What is the Sine Rule in Geometry and Trigonometry?

The Sine Rule, also known as the Law of Sines, is a formula used in Geometry and Trigonometry to find unknown lengths or angles in any triangle. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides of the triangle. Mathematically, it is expressed as a/sin(A) = b/sin(B) = c/sin(C), where a, b, and c are the sides of the triangle, and A, B, and C are the respective opposite angles.

How do you apply the Cosine Rule in solving triangles?

The Cosine Rule, or Law of Cosines, is used in Geometry and Trigonometry to find a side or angle in a triangle when you know either two sides and the included angle or all three sides. It is expressed as c² = a² + b² - 2ab * cos(C), where a, b, and c are the sides of the triangle, and C is the angle opposite side c. This rule is particularly useful in non-right angled triangles.

When should I use the Sine Rule versus the Cosine Rule?

In the IB MYP Mathematics curriculum, the Sine Rule is typically used when you have either two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). The Cosine Rule is more suitable when you have two sides and the included angle (SAS) or all three sides (SSS). Choosing the correct rule depends on the given information in the problem.

Can the Sine and Cosine Rule be used in right-angled triangles?

While the Sine and Cosine Rules can be applied to right-angled triangles, they are not necessary since right-angled triangles can be solved using basic trigonometric ratios (sine, cosine, and tangent). However, these rules are essential for solving non-right angled triangles, where the basic trigonometric ratios are not applicable.

What are some common mistakes to avoid when using the Sine and Cosine Rule?

Common mistakes include using the wrong rule for the given information, not ensuring the angle is opposite the side used in the ratio, and incorrect calculations of the sine or cosine values. It's also important to ensure angle measures are in the correct units (degrees or radians) as required by the problem. Carefully analyzing the triangle and double-checking calculations can help avoid these errors.

Why is "Using sine rule when you don't have a side-angle pair." a common mistake in Sine and Cosine Rule?

Why it happens: Defaulting to sine rule. How to avoid it: Without a side and its opposite angle, sine rule has no anchor. Use cosine rule when you have 3 sides or 2 sides + included angle.

Why is "Forgetting the minus sign in $-2bc\cos A$." a common mistake in Sine and Cosine Rule?

Why it happens: Slip. How to avoid it: Always write the formula in full: a^2 = b^2 + c^2 - 2bc A. The minus is essential.

Why is "Using the wrong angle in the area formula." a common mistake in Sine and Cosine Rule?

Why it happens: Confusing 'included' with 'opposite'. How to avoid it: 12 ab C uses the angle BETWEEN sides a and b — i.e. opposite to side c (the not-used one).

Geometry and TrigonometryIB MYP Maths Extended Level

Sine and Cosine Rule

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Sine and Cosine Rule — IB MYP Mathematics (Extended): trigonometry for ANY triangle

SOHCAHTOA only works for right triangles. The sine and cosine rules unlock any triangle. This MYP Mathematics Extended note covers both rules and when to use each.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Apply sine and cosine rules.
MYP Mathematics A — Find triangle area using the $\tfrac{1}{2}ab\sin C$ formula.
MYP Mathematics C — Choose the correct rule based on given info.
MYP Mathematics D — Apply to real situations (navigation, surveying).

The sine rule

Side/sin(opposite angle) is constant.

In any triangle with sides $a, b, c$ opposite angles $A, B, C$ respectively:

$\boxed{\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}}$

The ratio of EACH SIDE to the SINE of its OPPOSITE ANGLE is the same.

Use this when:

You know a side and its opposite angle (gives the common ratio).
You want to find another side or angle.

Worked example. Triangle with $A = 40°$ , $a = 8$ cm, $B = 65°$ . Find $b$ .

$\dfrac{a}{\sin A} = \dfrac{8}{\sin 40°} \approx \dfrac{8}{0.643} \approx 12.44$ .
$\dfrac{b}{\sin B} = \dfrac{b}{\sin 65°}$ .
Set equal: $b = 12.44 \times \sin 65° \approx 12.44 \times 0.906 \approx 11.27$ cm.

