Triangle Trigonometry — SL 3.2 Trigonometry in right triangles and non-right triangles (sine/cosine rules; area) - IB Mathematics Applications and Interpretation

SECTION 1

Historical Context & Motivation

Humans have been measuring triangles for thousands of years. Ancient civilizations needed to survey land, navigate oceans, and build monuments — all tasks that boil down to calculating unknown sides and angles of triangles. The word trigonometry itself comes from the Greek words trigonon (triangle) and metron (measure). Long before calculators existed, mathematicians across cultures developed increasingly powerful tools for solving triangles.

~1800 BCE • Babylonian Clay Tablets

The Plimpton 322 tablet shows that Babylonians already understood relationships between the sides of right triangles, listing Pythagorean triples centuries before Pythagoras.

~150 CE • Ptolemy's Chord Tables

Greek astronomer Claudius Ptolemy created detailed tables of chord lengths in circles, effectively building the first trigonometric reference tables for use in astronomy and navigation.

~800 CE • Islamic Golden Age — Sine & Cosine

Persian and Arab mathematicians such as al-Khwārizmī and al-Battānī replaced Ptolemy's chord system with the sine and cosine functions we still use today, making calculations far more efficient.

1596 • Law of Cosines Published

François Viète formalized the cosine rule for non-right triangles, extending trigonometry beyond the limitations of right-angle methods and opening the door to general triangle solving.

Modern Era • IB Mathematics — SL 3.2

Today the sine rule, cosine rule, and the area formula form the core toolkit tested in the IB Applications and Interpretation course, enabling students to solve any triangle in real-world contexts.

The central question this topic answers is deceptively simple: given some information about a triangle — a few sides, a few angles — how do you find everything else? Right-triangle trigonometry (SOH CAH TOA) handles the special case with a 90° angle. But most real-world triangles are non-right triangles, and that's where the sine rule, cosine rule, and the area formula become essential.

SECTION 2

Core Principles & Definitions

Before diving into formulas, it helps to establish the conventions used throughout this topic. In any triangle, we label vertices with uppercase letters A, B, C and the sides opposite those vertices with the corresponding lowercase letters a, b, c. So side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. This convention keeps formulas clean and makes it easy to identify which rule to apply.

Right-Triangle Trigonometry (SOH CAH TOA)

In a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. This only works when one angle is exactly 90°.

The Sine Rule

Relates each side to the sine of its opposite angle: a/sin A = b/sin B = c/sin C. Used when you know an angle–side opposite pair plus one more piece of information.

The Cosine Rule

Generalizes the Pythagorean theorem: c² = a² + b² − 2ab cos C. Used when you know two sides and the included angle (SAS), or all three sides (SSS).

Area Formula (½ ab sin C)

Calculates a triangle's area using two sides and the included angle, without needing the perpendicular height: Area = ½ × a × b × sin C.

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KEY TAKEAWAY

Think of your trigonometry toolkit like a car's gearbox. SOH CAH TOA is first gear — it only works in right triangles. The sine rule and cosine rule are higher gears that handle any triangle. Choosing the right gear depends on which pieces of information you already have.

SECTION 3

Visual Explanation — Triangle Labelling & Relationships

The triangle above shows the standard IB convention: vertex A sits at the top with angle A, and the side directly across from it (connecting B and C) is labelled a in pink. The same logic applies to sides b (amber) and c (emerald).

This labelling system is not arbitrary — it directly powers the formulas you'll use. When the sine rule says a / sin A = b / sin B, it's pairing each side with the angle that faces it. If you mix up which side goes with which angle, the formula will give you the wrong answer. Take a moment to internalize this: each lowercase side letter shares its name with the uppercase angle directly across the triangle.

SECTION 4

Mathematical Framework

Right-Triangle Ratios (SOH CAH TOA)

When a triangle contains a 90° angle, you can use the basic trigonometric ratios. These are your starting point and remain useful even within non-right triangle problems (since you can sometimes drop a perpendicular to create two right triangles).

RIGHT-TRIANGLE RATIOS

sin θ = opposite / hypotenuse cos θ = adjacent / hypotenuse tan θ = opposite / adjacent

θ is the acute angle of interest; the hypotenuse is always the longest side (opposite the 90° angle); opposite faces θ; adjacent sits next to θ (and is not the hypotenuse).

The Sine Rule

SINE RULE

a / sin A = b / sin B = c / sin C

Use this when you know an angle and its opposite side (an angle–side pair) plus at least one more piece of information. Common setups: AAS (two angles + a non-included side) or ASA (two angles + the included side).

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The Ambiguous Case (SSA)

When you know two sides and a non-included angle (SSA), the sine rule can sometimes yield two possible triangles. This happens because sin θ = sin(180° − θ). Always check whether the second solution is valid by verifying that the sum of angles doesn't exceed 180°.

The Cosine Rule

COSINE RULE — FINDING A SIDE

c² = a² + b² − 2ab cos C

Use when you know two sides and the included angle (SAS). The side you're finding (c) is opposite the known angle (C). Notice that if C = 90°, then cos 90° = 0, and the formula collapses to the Pythagorean theorem: c² = a² + b².

