Non-right triangle trigonometry

IB Mathematics AA HL· 6 min read

1. The Sine Rule and Ambiguous Case★★☆☆☆⏱ 15 min

📘 Definition

Sine Rule

For triangle $A B C$ , with side $a$ opposite angle $A$ , $b$ opposite $B$ , $c$ opposite $C$

The ratio of any side to the sine of its opposite angle is constant for all sides of the triangle

Example:

$\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}$

The sine rule is used when we know either (1) two angles and one side (AAS/ASA), or (2) two sides and a non-included angle (SSA). The second SSA case is called the ambiguous case, because two different triangles can satisfy the given measurements.

📐 Worked Example

Find all possible values of angle $B$ in triangle $A B C$ where $a = 10$ cm, $A = 3 0^{\circ}$ , $b = 12$ cm

1
Apply the sine rule, rearrange to solve for $sin B$ :
2
$\frac{10}{\sin 30^\circ} = \frac{12}{\sin B}$
3
Calculate the left-hand side, since $sin 3 0^{\circ} = 0.5$ :
4
$\sin B = \frac{12 \times 0.5}{10} = 0.6$
5
The first solution from the calculator is the acute angle:
6
$B_1 = \arcsin(0.6) \approx 36.9^\circ$
7
The second possible solution is obtuse, since $sin θ = sin (18 0^{\circ} - θ)$ :
8
$B_2 = 180^\circ - 36.9^\circ = 143.1^\circ$
9
Check validity: $143. 1^{\circ} + 3 0^{\circ} = 173. 1^{\circ} < 18 0^{\circ}$ , so both solutions are valid.

2. The Cosine Rule★★☆☆☆⏱ 15 min

📘 Definition

Cosine Rule

Same notation for triangle $A B C$ as before

Relates the lengths of sides to the cosine of an angle, with two common forms for finding sides or angles

Example:

Side form: $c^{2} = a^{2} + b^{2} - 2 ab cos C$ ; Angle form: $cos C = \frac{a ^{2} + b ^{2} - c ^{2}}{2 ab}$

The cosine rule is used when we know either (1) three sides (SSS), or (2) two sides and the included angle (SAS). Unlike the sine rule, there is no ambiguous case here, because the cosine of an obtuse angle is negative, giving a unique solution.

📐 Worked Example

Find the length of side $c$ in triangle $A B C$ where $a = 5$ cm, $b = 7$ cm, and the included angle $C = 6 0^{\circ}$

1
Use the side form of the cosine rule:
2
$c^2 = a^2 + b^2 - 2ab\cos C$
3
Substitute the known values, $cos 6 0^{\circ} = 0.5$ :
4
$c^2 = 5^2 + 7^2 - 2(5)(7)(0.5) = 25 + 49 - 35 = 39$
5
Take the positive square root for side length:
6
$c = \sqrt{39} \approx 6.25 \text{ cm (3 s.f.)}$

✓ Quick check

Test your understanding of when to use the cosine rule:

Which of the following cases requires the cosine rule?
- Two angles and one side
- Two sides and non-included angle
- Two sides and included angle
- All of the above
Reveal answer
2 —
Correct. Only SAS or SSS cases require the cosine rule; other cases are solved faster with the sine rule.

3. Area of a Non-Right Triangle★☆☆☆☆⏱ 10 min

📘 Definition

Area Formula for Any Triangle

The area of any triangle is half the product of two sides multiplied by the sine of the included angle between them

Example:

$Area = \frac{1}{2} ab sin C$

This formula works for all triangles, right and non-right. If you know all three sides but no angles, you can first use the cosine rule to find any included angle, then apply the area formula.

📐 Worked Example

Find the area of triangle $A B C$ with $a = 4$ cm, $b = 6$ cm, and included angle $C = 3 5^{\circ}$

1
Substitute into the area formula:
2
$\text{Area} = \frac{1}{2} (4)(6) \sin 35^\circ$
3
Calculate the result, rounding to 3 significant figures:
4
$\text{Area} = 12 \times 0.5736 \approx 6.88 \text{ cm}^2$

4. Choosing the Correct Rule★★★☆☆⏱ 20 min

Methods compared

The rule you choose depends entirely on the information you are given. Here is a quick comparison:

Sine Rule

Use for AAS, ASA, or SSA (two sides + non-included angle)

+ Pros: Simpler calculations than the cosine rule

− Cons: Requires checking for ambiguous case in SSA

Cosine Rule

Use for SAS (two sides + included angle) or SSS (three sides)

+ Pros: No ambiguous case, always gives a unique solution

− Cons: More arithmetic steps, higher chance of substitution error

Area Formula

Use to find area when you know two sides and included angle

+ Pros: Fast calculation once the angle is known

− Cons: Cannot be used directly without an included angle

5. Common Pitfalls

Wrong move:

Forgetting to check the second solution in SSA sine rule cases

Why:

Calculators only return the acute solution for arcsine, so you will miss the second valid obtuse solution

Correct move:

Always calculate $18 0^{\circ} - arcsin (sin B)$ and check if the sum with the known angle is less than $18 0^{\circ}$ ; add it to your answer if valid

Wrong move:

Matching the wrong side to the wrong angle in the sine/cosine rule

Why:

Mislabeling the triangle leads to completely incorrect calculations

Correct move:

Always label the triangle clearly before starting, confirm side $a$ is always opposite angle $A$

Wrong move:

Rounding intermediate values too early

Why:

Early rounding leads to inaccurate final answers that lose accuracy marks

Correct move:

Keep full calculator precision for all intermediate steps, only round your final answer

Wrong move:

Using the sine rule for an included angle between two known sides

Why:

This adds unnecessary steps and increases the chance of error

Correct move:

Use the cosine rule directly for SAS cases to find the unknown side immediately

Wrong move:

Wasting time calculating an unknown height for area

Why:

Many students try to find height to use $\frac{1}{2} \times base \times height$ when the simpler formula works

Correct move:

If you know two sides and the included angle, use $\frac{1}{2} ab sin C$ directly

6. Quick Reference Cheatsheet

Given Information	Rule to Use
Two angles + one side	Sine Rule
Two sides + non-included angle	Sine Rule (check ambiguous case)
Two sides + included angle (find side)	Cosine Rule
Two sides + included angle (find area)	$\frac{1}{2} ab sin C$
Three sides (find angles)	Cosine Rule

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Find side length using sine rule
2021 · 2
Calculate area of oblique triangle
2023 · 1
Solve ambiguous sine rule case

Going deeper

referenceIB AA HL Formula BookletAll rules in this topic are included in the formula booklet

What's Next

Non-right triangle trigonometry is the foundation for all advanced trigonometric and vector topics in IB AA HL. You will next apply these rules to 3D geometry, solving for distances and angles between lines and planes. The cosine rule is also directly related to the vector dot product formula, which you will use extensively when working with vectors in 2D and 3D. Mastering the skill of choosing the right rule and checking for the ambiguous case will help you avoid common mistakes in these more complex topics later in the course.