IB Maths: Analysis and Approaches HL 166711

The distance between two points in three-dimensional space, and their midpoint.

In SL examinations, only right-angled trigonometry questions will be set in reference to three-dimensional shapes.

Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids.

The size of an angle between two intersecting lines or between a line and a plane.

In problems related to these topics, students should be able to identify relevant right-angled triangles in three-dimensional objects and use them to find unknown lengths and angles.

Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.

In all areas of this topic, students should be encouraged to sketch well-labelled diagrams to support their solutions.

Link to: inverse functions (SL 2.2) when finding angles.

The sine rule: $\dfrac{a}{\sin{A}}=\dfrac{b}{\sin{B}}=\dfrac{c}{\sin{C}}$.

The cosine rule: $c^2=a^2+b^2−2ab\cos{C}$;

$\cos{C}=\dfrac{a^2+b^2−c^2}{2ab}$.

Area of a triangle as $\dfrac{1}{2}ab\sin{C}$.

This section does not include the ambiguous case of the sine rule.

Applications of right and non-right angled trigonometry, including Pythagoras’s theorem.

Angles of elevation and depression.

Contexts may include use of bearings.

Construction of labelled diagrams from written statements.

The circle: radian measure of angles; length of an arc; area of a sector.

Radian measure may be expressed as exact multiples of $π$, or decimals.

Definition of $\cos{θ}$, $\sin{θ}$ in terms of the unit circle.

Includes the relationship between angles in different quadrants.

Examples:

• $\cos{x}=\cos{(−x)}$
• $\tan{(3π−x)}=−\tan{x}$
• $\sin{(π+x)}=−\sin{x}$

Definition of $\tan{θ}$ as $\dfrac{\sin{θ}}{\cos{θ}}$.

The equation of a straight line through the origin is $y=x\tan{θ}$, where $θ$ is the angle formed between the line and positive x-axis.

Exact values of trigonometric ratios of 0, $\dfrac{π}{6}$, $\dfrac{π}{4}$, $\dfrac{π}{3}$, $\dfrac{π}{2}$ and their multiples.

$\sin{\dfrac{π}{3}}=\dfrac{\sqrt{3}}{2}$, $\cos{\dfrac{3π}{4}}=−\dfrac{1}{\sqrt{2}}$, $\tan{210°}=\dfrac{\sqrt{3}}{3}$

Extension of the sine rule to the ambiguous case.

The Pythagorean identity $\cos^2{θ}+\sin^2{θ}=1$.

Double angle identities for sine and cosine.

Simple geometrical diagrams and dynamic graphing packages may be used to illustrate the double angle identities (and other trigonometric identities).

The relationship between trigonometric ratios.

Examples:

Given $\sin{θ}$, find possible values of $\tan{θ}$, (without finding $θ$).

Given $\cos{x}=\dfrac{3}{4}$ and $x$ is acute, find $\sin{2x}$, (without finding $x$).

The circular functions $\sin{x}$, $\cos{x}$, and $\tan{x}$; amplitude, their periodic nature, and their graphs

Composite functions of the form:

$f(x)=a\sin{(b(x+c))}+d$.

Trigonometric functions may have domains given in degrees or radians.

Examples:

• $f(x)=\tan{(x−\dfrac{π}{4})}$,
• $f(x)=2\cos{(3(x−4))}+1$.

Transformations.

Example: $y=\sin{x}$ used to obtain $y=3\sin{2x}$ by a stretch of scale factor 3 in the $y$ direction and a stretch of scale factor $\dfrac{1}{2}$ in the $x$ direction.

Link to: transformations of graphs (SL 2.11).

Real-life contexts.

Examples: height of tide, motion of a Ferris wheel.

Students should be aware that not all regression technology produces trigonometric functions in the form $f(x)=a\sin{(b(x+c))}+d$.

Solving trigonometric equations in a finite interval, both graphically and analytically.

Examples:

• $2\sin{x}=1, 0≤x≤2π$
• $2\sin{2x}=3\cos{x}, 0°≤x≤180°$
• $2\tan{(3(x−4))}=1, −π≤x≤3π$

Equations leading to quadratic equations in $\sin{x}$, $\cos{x}$ or $\tan{x}$.

Examples:

• $2\sin{2x}+5\cos{x}+1=0, 0≤x≤4π$,
• $2\sin{x}=\cos{2x}, −π≤x≤π$

Not required: The general solution of trigonometric equations.

Definition of the reciprocal trigonometric ratios $\sec{θ}$, $\cosec{θ}$ and $\cot{θ}$.

Pythagorean identities:

• $1+\tan^2{θ}=\sec^2{θ}$
• $1+\cot^2{θ}=\cosec^2{θ}$

The inverse functions $f(x)=\arcsin{x}$, $f(x)=\arccos{x}$, $f(x)=\arctan{x}$; their domains and ranges; their graphs.

Compound angle identities.

Double angle identity for tan.

Derivation of double angle identities from compound angle identities.

