Sine Rule and Cosine Rule

Overview

The sine rule and cosine rule extend trigonometry beyond right-angled triangles. In earlier work, Trigonometric Ratios used $\sin$, $\cos$, and $\tan$ inside right triangles. This chapter asks a broader question: what can we do when a triangle has no right angle?

The answer is to use relationships that work in any triangle. The sine rule connects each side with the sine of its opposite angle. The cosine rule connects three sides and one angle, and it is especially useful when the known angle is between two known sides.

This topic is important in surveying, navigation, map work, construction, and many exam-style geometry problems. A learner who can choose the correct rule, draw a clear triangle, and check whether an answer is sensible can solve many distance and angle questions without forcing a false right triangle into the diagram.

Prerequisites

• Trigonometric Ratios - Learners need the meanings of $\sin \theta$ and $\cos \theta$.
• Angles of Elevation and Depression - Many practical distance problems begin with a diagram and angle labels.
• Triangle angle sum - The interior angles of a triangle add to $180^\circ$.
• Pythagoras' theorem - The cosine rule generalizes Pythagoras' theorem when the included angle is not $90^\circ$.
• Basic algebra - Learners rearrange formulas and solve for unknown sides or angles.
• Calculator skills - Degree mode, inverse sine, and inverse cosine must be used carefully.
• Rounding and estimation - Final answers often need a stated degree or length accuracy.

Learning Scope

This chapter covers naming a general triangle, the sine rule, the cosine rule, choosing between the two rules, finding missing sides, finding missing angles, handling the possible ambiguous sine-rule case, and checking whether results fit the triangle.

This page does not teach full bearings, vectors, circle theorems, or proof of every identity. It also does not replace Compound Angles, although compound-angle identities can later explain some exact trigonometric values that appear in advanced examples.

Subtopics

Naming A General Triangle

In a triangle $ABC$, the side opposite angle $A$ is usually named $a$, the side opposite angle $B$ is named $b$, and the side opposite angle $C$ is named $c$.

That convention gives:

$$ a \leftrightarrow A, \quad b \leftrightarrow B, \quad c \leftrightarrow C $$

Key insight: A side and its opposite angle are a pair. Many errors begin when a learner pairs a side with an adjacent angle instead of the opposite angle.

The Sine Rule

The sine rule states that, in any triangle:

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

It can also be written as:

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

Both forms are correct. The best form depends on what is unknown:

• Use side-over-sine when finding a side.
• Use sine-over-side when finding an angle.

Key insight: The sine rule needs at least one complete opposite pair, such as $a$ and $A$, plus another side or angle from a second pair.

When To Use The Sine Rule

Use the sine rule when the information includes an opposite side-angle pair.

Common cases:

• Two angles and one side are known, and another side is required.
• Two sides and a non-included angle are known, and another angle is required.
• One complete pair, such as $a$ and $A$, is known.

For example, if $A = 40^\circ$, $B = 70^\circ$, and $a = 12$ cm, then the pair $(a, A)$ is complete. The side $b$ can be found using:

$$ \frac{b}{\sin B} = \frac{a}{\sin A} $$

The Cosine Rule

The cosine rule states:

$$ a^2 = b^2 + c^2 - 2bc\cos A $$

The same pattern works for the other sides:

$$ b^2 = a^2 + c^2 - 2ac\cos B $$

$$ c^2 = a^2 + b^2 - 2ab\cos C $$

The angle in the cosine term is always opposite the side on the left.

Key insight: The cosine rule is Pythagoras' theorem with an angle correction. If $A = 90^\circ$, then $\cos 90^\circ = 0$, so $a^2 = b^2 + c^2$.

When To Use The Cosine Rule

Use the cosine rule when:

• two sides and the included angle are known, and the third side is required;
• all three sides are known, and an angle is required;
• there is no complete opposite side-angle pair for the sine rule.

The word included angle means the angle between the two known sides. For example, if sides $b$ and $c$ are known and angle $A$ is between them, use:

$$ a^2 = b^2 + c^2 - 2bc\cos A $$

Finding An Angle With The Cosine Rule

When three sides are known, rearrange the cosine rule.

From:

$$ a^2 = b^2 + c^2 - 2bc\cos A $$

we get:

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

Then:

$$ A = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right) $$

Key insight: The angle found is opposite the side used as the left-hand side in the original formula.

The Ambiguous Sine-Rule Case

When two sides and a non-included angle are known, the sine rule may produce one triangle, two possible triangles, or no triangle. This is sometimes called the ambiguous case.

