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Trigonometry & Geometry SL

IB Mathematics: Applications & Interpretation · Topic 3: Geometry & Trigonometry
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1
Right-Angled Trigonometry (SOH CAH TOA)
For a right-angled triangle, label the sides relative to angle \( \theta \): Opposite, Adjacent, Hypotenuse.
adjacent (A) opposite (O) hypotenuse (H) θ SOH CAH TOA
Trigonometric ratios
\( \sin \theta = \frac{O}{H} \quad \cos \theta = \frac{A}{H} \quad \tan \theta = \frac{O}{A} \)
Finding an angle
\( \theta = \sin^{-1}(O / H) \)
Similarly for \( \cos^{-1} \) and \( \tan^{-1} \)
Worked Example
In a right triangle, the side opposite \( \theta \) is 7 cm and the hypotenuse is 13 cm. Find \( \theta \).
\( \sin \theta = 7 / 13 = 0.5385... \)
\( \theta = \sin^{-1}(0.5385) = 32.6^\circ \)
Answer: \( \theta = 32.6^\circ \)
2
Sine Rule
Use the sine rule for non-right-angled triangles when you know an angle and its opposite side, plus one more piece of information.
B C A c a b B C A Side a is opposite angle A, etc.
Sine rule (given in formula booklet)
\( a / \sin A = b / \sin B = c / \sin C \)
Use to find a missing side or a missing angle. At SL, you do NOT need the ambiguous case.
Worked Example
In triangle ABC, \( A = 40 \)°, \( B = 73 \)°, and \( a = 15 \) cm. Find side \( b \).
\( a / \) sin \( A = b / \) sin \( B \)
15 / sin 40° = \( b / \) sin 73°
\( b = 15 \times \) sin 73° / sin 40° = 22.3 cm
Answer: \( b = 22.3 \) cm
3
Cosine Rule
Use the cosine rule when you know two sides and the included angle (SAS) or all three sides (SSS).
Finding a side (given in formula booklet)
\( c^2 = a^2 + b^2 - 2ab \cos C \)
Finding an angle
\( \cos C = (a^2 + b^2 - c^2) / (2ab) \)
Worked Example
In triangle PQR, \( p = 8 \) cm, \( q = 11 \) cm, and angle \( R = 52 \)°. Find side \( r \).
\( r^2 = 8^2 + 11^2 - 2(8)(11) \) cos 52°
\( r^2 = 64 + 121 - 176 \times 0.6157 = 185 - 108.4 = 76.6 \)
\( r = \sqrt{76}.6 = 8.75 \) cm
Answer: \( r = 8.75 \) cm
4
Area of a Triangle
Area formula (given in formula booklet)
\( Area = \frac{1}{2} ab \sin C \)
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 \( \to \) \( \frac{1}{2} ab \sin C \).
JMaths · www.jmaths.xyz Trigonometry & Geometry · Page 1 of 3

Trigonometry & Geometry SL

IB Mathematics: Applications & Interpretation · Topic 3: Geometry & Trigonometry
www.jmaths.xyz
5
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
\( V = \frac{1}{3} \times base area \times h \)
Prism
\( V = cross-section area \times length \)
3D distance: Distance between two points in 3D = \( \sqrt{(x_2-x_1}^2 + (y_2-y_1)^2 + (z_2-z_1)^2) \)
Worked Example
A cone has radius 5 cm and height 12 cm. Find the volume and the slant height.
V = \( \frac{1}{3} \pi (5)^2(12) = \frac{1}{3} \times 300\pi = 100\pi = 314 \) cm\( ^3 \)
Slant height: \( l = \sqrt{r^2 + h^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \) cm
Answer: V = 314 cm\( ^3, l = 13 \) cm
6
Bearings
A bearing is measured clockwise from North, always written as a three-digit number (e.g. 045°, not 45°).
N 045° P Bearing measured clockwise from N
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 North lines at each turning point.
The angle between the two paths = 180° \( - 65 \)° + (150° \( - 180 \)°) ... use the diagram to find the included angle = 95°
Cosine rule: \( d^2 = 20^2 + 15^2 - 2(20)(15) \) cos 95°
\( d^2 = 400 + 225 - 600(-0.0872) = 625 + 52.3 = 677.3 \)
\( d = \sqrt{677}.3 = 26.0 \) km
Answer: 26.0 km
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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.
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Trigonometry & Geometry SL

IB Mathematics: Applications & Interpretation · Topic 3: Geometry & Trigonometry
www.jmaths.xyz
7
Arc Length & Sector Area (Degrees)
At SL, arcs and sectors use degrees (NOT radians). The formulae find the fraction of the full circle.
Arc length
\( l = (\theta / 360°) \times 2\pi r \)
Sector area
\( A = (\theta / 360°) \times \pi r^2 \)
Worked Example
A sector has radius 8 cm and angle 75°. Find the arc length and area.
Arc length = (75 / 360) \( \times 2\pi (8) = (75 / 360) \times 50.27 = 10.5 \) cm
Area = (75 / 360) \( \times \pi (8)^2 = (75 / 360) \times 201.1 = 41.9 \) cm\( ^2 \)
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\( r \).
8
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] → Angle: Degree
Or type the degree symbol: press [CTRL] + catalogue → °
\( \sin^{-1} \)(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]
9
Common Errors & Exam Traps
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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).
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