Trigonometry — Sine Rule and Cosine Rule (Extended)
SOHCAHTOA only works for right-angled triangles. The sine and cosine rules extend trigonometry to any triangle. At Year 4 Advanced, you must choose and justify the appropriate rule for each problem.
What You'll Learn
Apply the sine rule to find unknown sides and angles in non-right triangles
Apply the cosine rule when given SAS (two sides and included angle) or SSS (three sides)
Calculate the area of any triangle using ½ab sin C
Recognise and handle the ambiguous case of the sine rule
Choose between sine rule and cosine rule with justification
Solve real-world problems involving bearings, navigation, and surveying
Criterion B: Justify the choice of rule; explain why the cosine rule reduces to Pythagoras when the angle is 90°.
Criterion C: Label diagrams clearly; use correct notation; show logical working.
Criterion D: Round appropriately and justify accuracy in context (nearest degree, 1 d.p., etc.).
Key Vocabulary
Term
Definition
Sine rule
a/sin A = b/sin B = c/sin C — relates sides to their opposite angles
Cosine rule
a² = b² + c² − 2bc cos A — a generalisation of Pythagoras
Included angle
The angle between two known sides
Opposite side
The side directly across from a given angle in a triangle
Ambiguous case
When using the sine rule to find an angle, two solutions may exist (sin θ = sin(180° − θ))
Bearing
A direction measured clockwise from north, expressed as a three-figure angle (e.g., 045°)
Non-right triangle
A triangle with no 90° angle — SOHCAHTOA does not apply directly
Mathematics · Topic 1.3
The Sine Rule
The sine rule connects each side of a triangle with the sine of its opposite angle. Use it when you know a side and its opposite angle plus one more piece of information.
Side a is opposite angle A, side b opposite B, side c opposite C
Sine Rule
asin A = bsin B = csin C
When to Use the Sine Rule
AAS / ASA: Two angles and one side known → find the remaining side
SSA: Two sides and an angle opposite one of them → find the opposite angle (watch for ambiguous case)
Example: In triangle ABC, A = 42°, B = 73°, a = 8 cm. Find b.
Worked Example — Sine Rule (Find a Side)
asin A = bsin B → 8sin 42° = bsin 73°sine rule setup
b = 8 × sin 73°sin 42° = 8 × 0.95630.6691rearrange for b
b = 11.43 cm (2 d.p.)
Finding an Angle
Example: In triangle PQR, p = 12 cm, q = 9 cm, P = 65°. Find angle Q.
Worked Example — Sine Rule (Find an Angle)
sin Q = q × sin Pp = 9 × sin 65°12rearrange sine rule
The cosine rule is a generalisation of Pythagoras’ theorem. When the included angle is 90°, the −2bc cos A term vanishes (since cos 90° = 0), giving a² = b² + c².
Cosine Rule (find a side)
a² = b² + c² − 2bc cos A
Cosine Rule (find an angle)
cos A = b² + c² − a²2bc
When to Use the Cosine Rule
SAS: Two sides and the included angle → find the third side
SSS: Three sides known → find any angle
Example (find a side): b = 7, c = 10, A = 58°. Find a.
Example (find an angle): a = 5, b = 8, c = 9. Find angle A.
Worked Example — Cosine Rule (Find an Angle)
cos A = 8² + 9² − 5²2 × 8 × 9 = 64 + 81 − 25144 = 120144 = 0.8333substitute into formula
A = cos−1(0.8333)inverse cosine
A = 33.6° (1 d.p.)
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Key Insight: When A = 90°, cos 90° = 0, so a² = b² + c² − 0 = b² + c². The cosine rule reduces to Pythagoras’ theorem. This is why the cosine rule is called a generalisation of Pythagoras.
Mathematics · Topic 1.3
Area of Any Triangle
When you know two sides and the included angle, use the sine area formula instead of base × height.
Area Formula
Area = ½ ab sin C
Example: Find the area of triangle with sides a = 6, b = 9 and included angle C = 40°.
Worked Example — Area Formula
Area = ½ × 6 × 9 × sin 40°substitute a, b, C
= 27 × 0.6428evaluate
Area = 17.36 cm² (2 d.p.)
Choosing the Right Formula
Information Given
Use
Base and perpendicular height
Area = ½ × base × height
Two sides and included angle
Area = ½ ab sin C
Three sides (no angle)
Use cosine rule first to find an angle, then use ½ ab sin C
Mathematics · Topic 1.3
The Ambiguous Case
When using the sine rule to find an angle (SSA configuration), there can be two valid triangles. This happens because sin θ = sin(180° − θ).
Example: In triangle ABC, a = 7, b = 10, A = 40°. Find angle B.
sin B = b sin A / a = 10 × sin 40° / 7 = 10 × 0.6428 / 7 = 0.9183.