For example, if $\sin B = 0.7$, a calculator gives one acute angle:

$$ B \approx 44.4^\circ $$

But another angle can have the same sine:

$$ 180^\circ - 44.4^\circ = 135.6^\circ $$

Both must be checked against the triangle angle sum and the given measurements.

Key insight: Sine is positive for both acute and obtuse angles. An inverse sine answer may not be the only possible angle in a triangle.

Choosing The Correct Rule

A useful decision routine is:

1. Label the triangle with sides opposite their angles.
2. Look for a complete side-angle pair.
3. If a complete pair is present, try the sine rule.
4. If two sides and the included angle are known, use the cosine rule.
5. If three sides are known, use the cosine rule to find an angle.
6. After finding one angle, use the triangle angle sum before doing extra calculations.

Warning sign: If the sine rule setup has no complete opposite pair, the rule has been chosen too early.

Checking A Solution

Every answer should pass basic checks:

• The largest side should be opposite the largest angle.
• The three angles should add to $180^\circ$.
• A side length should be positive.
• If a side is very long, its opposite angle should usually be comparatively large.
• Calculator mode should be degrees unless the question clearly uses radians.

These checks do not replace calculation, but they catch many slips before the final answer is written.

Key Terms

• General triangle: A triangle that may or may not contain a right angle.
• Opposite pair: A side and the angle across from it, such as $a$ and $A$.
• Sine rule: A relationship connecting sides of a triangle with sines of their opposite angles.
• Cosine rule: A relationship connecting three sides of a triangle and one angle.
• Included angle: The angle between two given sides.
• Non-included angle: A given angle that is not between the two given sides.
• Ambiguous case: A sine-rule situation where more than one triangle may satisfy the data.
• Inverse sine: The calculator operation $\sin^{-1}$ used to find an angle from a sine value.
• Inverse cosine: The calculator operation $\cos^{-1}$ used to find an angle from a cosine value.
• Degree mode: Calculator angle setting used for answers in degrees.

Worked Examples

Example 1: Use The Sine Rule To Find A Side

In triangle $ABC$, $A = 40^\circ$, $B = 70^\circ$, and $a = 12$ cm. Find $b$ to one decimal place.

Step 1: Identify the complete pair.

The side $a = 12$ cm is opposite $A = 40^\circ$.

Step 2: Write the sine rule using $b$ and the known pair.

$$ \frac{b}{\sin 70^\circ} = \frac{12}{\sin 40^\circ} $$

Step 3: Rearrange.

$$ b = \frac{12\sin 70^\circ}{\sin 40^\circ} $$

Step 4: Calculate.

$$ b \approx \frac{12(0.9397)}{0.6428} $$

$$ b \approx 17.5\ \text{cm} $$

Check: $B = 70^\circ$ is larger than $A = 40^\circ$, so $b$ should be larger than $a$. Since $17.5 > 12$, the answer is reasonable.

Example 2: Use The Sine Rule To Find An Angle

In triangle $PQR$, $p = 8$ cm, $q = 10$ cm, and $P = 35^\circ$. Find angle $Q$ if the triangle is the acute-angle case.

Step 1: Write the sine rule in sine-over-side form.

$$ \frac{\sin Q}{10} = \frac{\sin 35^\circ}{8} $$

Step 2: Rearrange.

$$ \sin Q = \frac{10\sin 35^\circ}{8} $$

Step 3: Calculate the sine value.

$$ \sin Q \approx \frac{10(0.5736)}{8} $$

$$ \sin Q \approx 0.7170 $$

Step 4: Use inverse sine.

$$ Q \approx \sin^{-1}(0.7170) $$

$$ Q \approx 45.8^\circ $$

Step 5: Note the possible ambiguous angle.

Another angle with the same sine is:

$$ 180^\circ - 45.8^\circ = 134.2^\circ $$

But the question asks for the acute-angle case, so:

$$ Q \approx 45.8^\circ $$

Example 3: Use The Cosine Rule To Find A Side

In triangle $ABC$, $b = 9$ cm, $c = 7$ cm, and $A = 60^\circ$. Find $a$.

Step 1: Check the information type.

Two sides and the included angle are known. Use the cosine rule.

Step 2: Substitute into the formula.

$$ a^2 = b^2 + c^2 - 2bc\cos A $$

$$ a^2 = 9^2 + 7^2 - 2(9)(7)\cos 60^\circ $$

Step 3: Use $\cos 60^\circ = \frac{1}{2}$.

$$ a^2 = 81 + 49 - 126\left(\frac{1}{2}\right) $$

$$ a^2 = 130 - 63 = 67 $$

Step 4: Take the positive square root.

$$ a = \sqrt{67}\ \text{cm} $$

$$ a \approx 8.2\ \text{cm} $$

Check: Since the included angle is $60^\circ$, the third side being between $7$ cm and $9$ cm is plausible.

