IB Math AA HL — Geometry & Trigonometry

Complete Study Guide

Topics Covered

1. 3D Geometry — distance, midpoint, volume, surface area
2. Right-Angle and Non-Right-Angle Trigonometry — sine rule, cosine rule, area
3. Applications — bearings, angles of elevation and depression
4. Radian Measure, Arc Length, Sector Area
5. The Unit Circle and Exact Trigonometric Values
6. Trigonometric Identities — Pythagorean, double angle, compound angle HL
7. Trigonometric Functions and Their Graphs
8. Solving Trigonometric Equations
9. Reciprocal and Inverse Trigonometric Functions HL
10. Vector Algebra — notation, addition, dot product, cross product HL
11. Vector Equations of Lines in 2D and 3D HL
12. Vector Equations of Planes HL
13. Intersections of Lines and Planes HL
14. Practice Exam Questions

Topic 3 of the IB Math AA HL syllabus — 25 SL hours, 51 HL hours — Paper 1 and Paper 2

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Section 1: 3D Geometry

1.1 Distance and Midpoint in 3D

In three-dimensional space, points are described by coordinates $(x, y, z)$. The distance between points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ is a direct extension of the 2D Pythagorean theorem:

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

The midpoint $M$ of segment $AB$ is:

$M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2},\ \frac{z_1 + z_2}{2}\right)$

1.2 Volumes and Surface Areas of 3D Solids

The IB expects you to apply standard formulae to composite 3D problems. All the following formulas are in the formula booklet.

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Quick Recall — Section 1

Try to answer without scrolling up:

1. Write the distance formula between $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$.
2. What is the volume of a cone with base radius $r$ and height $h$?
3. Why is the shared circle between a cone and a cylinder not counted in surface area?

Reveal answers

1. $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$
2. $V = \dfrac{1}{3}\pi r^2 h$
3. That circle is an internal face — it lies inside the composite solid and is not exposed to the exterior.

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Section 2: Right-Angle and Non-Right-Angle Trigonometry

2.1 Right-Triangle Trigonometry (SOH CAH TOA)

For a right-angled triangle with acute angle $\theta$, hypotenuse $h$, opposite side $o$, and adjacent side $a$:

$\sin\theta = \frac{o}{h} \qquad \cos\theta = \frac{a}{h} \qquad \tan\theta = \frac{o}{a}$

These definitions apply directly to right-triangle problems (ladders, building heights, simple bearings).

2.2 The Sine Rule

For any triangle with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively:

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

Use the sine rule when you know:

• Two angles and one side (AAS or ASA), or
• Two sides and a non-included angle (SSA — watch for the ambiguous case)

2.3 The Cosine Rule

For a triangle with sides $a$, $b$, $c$ and angle $C$ opposite side $c$:

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

Rearranged to find angle $C$ when all three sides are known:

$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$

Use the cosine rule when you know:

• Two sides and the included angle (SAS), or
• All three sides (SSS)

2.4 Area of a Triangle

$\text{Area} = \frac{1}{2}ab\sin C$

where $a$ and $b$ are two sides and $C$ is the included angle between them.

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Quick Recall — Section 2

Try to answer without scrolling up:

1. State the sine rule.
2. State the cosine rule (finding side $c$).
3. What is the area of a triangle with sides $a$ and $b$ and included angle $C$?
4. Under what two conditions do you use the cosine rule (not the sine rule)?

Reveal answers

1. $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$
2. $c^2 = a^2 + b^2 - 2ab\cos C$
3. $\dfrac{1}{2}ab\sin C$
4. SAS (two sides and included angle) or SSS (all three sides known).

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Section 3: Applications — Bearings and Angles of Elevation/Depression

3.1 Bearings

A bearing is a direction measured as a three-digit angle clockwise from North. For example, a bearing of $065°$ means $65°$ clockwise from North (i.e., roughly north-east).

Key conventions:

• Bearings are always measured clockwise from North
• They are written as three digits: $005°$, $090°$, $180°$, $270°$
• A bearing of $090°$ is due East; $180°$ is due South; $270°$ is due West

Reverse bearing: If the bearing from $A$ to $B$ is $\theta$, the bearing from $B$ to $A$ is $\theta + 180°$ (mod $360°$).