COSINE RULE — FINDING AN ANGLE

cos C = (a² + b² − c²) / (2ab)

Rearranged form. Use when you know all three sides (SSS) and need to find an angle. Take the inverse cosine (cos⁻¹) of the result to get the angle in degrees.

Area of a Triangle

AREA FORMULA

Area = ½ × a × b × sin C

Here a and b are any two sides, and C is the angle between those two sides (the included angle). This replaces the base × height formula when height is hard to measure directly.

SECTION 5

Decision Guide — Which Rule to Use

One of the most common mistakes students make is grabbing the wrong formula. The key is to look at what information you already have and what you need to find. The flowchart below makes the decision almost automatic.

Follow the flowchart from top to bottom. First ask whether the triangle has a right angle. If not, determine whether you have an angle–side opposite pair (→ sine rule) or SAS/SSS information (→ cosine rule). The area formula works whenever you know two sides and the included angle.

Given Information	Configuration	Rule to Use
Right angle + a side + another side or angle	Right triangle	SOH CAH TOA
Two angles + one side	AAS or ASA	Sine Rule
Two sides + a non-included angle	SSA (ambiguous)	Sine Rule — check for two solutions
Two sides + the included angle	SAS	Cosine Rule
Three sides	SSS	Cosine Rule
Two sides + included angle (area needed)	SAS	Area = ½ab sin C

Quick-reference table for rule selection

SECTION 6

Worked Example — Cosine Rule & Area Formula

A surveyor measures two sides of a triangular plot of land as a = 120 m and b = 85 m, with the included angle C = 62°. Find the length of the third side and the area of the plot.

Finding Side c and the Area

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Step 1 — Identify Given Values and Choose a Rule

We know two sides (a = 120, b = 85) and the included angle (C = 62°). This is an SAS configuration, so we use the cosine rule to find side c.
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Step 2 — Apply the Cosine Rule

c² = a² + b² − 2ab cos C c² = 120² + 85² − 2(120)(85) cos 62° c² = 14400 + 7225 − 20400 × cos 62° c² = 21625 − 20400 × 0.46947 c² = 21625 − 9577.2 c² = 12047.8

c = √12047.8 ≈ 109.8 m
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Step 3 — Calculate the Area

We already have two sides and the included angle, so we apply the area formula: Area = ½ × a × b × sin C Area = ½ × 120 × 85 × sin 62° Area = ½ × 120 × 85 × 0.88295 Area = 5100 × 0.88295

Area ≈ 4503 m²
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Step 4 — Verify Reasonableness

The third side (109.8 m) is between 35 m (|120 − 85|) and 205 m (120 + 85), satisfying the triangle inequality. The area of roughly 4503 m² (about 0.45 hectares) is reasonable for a plot with sides of ~85–120 m. Our answers make sense.

SECTION 7

Comparing the Rules — Strengths & Limitations

Feature	Sine Rule	Cosine Rule
Best for	Finding a missing side or angle when you have an angle–side opposite pair	Finding a side (SAS) or angle (SSS) without needing an opposite pair
Ease of use	Simpler algebra — just a proportion to solve	More complex — requires squaring, subtracting, then square-rooting
Ambiguity risk	Yes — the SSA ambiguous case can give 0, 1, or 2 solutions	No ambiguity — always gives a unique answer
Connection to Pythagoras	No direct connection	Reduces to Pythagorean theorem when the angle is 90°
IB exam frequency	Very common — often in Paper 1 and Paper 2	Very common — especially in multi-step contextual problems

Side-by-side comparison of the sine and cosine rules

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KEY TAKEAWAY

Neither rule is "better" — they're complementary tools. Think of it like a Phillips-head versus a flat-head screwdriver: each is perfect for its specific screw type. The sine rule shines when you already have an opposite angle–side pair; the cosine rule steps in when you don't. On the IB exam, the most efficient strategy is to quickly identify your configuration (AAS, SAS, SSS, etc.) and then reach for the matching tool.

SECTION 8

Connection to Advanced Theory

The rules you've learned in SL 3.2 form the foundation for more advanced topics you may encounter later in mathematics and science. Understanding where these ideas lead can deepen your appreciation of why they matter.

SL 3.2 Concept	Where It Leads
Sine Rule	The Extended Law of Sines (a/sin A = 2R, where R is the circumradius) connects triangle trigonometry to circles. In HL, you'll also explore proofs using the unit circle.
Cosine Rule	Generalizes to the dot product of vectors in linear algebra: a · b = \|a\|\|b\| cos θ. This is fundamental in physics, computer graphics, and machine learning.
Area = ½ab sin C	Leads to the cross product magnitude in 3D vectors (\|a × b\| = \|a\|\|b\| sin θ) and to Heron's formula for area using only side lengths.
Triangle solving	Trigonometric identities (sum/difference, double angle) build on these foundations. Applications extend to surveying, GPS triangulation, and 3D modelling.