Example 4: Use The Cosine Rule To Find An Angle

In triangle $ABC$, $a = 13$ cm, $b = 14$ cm, and $c = 15$ cm. Find angle $A$ to the nearest degree.

Step 1: Use the rearranged cosine rule.

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

Step 2: Substitute values.

$$ \cos A = \frac{14^2 + 15^2 - 13^2}{2(14)(15)} $$

$$ \cos A = \frac{196 + 225 - 169}{420} $$

$$ \cos A = \frac{252}{420} = 0.6 $$

Step 3: Use inverse cosine.

$$ A = \cos^{-1}(0.6) $$

$$ A \approx 53^\circ $$

Check: Side $a = 13$ is the smallest of the three sides, so angle $A$ should be the smallest of the three angles. An angle of about $53^\circ$ is possible because the other two angles can still add to $127^\circ$.

Example 5: Model A Practical Distance

Two points $A$ and $B$ are $120$ m apart. A third point $C$ is observed so that $\angle A = 48^\circ$ and $\angle B = 62^\circ$. Find $AC$ to the nearest metre.

Step 1: Find angle $C$.

$$ C = 180^\circ - 48^\circ - 62^\circ = 70^\circ $$

Step 2: Match opposite pairs.

Side $AB = 120$ m is opposite angle $C = 70^\circ$.

Side $AC$ is opposite angle $B = 62^\circ$.

Step 3: Use the sine rule.

$$ \frac{AC}{\sin 62^\circ} = \frac{120}{\sin 70^\circ} $$

Step 4: Calculate.

$$ AC = \frac{120\sin 62^\circ}{\sin 70^\circ} $$

$$ AC \approx 113\ \text{m} $$

Check: Since angle $62^\circ$ is slightly smaller than $70^\circ$, its opposite side should be slightly shorter than $120$ m. The answer fits.

Common Mistakes

• Pairing a side with the angle next to it instead of the angle opposite it.
• Using the sine rule when no complete opposite side-angle pair is known.
• Using the cosine rule with the wrong included angle.
• Forgetting to take the square root after finding $a^2$.
• Rounding too early, then carrying a rounded value into the next step.
• Treating $\sin^{-1}$ as $\frac{1}{\sin}$ instead of inverse sine.
• Ignoring the possible second angle in a sine-rule angle problem.
• Leaving calculator mode in radians when the question uses degrees.
• Assuming the largest side can be opposite a small angle.
• Drawing a diagram that does not match the given angle positions.

Practice Tasks

Foundation

1. In triangle $ABC$, name the side opposite angle $A$.
2. State the sine rule using $a$, $b$, $A$, and $B$ only.
3. State the cosine rule for finding side $c$.
4. In a triangle, $A = 50^\circ$ and $B = 80^\circ$. Find $C$.
5. Decide whether the sine rule or cosine rule is more suitable when two sides and the included angle are known.
6. Explain why the side opposite the largest angle should be the longest side.

Skill-Building

1. In triangle $ABC$, $A = 35^\circ$, $B = 75^\circ$, and $a = 10$ cm. Find $b$.
2. In triangle $PQR$, $P = 42^\circ$, $Q = 68^\circ$, and $QR = 14$ cm. Find $PR$.
3. In triangle $ABC$, $b = 8$ cm, $c = 11$ cm, and $A = 45^\circ$. Find $a$.
4. In triangle $XYZ$, $x = 6$ cm, $y = 7$ cm, and $z = 9$ cm. Find angle $X$.
5. In triangle $ABC$, $a = 9$ cm, $A = 38^\circ$, and $B = 52^\circ$. Find $b$ and $C$.
6. A learner finds $a^2 = 49$ and writes $a = 49$ cm. Correct the mistake.

Exam-Style

1. A triangle has sides $12$ cm and $15$ cm with included angle $40^\circ$. Find the third side correct to one decimal place.
2. The sides of a triangle are $7$ cm, $10$ cm, and $12$ cm. Find the largest angle to the nearest degree.
3. Points $A$ and $B$ are $80$ m apart. From $A$, the angle to point $C$ is $56^\circ$, and from $B$ it is $44^\circ$. Find $AC$.
4. In triangle $ABC$, $a = 7$ cm, $b = 9$ cm, and $A = 35^\circ$. Find the possible values of $B$, then decide which values give a valid triangle.
5. A surveyor measures two sides of a triangular field as $45$ m and $60$ m, with included angle $72^\circ$. Estimate the third side.
6. A triangle has sides $5$ cm, $8$ cm, and $10$ cm. Use the cosine rule to decide whether the angle opposite the $10$ cm side is acute, right, or obtuse.