Worked example (finding an angle). Triangle with $a = 7$ , $b = 10$ , $A = 35°$ . Find $B$ .

$\dfrac{7}{\sin 35°} = \dfrac{10}{\sin B}$ .
$\sin B = \dfrac{10 \sin 35°}{7} \approx \dfrac{10 \times 0.574}{7} \approx 0.819$ .
$B = \sin^{-1}(0.819) \approx 55.0°$ .

Sine rule: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$ .
Use when you have side-angle PAIRS.
Form a pair from given info → solve.

The cosine rule

Generalised Pythagoras.

In any triangle:

$\boxed{a^2 = b^2 + c^2 - 2bc \cos A}.$

(And by symmetry, $b^2 = a^2 + c^2 - 2ac \cos B$ , etc.)

This is the cosine rule — a generalisation of Pythagoras. When $A = 90°$ , $\cos 90° = 0$ and the formula reduces to $a^2 = b^2 + c^2$ , the usual Pythagoras.

Use this when:

All three SIDES are known and you want an angle.
Two SIDES and the INCLUDED ANGLE are known and you want the third side.

Worked example. Two sides + included angle. $b = 5$ , $c = 7$ , $A = 60°$ . Find $a$ .

$a^2 = 5^2 + 7^2 - 2(5)(7)\cos 60° = 25 + 49 - 70 \times 0.5 = 39$ .
$a = \sqrt{39} \approx 6.24$ .

Worked example (finding an angle). $a = 6$ , $b = 8$ , $c = 10$ . Find $A$ .

Rearranged for angle: $\cos A = \dfrac{b^2 + c^2 - a^2}{2bc} = \dfrac{64 + 100 - 36}{160} = \dfrac{128}{160} = 0.8$ .

$A = \cos^{-1}(0.8) \approx 36.9°$ .

(Aside: this triangle has sides $6, 8, 10$ — a Pythagorean triple. Should be a right angle at $C$ . Check: $\cos C = \dfrac{36 + 64 - 100}{96} = 0$ , so $C = 90°$ ✓.)

Cosine rule: $a^2 = b^2 + c^2 - 2bc \cos A$ .
Use when 3 sides OR 2 sides + included angle.
Rearranged for angle: $\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}$ .

Choosing the right rule + area formula

What info do you have? Pick accordingly.

Decision tree:

Given	Use
2 sides + included angle → find 3rd side	COSINE rule
3 sides → find any angle	COSINE rule
Any side-angle PAIR + another piece → find unknown	SINE rule
Right triangle	SOHCAHTOA

Area of a triangle using two sides + included angle:

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

This is a slick alternative to $\tfrac{1}{2} \times \text{base} \times \text{height}$ — you don't need the height; just two sides and the angle between them.

Worked example. Triangle with $a = 6$ , $b = 9$ , included angle $C = 50°$ . Find the area.

$\text{Area} = \tfrac{1}{2} \times 6 \times 9 \times \sin 50° \approx 27 \times 0.766 \approx 20.7 \text{ square units}.$

Real-world (criterion D). A surveyor measures two distances from a point and the angle between them, then uses the cosine rule to compute the distance between the two far points. This is the basis of TRIANGULATION — historically used to map continents.

Sine rule: angle-side PAIRS.
Cosine rule: 3 sides OR 2 sides + angle.
Triangle area: $\tfrac{1}{2} ab \sin C$ (two sides + included angle).
Triangulation: real-world cosine-rule application.

How it’s examined

Criterion A: apply rules. Criterion C: justify choice of rule. Criterion D: real-context triangulation / navigation.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

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Step-by-step worked examples — Sine and Cosine Rule

Step-by-step solutions to past-paper-style questions on sine and cosine rule, written exactly the way a tutor would explain them at the board.