3.2 Angles of Elevation and Depression

• Angle of elevation: The angle measured upward from the horizontal to a line of sight.
• Angle of depression: The angle measured downward from the horizontal to a line of sight.

These angles are equal when the observer and the object are at the same horizontal level for alternate interior angles, but in general they are supplementary parts of the right-angle trig setup.

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Section 4: Radian Measure, Arc Length, and Sector Area

4.1 Radian Measure

One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.

$\pi \text{ radians} = 180°$

Conversions:

$\theta\text{ (radians)} = \theta\text{ (degrees)} \times \frac{\pi}{180°} \qquad \theta\text{ (degrees)} = \theta\text{ (radians)} \times \frac{180°}{\pi}$

4.2 Arc Length and Sector Area

For a circle of radius $r$ and a sector with central angle $\theta$ (in radians):

$\text{Arc length} \quad l = r\theta$

$\text{Sector area} \quad A = \frac{1}{2}r^2\theta$

Segment area = Sector area $-$ Triangle area:

$A_{\text{segment}} = \frac{1}{2}r^2\theta - \frac{1}{2}r^2\sin\theta = \frac{1}{2}r^2(\theta - \sin\theta)$

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Quick Recall — Section 4

Try to answer without scrolling up:

1. Convert $240°$ to radians.
2. Write the arc length formula.
3. Write the sector area formula.
4. What is the area of a circular segment?

Reveal answers

1. $240° \times \dfrac{\pi}{180°} = \dfrac{4\pi}{3}$ radians
2. $l = r\theta$ (with $\theta$ in radians)
3. $A = \dfrac{1}{2}r^2\theta$ (with $\theta$ in radians)
4. $A_{\text{segment}} = \dfrac{1}{2}r^2(\theta - \sin\theta)$

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Section 5: The Unit Circle and Exact Trigonometric Values

5.1 The Unit Circle

The unit circle has centre $(0, 0)$ and radius 1. For any angle $\theta$ measured anticlockwise from the positive $x$-axis, the point on the unit circle is $(\cos\theta, \sin\theta)$.

This gives a definition of $\sin$ and $\cos$ that extends beyond right triangles to all real angles.

$\cos\theta = x\text{-coordinate} \qquad \sin\theta = y\text{-coordinate} \qquad \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}$

5.2 Signs in Each Quadrant (CAST)

Quadrant	Angles	Positive functions
1st ($0$ to $\frac{\pi}{2}$)	$0°$ to $90°$	All ($\sin$, $\cos$, $\tan$)
2nd ($\frac{\pi}{2}$ to $\pi$)	$90°$ to $180°$	$\sin$ only
3rd ($\pi$ to $\frac{3\pi}{2}$)	$180°$ to $270°$	$\tan$ only
4th ($\frac{3\pi}{2}$ to $2\pi$)	$270°$ to $360°$	$\cos$ only

The acronym CAST (reading quadrants 4–1–2–3 anticlockwise) is a mnemonic: Cos, All, Sin, Tan.

5.3 Exact Trigonometric Values

5.4 Symmetry and Reference Angles

For any angle $\theta$ in quadrants 2, 3, or 4, find the reference angle $\alpha$ (the acute angle to the nearest $x$-axis), then apply the sign from CAST.

• Quadrant 2: $\alpha = \pi - \theta$
• Quadrant 3: $\alpha = \theta - \pi$
• Quadrant 4: $\alpha = 2\pi - \theta$

Example: $\sin\!\left(\dfrac{5\pi}{6}\right) = \sin\!\left(\pi - \dfrac{\pi}{6}\right) = \sin\!\dfrac{\pi}{6} = \dfrac{1}{2}$ (positive in Q2).

