AI is GDC-first: rather than rearrange by hand, let your calculator’s equation solver find the unknown. Make sure the calculator is in DEGREE mode first.
TI-84 Plus CE — Equation solver
First set [MODE] → DEGREE. [MATH] → scroll to Solver... (last item)
Enter the equation as \( =0 \), e.g. cosine rule: E1: 8^2+11^2-2(8)(11)cos(X)-7.5^2
Put cursor on X, press [ALPHA][ENTER] (SOLVE) to find the angle.
TI-Nspire CX II — Equation solver
Set [doc] → Settings → Document Settings → Angle: Degree.
In a Calculator page use nSolve(, e.g. nSolve(8^2+11^2-2·8·11·cos(x)=7.5^2, x)[Enter]
Or [menu] → Algebra → Solve for the exact form.
Casio fx-CG50 — Equation solver
Set [SHIFT][MENU] → Angle: Deg first.
From the main menu open Equation → F3 (SOLVER)
Type the equation, e.g. 8^2+11^2-2×8×11×cos(X)=7.5^2
Move to X and press F6 (SOLV) to find the angle.
Where \( a \) and \( b \) are two sides, and \( C \) is the included angle between them.
Worked Example
Find the area of a triangle with sides 9 cm and 14 cm and an included angle of 67°.
Area = \( \frac{1}{2} \times 9 \times 14 \times \) sin 67°
Area = 63 \( \times 0.9205 = 57.99... \)
Answer: Area = 58.0 cm\( ^2 \)
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Which rule to use? Know an angle + opposite side pair \( \to \) sine rule. Know SAS or SSS \( \to \) cosine rule. Know two sides + included angle and want area \( \to \) \( \frac{1}{2} ab \sin C \).
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3D Geometry: Volume & Surface Area
All volume and surface area formulae are given in the formula booklet. Focus on knowing which formula to use and how to extract values from the context.
Cuboid
\( V = lwh \) \( SA = 2(lw + lh + wh) \)
Cylinder
\( V = \pi r^2h \) \( SA = 2\pi r^2 + 2\pi rh \)
Cone
\( V = \frac{1}{3} \pi r^2h \) \( SA = \pi r^2 + \pi rl \)
\( l = \) slant height
Sphere
\( V = \frac{4}{3} \pi r^3 \) \( SA = 4\pi r^2 \)
Pyramid
\( V = \frac{1}{3} \times \text{base area} \times h \)
Prism
\( V = \text{cross-section area} \times \text{length} \)
3D distance (given in formula booklet)
\( d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} \)
3D midpoint
\( M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right) \)
Worked Example
A cone has radius 5 cm and height 12 cm. Find the volume and the slant height.
Slant height: \( l = \sqrt{r^2 + h^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \) cm
Answer: V = 314 cm\( ^3, l = 13 \) cm
Angle between a line and a plane (3D)
Drop the line onto the plane to form a right triangle, then use SOH CAH TOA. For a space diagonal of a cuboid making angle \( \theta \) with the base:
\( \tan\theta = \dfrac{h}{\sqrt{l^2+w^2}} \)
where \( \sqrt{l^2+w^2} \) is the base diagonal. The angle between two lines is found the same way: build the right triangle and apply trig.
Worked Example
A box is \( 6 \times 8 \times 5 \) cm. Find the angle the space diagonal makes with the base.
Base diagonal = \( \sqrt{6^2 + 8^2} = \sqrt{100} = 10 \) cm
A bearing is measured clockwise from North, always written as a three-digit number (e.g. 045°, not 45°).
Key facts
North = 000° East = 090° South = 180° West = 270°
Back bearing = bearing \( \pm 180 \)°
Worked Example
A ship sails 20 km on a bearing of 065°, then 15 km on a bearing of 150°. Find the direct distance from the starting point.
Draw a diagram with a North line at the turning point. The two legs make bearings 065° and 150°.
At the turning point, the incoming leg’s reverse direction and the new North line give co-interior angles. The included angle is \( 180^\circ - (150^\circ - 65^\circ) = 180^\circ - 85^\circ = 95^\circ \).
Elevation = angle looking up from the horizontal. Depression = angle looking down from the horizontal.
Between two observers they are equal (alternate angles between parallel horizontals). Set up a right triangle and use SOH CAH TOA.
Worked Example
From a point 40 m from the base of a tower, the angle of elevation to the top is 32°. Find the height of the tower.
\( \tan 32^\circ = \dfrac{h}{40} \)
\( h = 40 \times \tan 32^\circ = 40 \times 0.6249 = 25.0 \) m
Answer: \( h = 25.0 \) m
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IB Exam tip: ALWAYS draw a clear diagram with North arrows for bearing questions. Label angles carefully. Most marks come from the setup, not the calculation.
Answer: Arc length = 10.5 cm, Area = 41.9 cm\( ^2 \)
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Perimeter of a sector: Don't forget the two radii. Perimeter = arc length + 2\( r \).
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GDC Setup for Trigonometry
TI-84 Plus CE — Degree mode
[MODE] → set to DEGREE (not RADIAN)
Solving triangles: enter calculations directly, e.g.:
• \( \sin^{-1} \)(7/13) → type [2nd][SIN] ( 7 / 13 ) [ENTER]
• Cosine rule: type the full expression and press [ENTER]
TI-Nspire CX II — Degree mode
[doc] → Settings → Document Settings → Angle: Degree
For a one-off degree symbol, insert ° from the symbol palette ([ctrl][catalog])
\( \sin^{-1} \)(7/13) → type \( \sin^{-1} \)(7/13) [Enter]
Casio fx-CG50 — Degree mode
[SHIFT][MENU] → Angle: Deg
Look for the D indicator at the top of the screen
\( \sin^{-1} \)(7/13) → [SHIFT][sin] ( 7 / 13 ) [EXE]
\( y - 5 = -\dfrac{1}{2}(x - 4) \Rightarrow y = -\dfrac{1}{2}x + 7 \)
Answer: \( y = -\frac{1}{2}x + 7 \)
TI-84 Plus CE — Check / graph
Enter the bisector in [Y=], e.g. Y1 = -0.5X + 7 [GRAPH] to see the line; plot A and B with [2nd][Y=] (STAT PLOT) to confirm it sits midway.