1Use sine rule to find a side
Building confidence• sine rule
Question
Triangle with $A = 50°$ , $B = 70°$ , $a = 8$ . Find $b$ .
Step-by-step solution
1. Step 1
  Sine rule: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$ .
2. Step 2
  $\dfrac{8}{\sin 50°} = \dfrac{b}{\sin 70°}$ .
3. Step 3
  $b = \dfrac{8 \sin 70°}{\sin 50°} \approx \dfrac{8 \times 0.940}{0.766} \approx 9.81$ .
Answer
$b \approx 9.81$
2Use cosine rule to find a side
Building confidence• cosine rule
Question
Triangle with $b = 7$ , $c = 9$ , $A = 100°$ . Find $a$ .
Step-by-step solution
1. Step 1
  Cosine rule: $a^2 = b^2 + c^2 - 2bc \cos A$ .
2. Step 2
  $a^2 = 49 + 81 - 2(7)(9)\cos 100° = 130 - 126 \times (-0.174) \approx 130 + 21.9 = 151.9$ .
3. Step 3
  $a = \sqrt{151.9} \approx 12.3$ .
Answer
$a \approx 12.3$
3Triangle area
Building confidence• area
Question
Find the area of a triangle with sides 8 cm and 11 cm and included angle $40°$ .
Step-by-step solution
1. Step 1
  Area = $\tfrac{1}{2} ab \sin C = \tfrac{1}{2} \times 8 \times 11 \times \sin 40°$ .
2. Step 2
  $\sin 40° \approx 0.643$ . Area $\approx \tfrac{1}{2} \times 88 \times 0.643 \approx 28.3\,\text{cm}^2$ .
Answer
$\approx 28.3\,\text{cm}^2$
4Find an angle from 3 sides
Stretch• cosine rule
Question
Triangle with $a = 7$ , $b = 9$ , $c = 12$ . Find angle $A$ .
Step-by-step solution
1. Step 1
  Cosine rule rearranged: $\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}$ .
2. Step 2
  $\cos A = \dfrac{81 + 144 - 49}{216} = \dfrac{176}{216} \approx 0.815$ .
3. Step 3
  $A = \cos^{-1}(0.815) \approx 35.4°$ .
Answer
$A \approx 35.4°$

Key Definitions and Keywords — Sine and Cosine Rule

Definitions to memorise and the exact keywords mark schemes credit for sine and cosine rule answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Sine rule
Examiner keyword
$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$ . Each side over the sine of its opposite angle gives the same constant.
Cosine rule
Examiner keyword
$a^2 = b^2 + c^2 - 2bc \cos A$ . Generalises Pythagoras.
Triangle area (SAS formula)
Examiner keyword
Area = $\dfrac{1}{2} ab \sin C$ where $C$ is the angle between sides $a$ and $b$ .

Common Mistakes and Misconceptions — Sine and Cosine Rule

The traps other students keep falling into on sine and cosine rule questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Using sine rule when you don't have a side-angle pair.
Why it happens
Defaulting to sine rule.
How to avoid it
Without a side and its opposite angle, sine rule has no anchor. Use cosine rule when you have 3 sides or 2 sides + included angle.
✕Forgetting the minus sign in $-2bc\cos A$ .
Why it happens
Slip.
How to avoid it
Always write the formula in full: $a^2 = b^2 + c^2 - 2bc\cos A$ . The minus is essential.
✕Using the wrong angle in the area formula.
Why it happens
Confusing 'included' with 'opposite'.
How to avoid it
$\tfrac{1}{2} ab \sin C$ uses the angle BETWEEN sides $a$ and $b$ — i.e. opposite to side $c$ (the not-used one).

Practice questions

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Sine and Cosine Rule — frequently asked questions

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Sine and Cosine Rule

Short Study Notes in the form of Slides

Short Study Notes — Sine and Cosine Rule

Detailed Notes

What you’ll learn

How it’s examined

1Use sine rule to find a side

2Use cosine rule to find a side

3Triangle area

4Find an angle from 3 sides

Sine rule

Cosine rule

Triangle area (SAS formula)

✕Using sine rule when you don't have a side-angle pair.

✕Forgetting the minus sign in −2bccos⁡A-2bc\cos A−2bccosA.

✕Using the wrong angle in the area formula.

Practice questions

Past paper style quiz

Assess your understanding

Sine and Cosine Rule — frequently asked questions

✕Forgetting the minus sign in $-2bc\cos A$ .