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Section 6: Trigonometric Identities

6.1 Pythagorean Identity

From the unit circle definition ($\cos^2\theta + \sin^2\theta = 1$ is immediate from $x^2 + y^2 = 1$):

$\sin^2\theta + \cos^2\theta = 1$

Two derived forms (divide through by $\cos^2\theta$ or $\sin^2\theta$):

$1 + \tan^2\theta = \sec^2\theta \qquad \cot^2\theta + 1 = \csc^2\theta$

6.2 Double Angle Identities

$\sin 2\theta = 2\sin\theta\cos\theta$

$\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$

$\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$

6.3 Compound Angle Identities HL

$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$

$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

Note the sign convention: for $\cos$, the signs are opposite to the signs in the angle.

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Quick Recall — Section 6

Try to answer without scrolling up:

1. State the Pythagorean identity.
2. Write all three forms of $\cos 2\theta$.
3. State the compound angle identity for $\sin(A + B)$.
4. State the compound angle identity for $\cos(A - B)$.

Reveal answers

1. $\sin^2\theta + \cos^2\theta = 1$
2. $\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
3. $\sin(A+B) = \sin A\cos B + \cos A\sin B$
4. $\cos(A-B) = \cos A\cos B + \sin A\sin B$

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Section 7: Trigonometric Functions and Their Graphs

7.1 Graphs of sin, cos, and tan

The three basic trig functions have the following key features:

Feature	$y = \sin x$	$y = \cos x$	$y = \tan x$
Period	$2\pi$	$2\pi$	$\pi$
Amplitude	$1$	$1$	undefined
Domain	$\mathbb{R}$	$\mathbb{R}$	$x \neq \frac{\pi}{2} + n\pi$, $n \in \mathbb{Z}$
Range	$[-1, 1]$	$[-1, 1]$	$\mathbb{R}$
$y$-intercept	$0$	$1$	$0$
Zeros	$n\pi$	$\frac{\pi}{2} + n\pi$	$n\pi$

7.2 Transformations of Trig Functions

The general form is $y = a\sin(b(x - c)) + d$ (and similarly for cos and tan):

• $|a|$ = amplitude (vertical stretch/compression; if $a < 0$, reflection in $x$-axis)
• $\dfrac{2\pi}{|b|}$ = period (for $\sin$ and $\cos$); $\dfrac{\pi}{|b|}$ for $\tan$
• $c$ = horizontal shift (phase shift; positive $c$ shifts right)
• $d$ = vertical shift (midline is at $y = d$)

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Section 8: Solving Trigonometric Equations

8.1 General Strategy

To solve a trig equation on a given interval:

1. Isolate the trig function (e.g., $\sin x = \frac{1}{2}$)
2. Find the principal value using inverse trig
3. Use the unit circle or CAST diagram to find all solutions in the interval
4. Check that all solutions satisfy any restrictions

General solutions (all solutions, no interval restriction):

$\sin x = k \implies x = \arcsin k + 2n\pi \quad \text{or} \quad x = \pi - \arcsin k + 2n\pi, \quad n \in \mathbb{Z}$

$\cos x = k \implies x = \pm\arccos k + 2n\pi, \quad n \in \mathbb{Z}$

$\tan x = k \implies x = \arctan k + n\pi, \quad n \in \mathbb{Z}$

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Section 9: Reciprocal and Inverse Trigonometric Functions HL

9.1 Reciprocal Trig Functions

$\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$

These are defined wherever the denominator is non-zero. Key identities:

$1 + \tan^2\theta = \sec^2\theta \qquad \cot^2\theta + 1 = \csc^2\theta$

9.2 Inverse Trigonometric Functions HL

To make trig functions invertible, their domains are restricted:

Function	Restricted domain	Range (principal values)
$\arcsin x = \sin^{-1} x$	$[-1, 1]$	$\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$
$\arccos x = \cos^{-1} x$	$[-1, 1]$	$[0, \pi]$
$\arctan x = \tan^{-1} x$	$\mathbb{R}$	$\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$

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Section 10: Vector Algebra HL

10.1 Vector Notation and Basic Operations

A vector has both magnitude and direction. In 2D and 3D, vectors are written as column vectors or using unit vector notation:

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

where $\mathbf{i} = (1, 0, 0)$, $\mathbf{j} = (0, 1, 0)$, $\mathbf{k} = (0, 0, 1)$ are standard unit vectors.