Link to: De Moivre’s theorem (AHL 1.14).

Relationships between trigonometric functions and the symmetry properties of their graphs.

• $\sin{(π−θ)}=\sin{θ}$
• $\cos{(π−θ)}=−\cos{θ}$
• $\tan{(π−θ)}=−\tan{θ}$

Link to: the unit circle (SL 3.5), odd and even functions (AHL 2.14), compound angles (AHL 3.10).

Concept of a vector; position vectors; displacement vectors.

Representation of vectors using directed line segments.

Base vectors $\bold{i}$, $\bold{j}$, $\bold{k}$.

Components of a vector:

$v=\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}=v_1\bold{i}+v_2\bold{j}+v_3\bold{k}$.

Algebraic and geometric approaches to the following:

• the sum and difference of two vectors
• the zero vector $\bold{0}$, the vector $−\bold{v}$
• multiplication by a scalar, $k\bold{v}$, parallel vectors
• magnitude of a vector, $|\bold{v}|$; unit vectors, $\dfrac{\bold{v}}{|\bold{v}|}$
• position vectors $\overrightarrow{\text{OA}}=\bold{a}$, $\overrightarrow{\text{OB}}=\bold{b}$
• displacement vector $\overrightarrow{\text{AB}}=\bold{b}−\bold{a}$

Proofs of geometrical properties using vectors.

Distance between points $\text{A}$ and $\text{B}$ is the magnitude of $\overrightarrow{\text{AB}}$

The definition of the scalar product of two vectors.

The angle between two vectors.

Perpendicular vectors; parallel vectors.

Applications of the properties of the scalar product

• $\bold{v}·\bold{w}=\bold{w}·\bold{v}$;
• $\bold{u}·(\bold{v}+\bold{w})=\bold{u}·\bold{v}+\bold{u}·\bold{w}$;
• $(k\bold{v})·\bold{w}=k(\bold{v}·\bold{w})$;
• $\bold{v}·\bold{v}=|\bold{v}|^2$.
• $\bold{v}·\bold{w}=|\bold{v}||\bold{w}|\cos{θ}$, where $θ$ is the angle between $\bold{v}$ and $\bold{w}$.

For non-zero vectors $\bold{v}·\bold{w}=0$ is equivalent to the vectors being perpendicular; for parallel vectors $|\bold{v}⋅\bold{w}|=|\bold{v}||\bold{w}|$.

Vector equation of a line in two and three dimensions:

$\bold{r}=\bold{a}+λ\bold{b}$.

Relevance of $\bold{a}$ (position) and $\bold{b}$ (direction).

Knowledge of the following forms for equations of lines:

Parametric form:

$x=x_0+λl, y=y_0+λm, z=z_0+λn$.

Cartesian form:

$\dfrac{x−x_0}{l}=\dfrac{y−y_0}{m}=\dfrac{z−z_0}{n}$.

The angle between two lines.

Using the scalar product of the two direction vectors.

Simple applications to kinematics.

Interpretation of $λ$ as time and $\bold{b}$ as velocity, with $|\bold{b}|$ representing speed.

Coincident, parallel, intersecting and skew lines, distinguishing between these cases.

Points of intersection.

Skew lines are non-parallel lines that do not intersect in three-dimensional space.

The definition of the vector product of two vectors.

The vector product is also known as the “cross product”. $\bold{v}×\bold{w}=|\bold{v}||\bold{w}|\sin{θ}\bold{n}$, where $θ$ is the angle between $\bold{v}$ and $\bold{w}$, and $\bold{n}$ is the unit normal vector whose direction is given by the right-hand screw rule.

Properties of the vector product.

• $\bold{v}×\bold{w}=−\bold{w}×\bold{v}$;
• $\bold{u}×(\bold{v}+\bold{w})=\bold{u}×\bold{v}+\bold{u}×\bold{w}$;
• $(k\bold{v})×\bold{w}=k(\bold{v}×\bold{w})$;
• $\bold{v}×\bold{v}=\bold{0}$.

For non-zero vectors $\bold{v}×\bold{w}=\bold{0}$ is equivalent to the vectors being parallel.

Geometric interpretation of $|\bold{v}×\bold{w}|$

Use of $|\bold{v}×\bold{w}|$ to find the area of a parallelogram (and hence a triangle).

Vector equations of a plane:

$\bold{r}=\bold{a}+λ\bold{b}+μ\bold{c}$, where $\bold{b}$ and $\bold{c}$ are non-parallel vectors within the plane.

$\bold{r}·\bold{n}=\bold{a}·\bold{n}$, where $\bold{n}$ is a normal to the plane and $\bold{a}$ is the position vector of a point on the plane.

Cartesian equation of a plane $ax+by+cz=d$.

Intersections of: a line with a plane; two planes; three planes.

Angle between: a line and a plane; two planes.

Finding intersections by solving equations; geometrical interpretation of solutions.

Link to: solutions of systems of linear equations (AHL 1.16).