How SL 3.2 topics connect to more advanced mathematics

For now, the most important thing is fluency with the three main tools — sine rule, cosine rule, and area formula. Master these, and you'll be well prepared not just for IB exams but for any future course that involves spatial reasoning and measurement.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

In triangle PQR, you know angle P = 48°, angle Q = 67°, and side q = 14 cm. Explain which rule you would use to find side p, and why.

Show answer

You should use the sine rule. You already know angle Q and its opposite side q (an angle–side opposite pair), and you know angle P, so you can write p/sin P = q/sin Q. This gives p/sin 48° = 14/sin 67°. The cosine rule is not needed here because the sine rule is simpler when an opposite pair is available.

Question 2: BASIC CALCULATION

In triangle ABC, angle A = 35°, angle B = 78°, and side a = 10 cm. Find side b using the sine rule. Give your answer correct to 1 decimal place.

Show answer

Using a/sin A = b/sin B: 10/sin 35° = b/sin 78°. So b = 10 × sin 78° / sin 35° = 10 × 0.97815 / 0.57358 ≈ 17.1 cm. The sine rule makes this a straightforward proportion. Note that angle C = 180° − 35° − 78° = 67°, confirming the triangle is valid.

Question 3: INTERMEDIATE

In triangle DEF, d = 9 cm, e = 12 cm, and f = 15 cm. Find angle F to the nearest degree. Then calculate the area of the triangle.

Show answer

Since all three sides are known (SSS), use the cosine rule rearranged for angle F: cos F = (d² + e² − f²) / (2de) = (81 + 144 − 225) / (2 × 9 × 12) = 0 / 216 = 0. Therefore F = cos⁻¹(0) = 90°. This is actually a right triangle (a 3-4-5 triple scaled by 3). For the area: Area = ½ × d × e × sin F = ½ × 9 × 12 × sin 90° = ½ × 9 × 12 × 1 = 54 cm². Alternatively, since it's a right triangle, Area = ½ × 9 × 12 = 54 cm².

Question 4: APPLIED

Two fire towers are 24 km apart. From tower A, a fire is observed at a bearing of 058°. From tower B (due east of A), the same fire is observed at a bearing of 320°. Find the distance from tower A to the fire, correct to 1 decimal place.

Show answer

Let F be the fire. Tower B is due east of A, so the baseline AB runs along 090°. At A, the angle between AB and AF is 90° − 58° = 32°. At B, the bearing of 320° means the angle between BA (bearing 270°) and BF is 320° − 270° = 50°. The angle at F = 180° − 32° − 50° = 98°. Using the sine rule: AF/sin B = AB/sin F → AF/sin 50° = 24/sin 98° → AF = 24 × sin 50° / sin 98° = 24 × 0.76604 / 0.99027 ≈ 18.6 km. The fire is approximately 18.6 km from tower A.

Question 5: CRITICAL THINKING

In triangle XYZ, x = 8, y = 5, and angle X = 40°. Show that there are two possible triangles satisfying these conditions. Find both possible values of angle Y and the corresponding areas.

Show answer

This is an SSA (ambiguous case) problem. Using the sine rule: sin Y / y = sin X / x → sin Y = 5 × sin 40° / 8 = 5 × 0.64279 / 8 ≈ 0.40174. So Y = sin⁻¹(0.40174) ≈ 23.7° (first solution). The second solution is Y' = 180° − 23.7° = 156.3°. Check validity: X + Y' = 40° + 156.3° = 196.3° > 180°, so the second solution is invalid. Wait — let's recheck: 40° + 156.3° = 196.3°, which exceeds 180°. Therefore only one triangle exists (Y ≈ 23.7°), and Z = 180° − 40° − 23.7° = 116.3°. Area = ½ × x × y × sin Z = ½ × 8 × 5 × sin 116.3° = 20 × 0.8957 ≈ 17.9 square units. In this case, although SSA can be ambiguous, only one valid triangle is possible because the second angle Y' makes the angle sum exceed 180°. This demonstrates why you must always verify both candidate solutions.

SUMMARY

Lesson Summary

Summary

Triangle trigonometry at the SL 3.2 level equips you with three powerful tools for solving any triangle. For right triangles, use SOH CAH TOA (sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent). For non-right triangles, the sine rule (a/sin A = b/sin B = c/sin C) works when you have an angle–side opposite pair, while the cosine rule (c² = a² + b² − 2ab cos C) handles SAS and SSS configurations. The area formula (Area = ½ab sin C) finds a triangle's area from two sides and the included angle.

The key to exam success is rule selection: identify what you know (right angle? opposite pair? SAS? SSS?), pick the matching formula, substitute carefully, and always check your answer is reasonable. Watch out for the ambiguous case (SSA) with the sine rule, where two solutions may exist. These three formulas — all provided in the IB formula booklet — are the backbone of triangle trigonometry and connect to advanced topics like vectors, surveying, and 3D geometry.