Knowledge organisers

Geometry and trigonometry: IB Maths AA SL 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

Radian: The angle at the centre of a circle when the arc is as long as the radius; π radians = 180°.
Unit circle: A circle of radius 1 centred at the origin: the point at angle θ is (cos θ, sin θ).
Sine rule: In any triangle, a/sin A = b/sin B = c/sin C. Use it with a side and its opposite angle.
Ambiguous case: Given two sides and a non-included angle, the sine rule can give two possible triangles.

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 distanceIn the formula booklet	\(d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}\)
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; coneIn the formula booklet	\(V=\tfrac13Ah;\) \(V=\tfrac13\pi r^2h,\) \(\text{curved }A=\pi rl\)
SphereIn the formula booklet	\(V=\tfrac43\pi r^3,\) \(A=4\pi r^2\)
Sine ruleIn the formula booklet	\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
Cosine ruleIn the formula booklet	\(c^2=a^2+b^2-2ab\cos C,\) \(\cos C=\frac{a^2+b^2-c^2}{2ab}\)
Area of a triangleIn the formula booklet	\(A=\tfrac12ab\sin C\)
Ambiguous case: given \(a,b,A\) with \(A\) acute, two triangles if \(b\sin A<a<b\)
Arc; sector (\(\theta\) in radians)In the formula booklet	\(l=r\theta,\) \(A=\tfrac12r^2\theta\)
Radians	\(\pi\text{ rad}=180^\circ\)
IdentitiesIn the formula booklet	\(\tan\theta=\frac{\sin\theta}{\cos\theta},\) \(\cos^2\theta+\sin^2\theta=1\)
Double angleIn the formula booklet	\(\sin2\theta=2\sin\theta\cos\theta,\) \(\cos2\theta=\cos^2\theta-\sin^2\theta\) \({}=2\cos^2\theta-1=1-2\sin^2\theta\)
Exact values at \(0,\frac\pi6,\frac\pi4,\frac\pi3,\frac\pi2\)	\(\sin:\ 0,\tfrac12,\tfrac{\sqrt2}2,\tfrac{\sqrt3}2,1;\) \(\cos\text{: reverse order};\) \(\tan:\ 0,\tfrac1{\sqrt3},1,\sqrt3,\text{undefined}\)
Symmetry	\(\sin(\pi-\theta)=\sin\theta,\) \(\cos(\pi-\theta)=-\cos\theta,\) \(\cos(-\theta)=\cos\theta,\) \(\sin(-\theta)=-\sin\theta\)
\(y=a\sin\big(b(x-c)\big)+d\), \(b>0\)	\(\text{amplitude }\|a\|,\) \(\text{period }\tfrac{2\pi}b,\) \(\text{principal axis }y=d\)
\(\sin x=k\): \(x=\arcsin k\) or \(\pi-\arcsin k\), then add \(2\pi n\); keep solutions in the interval

Worked example

A sector has radius 6 cm and angle 1.2 radians. Find its arc length and area.

Arc length s = rθ = 6 × 1.2
Area A = ½r²θ = ½ × 36 × 1.2

Answer: Arc = 7.2 cm and area = 21.6 cm²

Common mistakes

GDC in the wrong angle mode, and degrees in radian formulas
Trigonometric equations: one solution only, and exact values not known
Using the wrong right-angled triangle when isolating a 3D angle
Forgetting to set the calculator to 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 the cosine rule to find a side from two sides and the included angle, and an angle from three sides, choosing between the sine and cosine rules.
Use A = ½r²θ to find the area of a sector, and combine it with ½ab sin C to find the area of a segment.
Use cos²θ + sin²θ = 1 and the double-angle identities for sin 2θ and cos 2θ to find exact values and simplify expressions.
Read the amplitude, period, principal axis and horizontal shift from a graph or context and write its equation y = a sin(b(x − c)) + d.
Find the parameters of a sine or cosine model from data or a description and use it to answer questions in context.

The printable sheet

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

Revise it next

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