

, for example

\sin\left(-x\right)=-\sin x

and

\tan\left(-x\right)=-\tan x

opposite

The side that lies across from the chosen acute angle in a right-angled triangle.

parallel

Checks consistency by comparing results from two different versions designed to measure the same thing.

parametric equation

A representation of a line or plane using one or more parameters, so that varying the parameter values generates all points on the object.

parametric form

A way to describe a line by writing each coordinate as a function of a parameter, for example $x=x_0+λl$ , $y=y_0+λm$ , $z=z_0+λn$ .

period

The horizontal length of one complete cycle of a trigonometric graph; for '

y=\sin x

' and '

y=\cos x

' it is '

2\pi

', and for '

y=\tan x

' it is '

\pi

periodic

Repeating in a regular pattern, so the graph looks the same after a fixed horizontal distance called the period.

plane

A two-dimensional coordinate system used to represent complex numbers, with the horizontal axis as the real axis and the vertical axis as the imaginary axis.

position

The vector from the origin $O$ to a point, written for example as $\rightarrow{OA}=\textbf{a}$ .

position vector

A vector from the origin to a point, used to locate the point in space; in $r=a+\lambda b$ , $\textbf{a}$ is the position vector of a fixed point on the line.

pyramid

A solid with a polygon base and triangular faces that meet at a single vertex; for a right pyramid, $V=\frac{1}{3}Ah$ where $A$ is the base area and $h$ is the perpendicular height.

Pythagorean identity

The fundamental relationship between sine and cosine given by '

\cos^{2}{\theta}+\sin^{2}{\theta}=1

', valid for all angles where the functions are defined.

quadrant

One of the four regions of the coordinate plane determined by the

x

- and

y

-axes, used to determine the signs of

\sin\theta

\cos\theta

, and

\tan\theta

quadratic form

A rearrangement where an equation becomes quadratic in a single trigonometric function, for example '

a\sin^{2}x+b\sin x+c=0

' (or similarly in '

\cos x

' or '

\tan x

'), allowing standard quadratic methods to be used.

quadratic formula

A method for solving '

au^{2}+bu+c=0

' by using '

u=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}

', often after substituting '

u=\sin x

', '

u=\cos x

' or '

u=\tan x

radian

A unit for measuring angles defined so that $\theta$ radians equals the arc length divided by the radius, $\theta=\frac{l}{r}$ , giving the key conversion $\pi$ radians $=180^\circ$ .

radius

The distance from the centre of a circle to any point on the circle.

range

The set of possible outputs a function can produce.

reference angle

The acute angle between the terminal side of an angle and the

x

-axis, used with quadrant signs to find exact trigonometric values.

Reverse integration

Describes using the chain rule in reverse to integrate expressions matching the pattern '

g'(x)f\left(g(x)\right)

', giving '

F\left(g(x)\right)+C

' where '

F'(u)=f(u)

right pyramid

A pyramid whose apex is vertically above the centre of its base; its volume is $V=\frac{1}{3}Ah$ , where $A$ is the base area and $h$ is the perpendicular height.

scalar

A single number used to multiply a matrix by multiplying every element by that number.

scalar parameter

A real number (such as $λ$ or $μ$ ) that varies to generate different points on a line in a parametric or vector equation.

sec

A reciprocal trigonometric function whose derivative is

\frac{d}{dx}\left(\sec x\right)=\sec x\tan x

; for an input

f(x)

\frac{d}{dx}\left(\sec\left(f(x)\right)\right)=\sec\left(f(x)\right)\tan\left(f(x)\right)f'(x)

sector

The region enclosed by two radii and the arc between them.

sin

A trigonometric function whose derivative is '

\cos x

sine

A trigonometric ratio/function that can be expanded for compound angles using $\\sin\left(A\pm B\right)=\sin A\cos B\pm\cos A\sin B$ .

sine rule

A relationship for any triangle with sides

a

b

c

opposite angles

A

B

C

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}

skew

Describes two lines in three dimensions that are not parallel and do not meet, so no common solution exists when their parametric equations are solved together.

solution sets

The collection of all points that satisfy given equations, interpreted geometrically as an intersection point, a line of points, no common points, or infinitely many points.

sphere

A solid consisting of all points at a fixed distance $r$ from a centre; its volume is $V=\frac{4}{3}\pi r^3$ .

tan

Defined by

\tan\theta=\frac{\sin\theta}{\cos\theta}

, so it is undefined whenever

\cos\theta=0

tangent

A line that touches a curve at a point and has the same local slope as the curve there, used to interpret the derivative at that point.

trigonometric identity

An equation involving trigonometric functions that is true for all values of the variable for which both sides are defined, and is used to simplify expressions, prove results, and solve equations.

unit circle

A circle of radius

1

centred at the origin, with equation

x^2+y^2=1

, used to define trigonometric functions for any angle.

vector

An operation on two vectors that returns a vector perpendicular to both, with direction given by the right-hand rule and magnitude $|\textbf{v}||\textbf{w}|\text{sin}\theta$ .

vector equation

A representation of a line in 2D or 3D written as a position vector $\textbf{r}$ in terms of a fixed point and a scalar parameter, typically in the form $r = a + \lambda b$ .

vertical

Describing changes parallel to the

y

-axis, such as a stretch by factor

|a|

or a shift up or down by

d

Volume

A measure of how much space a three-dimensional solid occupies, usually expressed in cubic units such as cm³.

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