Magnitude (modulus):

$|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$

Unit vector in the direction of $\mathbf{v}$:

$\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}$

Vector between two points: $\overrightarrow{AB} = B - A = \begin{pmatrix} b_1 - a_1 \\ b_2 - a_2 \\ b_3 - a_3 \end{pmatrix}$

10.2 Vector Addition and Scalar Multiplication

$\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ u_3 + v_3 \end{pmatrix} \qquad k\mathbf{v} = \begin{pmatrix} kv_1 \\ kv_2 \\ kv_3 \end{pmatrix}$

Geometrically, vector addition follows the triangle or parallelogram law. Scalar multiplication stretches or compresses the vector (and flips it if $k < 0$).

Parallel vectors: $\mathbf{u}$ and $\mathbf{v}$ are parallel if $\mathbf{u} = k\mathbf{v}$ for some scalar $k$.

10.3 The Dot Product (Scalar Product)

$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 = |\mathbf{u}||\mathbf{v}|\cos\theta$

where $\theta$ is the angle between the vectors. Rearranging gives:

$\cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}$

Key property: $\mathbf{u} \cdot \mathbf{v} = 0$ if and only if $\mathbf{u}$ and $\mathbf{v}$ are perpendicular (provided both are non-zero).

10.4 The Cross Product (Vector Product) HL

The cross product $\mathbf{u} \times \mathbf{v}$ produces a vector perpendicular to both $\mathbf{u}$ and $\mathbf{v}$.

$\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} = \begin{pmatrix} u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 \\ u_1 v_2 - u_2 v_1 \end{pmatrix}$

Magnitude of the cross product:

$|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin\theta$

This equals the area of the parallelogram formed by $\mathbf{u}$ and $\mathbf{v}$. The area of the triangle formed by $\mathbf{u}$ and $\mathbf{v}$ is $\dfrac{1}{2}|\mathbf{u} \times \mathbf{v}|$.

Key property: $\mathbf{u} \times \mathbf{v} = \mathbf{0}$ if and only if $\mathbf{u}$ and $\mathbf{v}$ are parallel.

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Quick Recall — Section 10

Try to answer without scrolling up:

1. How do you find the angle between two vectors?
2. What does $\mathbf{u} \cdot \mathbf{v} = 0$ tell you (assuming both vectors are non-zero)?
3. What is the geometric interpretation of $|\mathbf{u} \times \mathbf{v}|$?
4. How do you find a normal vector to a plane containing points $A$, $B$, $C$?

Reveal answers

1. $\cos\theta = \dfrac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}$, then $\theta = \arccos(\cdots)$.
2. $\mathbf{u}$ and $\mathbf{v}$ are perpendicular (orthogonal).
3. The area of the parallelogram formed by $\mathbf{u}$ and $\mathbf{v}$.
4. Compute $\overrightarrow{AB} \times \overrightarrow{AC}$.

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Section 11: Vector Equations of Lines HL

11.1 Vector Equation of a Line

A line in 2D or 3D is determined by a point on the line and a direction vector. If the line passes through point $A$ with position vector $\mathbf{a}$ and has direction vector $\mathbf{d}$, then any point $P$ on the line satisfies:

$\mathbf{r} = \mathbf{a} + t\mathbf{d}, \qquad t \in \mathbb{R}$

where $t$ is a scalar parameter. In component form (3D):

$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} + t\begin{pmatrix} d_1 \\ d_2 \\ d_3 \end{pmatrix}$

11.2 Parametric and Cartesian Forms

Parametric equations:

$x = a_1 + td_1 \qquad y = a_2 + td_2 \qquad z = a_3 + td_3$

Cartesian (symmetric) form (provided $d_i \neq 0$):

$\frac{x - a_1}{d_1} = \frac{y - a_2}{d_2} = \frac{z - a_3}{d_3} = t$

11.3 Angle Between Two Lines

The angle $\theta$ between lines with direction vectors $\mathbf{d}_1$ and $\mathbf{d}_2$:

$\cos\theta = \frac{|\mathbf{d}_1 \cdot \mathbf{d}_2|}{|\mathbf{d}_1||\mathbf{d}_2|}$

The absolute value ensures $\theta \in \left[0°, 90°\right]$ (we take the acute angle between lines).