TI-Nspire CX II — Geometry
On a Graphs/Geometry page, plot points A and B, then [menu] → Construction → Perpendicular Bisector and click the two points — the bisector is drawn automatically.
Casio fx-CG50 — Graph
Open Graph, enter Y1 = -0.5X + 7, press F6 (DRAW).
Use Statistics to scatter-plot A and B and check the line bisects the segment.
10
Voronoi Diagrams
A Voronoi diagram partitions a plane around sites so every point in a cell is closest to that cell’s site.
Edges
Each edge is a perpendicular bisector between two neighbouring sites — points on it are equidistant from both.
Vertices
A vertex is where 3+ cells meet — it is equidistant from its 3 surrounding sites.
Key applications
Adding a site: draw perpendicular bisectors to neighbouring sites; the new cell is bounded by them. Nearest-neighbour interpolation: an unknown point takes the value of the site whose cell it falls in. Toxic-waste-dump (largest empty circle): the “best” (farthest) location is at a Voronoi vertex (or on a boundary edge) — test each vertex’s distance to its nearest site and choose the largest.
Worked Example
Sites are at \( A(0,0) \) and \( B(4,0) \). Find the edge separating their cells.
The edge is the perpendicular bisector of AB. Midpoint \( = (2, 0) \).
AB is horizontal (\( m_{AB} = 0 \)), so the bisector is vertical: \( x = 2 \).
Answer: the boundary is the line \( x = 2 \)
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Distance check: to find the nearest site to a point, compute the distance to each site with \( d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \) and pick the smallest — that site’s cell contains the point.
Calculator in wrong mode. The number one error in trig questions. If your answer to sin 30° is not 0.5, you are in radian mode. Switch to degrees immediately.
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Missing or incomplete diagrams. Many marks are awarded for the diagram itself. Label all known sides, angles, and North arrows. An unlabelled diagram earns zero method marks.
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Using SOH CAH TOA on non-right triangles. SOH CAH TOA only works for right-angled triangles. For all others, use the sine rule or cosine rule.
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Cosine rule sign error. It is \( a^2 + b^2 \) minus 2\( ab \) cos \( C \). The minus sign is part of the formula. If the angle is obtuse, cos \( C \) is negative, so you end up adding.
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Rounding mid-calculation. Keep full GDC precision until the final answer. Rounding the cosine value before finding the square root can lose accuracy marks.
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Formula booklet: Sine rule, cosine rule, area = \( \frac{1}{2} \) ab sin C, volume and surface area formulae are all given. You do NOT need to memorise them — but practise using them quickly.
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3 sf rule: Give answers to 3 significant figures unless the question says otherwise. For angles, give to 1 decimal place (or 3 sf — whichever the question requests).
Trigonometry & Geometry HL
IB Mathematics: Applications & Interpretation · Topic 3: Radians & the Unit Circle (HL only)
JMaths
www.jmaths.xyz
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SL students: skip this page. Everything below is HL only (AI HL syllabus 3.7 & 3.8). SL uses degrees throughout.
12
Radians HL
A radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. A full turn is \( 2\pi \) radians, so \( 360^\circ = 2\pi \) rad and \( 180^\circ = \pi \) rad.
Answer: \( l = 9.6 \) cm, \( A = 38.4 \) cm\( ^2 \)
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The Unit Circle HL
On a circle of radius 1 centred at the origin, the point at angle \( \theta \) (measured anticlockwise from the positive \( x \)-axis) has coordinates \( (\cos\theta, \sin\theta) \). So \( \cos\theta \) is the \( x \)-coordinate and \( \sin\theta \) is the \( y \)-coordinate.
For any radian work the calculator must be in RADIAN mode, or every trig value will be wrong.
TI-84 Plus CE — Radian mode
[MODE] → set to RADIAN (not DEGREE)
Then evaluate directly, e.g. arc length: 8 × 1.2[ENTER]
Check: [SIN] ( π / 6 ) [ENTER] should give 0.5.
TI-Nspire CX II — Radian mode
[doc] → Settings → Document Settings → Angle: Radian
Then \( \sin(\pi/6) \) [Enter] returns 0.5.
Tip: a trailing ° or \( ^r \) symbol forces a one-off unit if your default differs.
Casio fx-CG50 — Radian mode
[SHIFT][MENU] (SET UP) → Angle: Rad
Look for the R indicator at the top of the screen.
Check: [sin] ( π / 6 ) [EXE] should give 0.5.
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Degree/radian mode mismatch — the classic error. Using degree mode for radian work (or vice versa) gives completely wrong trig values with no warning. Before any radian question, confirm the mode: \( \sin\dfrac{\pi}{6} \) must equal \( 0.5 \). If it doesn't, you are in the wrong mode.
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Match the units to the formula. \( l = r\theta \) and \( A = \frac{1}{2} r^2\theta \) need \( \theta \) in radians; the SL versions with \( /360 \) need degrees. Decide which form you are using, then set the GDC to match.
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