Challenge

1. Show that the cosine rule becomes Pythagoras' theorem when the included angle is $90^\circ$.
2. Create two different triangles that share two sides and a non-included angle. Explain why this is the ambiguous sine-rule case.
3. A triangle has $a = 20$ cm, $A = 30^\circ$, and $b = 25$ cm. Investigate how many possible values of $B$ can exist.
4. Derive the formula $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$ from $a^2 = b^2 + c^2 - 2bc\cos A$.
5. A learner chooses the cosine rule for a problem with two angles and one side. Explain a more efficient method and solve a sample problem of that type.

Generated Question Layer

Future generated practice questions for this topic should include:

• Pair-matching questions that ask learners to identify the side opposite a given angle.
• Rule-selection questions where learners explain whether the sine rule or cosine rule is appropriate before calculating.
• Sine-rule side questions using one complete opposite pair.
• Sine-rule angle questions that include an ambiguous-case warning.
• Cosine-rule side questions with two sides and the included angle.
• Cosine-rule angle questions with three known sides.
• Practical surveying and distance questions that require a labelled triangle.
• Error-analysis questions involving wrong side-angle pairing, missing square root, and calculator mode.
• Estimation questions where learners check largest side against largest angle.

Generated questions should be original learner practice. They should not copy past-paper wording, even when they are calibrated to similar skills.

Learner Aid Opportunities

• diagram: Show a triangle labelled $A$, $B$, $C$ with opposite sides $a$, $b$, $c$.
• chart: Compare sine-rule and cosine-rule trigger conditions.
• animation: Highlight how a complete opposite pair unlocks the sine rule.
• interactive: Let learners enter known sides and angles, then receive a rule-choice prompt before calculation.
• video: Demonstrate a surveying problem from diagram construction to final rounded distance.
• LLM tutor: Ask learners to justify the chosen rule and check side-angle pairing before solving.

Exam-Derived Signals

The automatic 2021-2025 topic-frequency file lists topic-sine-rule-and-cosine-rule in the Form IV trigonometry hub, but no reviewed past-question mapping is attached to this exact topic in the current extraction layer. Treat this as an unreviewed coverage signal only.

The 2022 examination-format crosswalk is an official-format-derived planning source, but the crosswalk record itself is marked needs_manual_review. It groups this topic under Trigonometry and Pythagoras theorem with related trigonometry topics.

| Source | Signal | Status | Review caution | | --- | --- | --- | --- | | data/topic_frequency_2021_2025.json | Lists topic-sine-rule-and-cosine-rule as a Form IV trigonometry topic. | Unreviewed extraction signal | Indicates topic presence, not a verified question trend. | | data/question_map_2021_2025.jsonl | No reliable exact-topic question records were found for topic-sine-rule-and-cosine-rule in the inspected mapping. | Unreviewed absence signal | Do not conclude the topic was absent from exams without manual paper review. | | data/exam_format_topic_crosswalk_2022.jsonl | Format group Trigonometry and Pythagoras theorem maps to topic-trigonometric-[[ratio|ratios]], topic-angles-of-[[elevation|elevation]]-and-depression, topic-sine-rule-and-cosine-rule, and topic-compound-angles. | Official-format crosswalk, record marked needs_manual_review | The crosswalk includes unmatched term Pythagoras theorem, so topic grouping needs reviewer confirmation. |

Source And Review Notes

• Official syllabus status: The topic identity, Form IV placement, competence, and hub come from data/curriculum_map.json, which cites raw/syllabuses/csee/2023/csee_mathematics_syllabus_2023.pdf.
• Official syllabus reference page: wiki/sources/csee-mathematics-syllabus-2023.md records the syllabus publisher, level, curriculum role, and assessment context.
• Assessment signal sources: data/topic_frequency_2021_2025.json, data/question_map_2021_2025.jsonl, and data/exam_format_topic_crosswalk_2022.jsonl.
• Review status: This is an unreviewed learner expansion from an official syllabus topic. The prose, examples, and practice tasks are original and should be checked by a mathematics reviewer before being marked reviewed.
• Exam-derived signals remain explicitly unreviewed unless a reviewer checks them against the original examination papers and marking schemes.
• Media status: This page uses text-only learner content. Future visuals should be created through reviewed learner-aid opportunities rather than embedded media files.
• Renderer QA: This page uses portable $...$ and $$...$$ math notation for future Obsidian, KaTeX, or MathJax rendering.