11.4 Relationships Between Lines

Two lines can be:

• Parallel: direction vectors are scalar multiples; lines do not intersect (unless they are the same line)
• Intersecting: lines meet at a unique point (solve simultaneously for parameters)
• Skew: lines are not parallel and do not intersect (only possible in 3D)

To check if two lines intersect: set the parametric equations equal and solve for the parameters $s$ and $t$. If all three equations are consistent with specific values of $s$ and $t$, the lines intersect. If the system is inconsistent, the lines are skew (or parallel).

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Section 12: Vector Equations of Planes HL

12.1 Equation of a Plane

A plane is determined by a point and a normal vector perpendicular to it.

If $\mathbf{n} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$ is the normal vector and the plane passes through point $(x_0, y_0, z_0)$, the Cartesian equation is:

$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$

Expanded as $ax + by + cz = d$ where $d = ax_0 + by_0 + cz_0$.

Vector form: $\mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}$, or equivalently $\mathbf{n} \cdot \mathbf{r} = d$.

Parametric form: Using two non-parallel vectors $\mathbf{b}$ and $\mathbf{c}$ in the plane:

$\mathbf{r} = \mathbf{a} + s\mathbf{b} + t\mathbf{c}, \qquad s, t \in \mathbb{R}$

12.2 Finding the Equation of a Plane

Given a normal vector and a point: Substitute directly into $ax + by + cz = d$.

Given three points: Find two vectors in the plane, take their cross product to get the normal, then use one point to find $d$.

12.3 Angle Between a Line and a Plane

Let $\mathbf{d}$ be the direction of the line and $\mathbf{n}$ be the normal to the plane. The angle $\phi$ between the line and the plane satisfies:

$\sin\phi = \frac{|\mathbf{d} \cdot \mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}$

(Note: this is the complement of the angle between the line and the normal.)

12.4 Angle Between Two Planes

The angle between two planes is the angle between their normal vectors:

$\cos\theta = \frac{|\mathbf{n}_1 \cdot \mathbf{n}_2|}{|\mathbf{n}_1||\mathbf{n}_2|}$

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Quick Recall — Section 12

Try to answer without scrolling up:

1. What is the normal vector of the plane $2x - y + 3z = 7$?
2. Given three points in a plane, how do you find the normal vector?
3. How do you find the angle between two planes?
4. A line has direction $\mathbf{d}$ and a plane has normal $\mathbf{n}$. How do you find the angle between the line and the plane?

Reveal answers

1. $\mathbf{n} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$
2. Find two vectors in the plane (using the points), then take their cross product.
3. $\cos\theta = \dfrac{|\mathbf{n}_1 \cdot \mathbf{n}_2|}{|\mathbf{n}_1||\mathbf{n}_2|}$
4. $\sin\phi = \dfrac{|\mathbf{d} \cdot \mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}$

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Section 13: Intersections of Lines and Planes HL

13.1 Line and Plane Intersection

To find where a line $\mathbf{r} = \mathbf{a} + t\mathbf{d}$ meets a plane $ax + by + cz = k$:

1. Substitute the parametric equations of the line into the plane equation
2. Solve for $t$
3. Substitute $t$ back to find the intersection point

If $\mathbf{d} \cdot \mathbf{n} = 0$, the line is parallel to the plane (no intersection unless the line lies in the plane).

13.2 Intersection of Two Planes

Two non-parallel planes intersect in a line. The line’s direction is given by $\mathbf{n}_1 \times \mathbf{n}_2$ (normal vectors of the two planes). To find a specific point on the line, set one variable to $0$ and solve the two plane equations simultaneously.

13.3 Intersection of Three Planes

Three planes can intersect in:

• A unique point (most common exam case — solve the $3 \times 3$ system)
• A line (two planes have the same intersection line, and the third contains it)
• No point (inconsistent system — parallel planes or other configurations)

Use row reduction (Gaussian elimination) to solve the $3 \times 3$ system. The number of solutions corresponds to the rank of the augmented matrix.

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Section 14: Practice Exam Questions

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