Knowledge organisers

Geometry and trigonometry: IB Maths AA HL knowledge organiser

Everything to know about geometry and trigonometry on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Scalar product: a · b = |a||b|cos θ = a₁b₁ + a₂b₂ + a₃b₃; it is zero when a and b are perpendicular.
Vector equation of a line: r = a + λb: a is a point on the line and b its direction.
Vector product: a × b is a vector perpendicular to both a and b, with length |a||b|sin θ.
Reciprocal trig functions: sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = 1/tan θ.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

3D distance (\(\Delta x=x_1-x_2\), …)In the formula booklet	\(d=\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}\)
3D midpointIn the formula booklet	\(\left(\tfrac{x_1+x_2}2,\tfrac{y_1+y_2}2,\tfrac{z_1+z_2}2\right)\)
Pyramid; cone; sphereIn the formula booklet	\(V=\tfrac13Ah;\) \(V=\tfrac13\pi r^2h,\) \(A=\pi rl;\) \(V=\tfrac43\pi r^3,\) \(A=4\pi r^2\)
Sine, cosine rules; areaIn the formula booklet	\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C},\) \(c^2=a^2+b^2-2ab\cos C,\) \(A=\tfrac12ab\sin C\)
Arc; sector (radians)In the formula booklet	\(l=r\theta,\) \(A=\tfrac12r^2\theta\)
IdentitiesIn the formula booklet	\(\tan\theta=\frac{\sin\theta}{\cos\theta},\) \(\cos^2\theta+\sin^2\theta=1\)
Reciprocal ratiosIn the formula booklet	\(\sec\theta=\frac1{\cos\theta},\) \(\cosec\theta=\frac1{\sin\theta},\) \(\cot\theta=\frac{1}{\tan\theta}\)
PythagoreanIn the formula booklet	\(1+\tan^2\theta=\sec^2\theta,\) \(1+\cot^2\theta=\cosec^2\theta\)
Compound anglesIn the formula booklet	\(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B,\) \(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\)
Compound tanIn the formula booklet	\(\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}\)
Double angleIn the formula booklet	\(\sin2\theta=2\sin\theta\cos\theta,\) \(\cos2\theta=2\cos^2\theta-1=1-2\sin^2\theta,\) \(\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta}\)
Exact values \(0,\frac\pi6,\frac\pi4,\frac\pi3,\frac\pi2\)	\(\sin:0,\tfrac12,\tfrac{\sqrt2}2,\tfrac{\sqrt3}2,1;\) \(\cos\text{: reverse};\) \(\tan:0,\tfrac1{\sqrt3},1,\sqrt3,\text{undef.}\)
Ranges	\(\arcsin x\in[-\tfrac\pi2,\tfrac\pi2],\) \(\arccos x\in[0,\pi],\) \(\arctan x\in(-\tfrac\pi2,\tfrac\pi2)\)
\(a\sin(b(x-c))+d\), \(b>0\)	\(\text{amplitude }\|a\|,\) \(\text{period }\tfrac{2\pi}b\)
Vector magnitudeIn the formula booklet	\(\|\boldsymbol v\|=\sqrt{v_1^2+v_2^2+v_3^2}\)
Scalar productIn the formula booklet	\(\boldsymbol v\cdot\boldsymbol w=v_1w_1+v_2w_2+v_3w_3=\|\boldsymbol v\|\|\boldsymbol w\|\cos\theta\)

More formulas are on the full IB Maths AA HL formula sheet.

Worked example

Find the angle between a = (1, 2, 2) and b = (2, −1, 2).

a · b = 2 − 2 + 4 = 4
|a| = 3 and |b| = 3
cos θ = 4/9

Answer: θ = 63.6° (3 s.f.)

Common mistakes

Trigonometric equations: missing solutions (obtuse, general or divided away)
Vectors: skew lines without the non-parallel check, and planes
Using the wrong right-angled triangle when isolating a 3D angle
Forgetting whether the calculator is in radians or degrees

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Find the distance between two points and the midpoint of a line segment in three dimensions, using the 3D form of Pythagoras' theorem.
Use area = ½ab sin C to find areas and unknown sides or angles, and apply it with the sine and cosine rules to problems in context.
Solve equations such as 2 sin x = 1 in a given interval, finding every solution with the unit circle or a graph, in degrees and radians.
Given one trigonometric ratio and the quadrant, find the exact values of the others without finding the angle.
Write the equation of a plane in vector, parametric and Cartesian forms, using a normal vector found with the vector product.
Use the compound-angle formulae, the double-angle formula for tan 2θ and the identities linking tan with sec and cot with cosec to simplify and prove results.

The printable sheet

Geometry and trigonometry knowledge organiser for IB Maths AA HL: one A4 page of key definitions, formulas, a worked example and common mistakes — Geometry and trigonometry knowledge organiser (IB Maths AA HL), A4. Download the PDF.

Revise it next

Other IB Maths AA HL topics: Number and algebra · Functions · Statistics and probability · Calculus · All IB Maths AA HL organisers