Both B = 66.8° and B = 113.2° produce valid triangles. This is the ambiguous case.
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Critical Rule: Always check the supplementary angle when finding an angle with the sine rule. If both angles give a valid triangle (sum of angles < 180°), both solutions must be stated. Only reject the supplementary angle if it makes the angle sum exceed 180°.
Mathematics · Topic 1.3
Worked Examples
Multi-step solutions showing the reasoning expected at Year 4 Advanced.
EXAMPLE 1In triangle ABC, a = 14, b = 9, C = 72°. Find side c.
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Full Solution
This is SAS (two sides and included angle) → cosine rule.
c² = a² + b² − 2ab cos C = 14² + 9² − 2(14)(9) cos 72°
= 196 + 81 − 252 × 0.3090 = 277 − 77.87 = 199.13.
c = √199.13 = 14.11 cm (2 d.p.).
EXAMPLE 2A ship sails 15 km on a bearing of 060°, then 20 km on a bearing of 150°. Find the direct distance from the starting point.
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Full Solution
The angle between the two paths: 150° − 060° = 90°... but we must consider the geometry. The angle at the turning point between the paths is 180° − (150° − 60°) = 180° − 90° = 90°. Wait — let me reconsider using the actual bearing angles.
At the turning point, the angle between the two legs: the ship was heading 060° (measuring from north clockwise). It then turns to 150°. The angle turned = 150° − 60° = 90°. The interior angle of the triangle at this point = 180° − 90° = 90°... Actually, the included angle = 150° − 60° = 90°.
Using cosine rule with included angle 90°:
d² = 15² + 20² − 2(15)(20) cos 90° = 225 + 400 − 0 = 625.
d = 25 km.
(When the included angle is 90°, the cosine rule reduces to Pythagoras.)
EXAMPLE 3Find the area of a triangle with sides 8, 11, and 14 cm.
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Full Solution
SSS — no angle given. First find an angle using the cosine rule.
Let a = 14, b = 8, c = 11.
cos A = (b² + c² − a²) / 2bc = (64 + 121 − 196) / (2 × 8 × 11) = −11/176 = −0.0625.
A = cos−1(−0.0625) = 93.58°.
Area = ½ × 8 × 11 × sin 93.58° = 44 × 0.9981 = 43.9 cm² (1 d.p.).
EXAMPLE 4Explain why the cosine rule reduces to Pythagoras’ theorem when angle A = 90°.
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Full Solution
The cosine rule states: a² = b² + c² − 2bc cos A.
When A = 90°: cos 90° = 0.
Therefore: a² = b² + c² − 2bc(0) = b² + c².
This is Pythagoras’ theorem: the square of the hypotenuse equals the sum of the squares of the other two sides. The cosine rule is therefore a generalisation that works for ALL triangles, not just right-angled ones. ∎
EXAMPLE 5In triangle DEF, d = 5, e = 8, D = 30°. Investigate the ambiguous case.
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Full Solution
sin E / e = sin D / d → sin E = 8 sin 30° / 5 = 8 × 0.5 / 5 = 0.8.
E = sin−1(0.8) = 53.1°.
Supplementary: E’ = 180° − 53.1° = 126.9°.
Triangle 1: D = 30°, E = 53.1°, F = 96.9°. Triangle 2: D = 30°, E = 126.9°, F = 23.1°.
Both are valid — this IS the ambiguous case.
Mathematics · Topic 1.3
Practice Q&A
Attempt each question before revealing the model answer. Justify your choice of rule.
CALCULATEIn triangle ABC, A = 50°, B = 70°, a = 12. Find b.
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Model Answer
AAS → sine rule. b = a sin B / sin A = 12 × sin 70° / sin 50° = 12 × 0.9397 / 0.7660 = 14.72 (2 d.p.).
CALCULATEFind angle C in a triangle with a = 6, b = 10, c = 13.
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Model Answer
SSS → cosine rule. cos C = (a² + b² − c²) / 2ab = (36 + 100 − 169) / 120 = −33/120 = −0.275.
C = cos−1(−0.275) = 106.0° (1 d.p.). The negative cosine confirms C is obtuse.
APPLYTwo boats leave port. Boat A sails 12 km on bearing 035°; Boat B sails 18 km on bearing 120°. Find the distance between them.
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Model Answer
The included angle at the port = 120° − 35° = 85°. Use cosine rule:
d² = 12² + 18² − 2(12)(18) cos 85° = 144 + 324 − 432 × 0.0872 = 468 − 37.67 = 430.33.
d = √430.33 = 20.74 km (2 d.p.).
AREAFind the area of a triangle with sides 7 cm and 11 cm and included angle 48°.
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Model Answer
Area = ½ × 7 × 11 × sin 48° = 38.5 × 0.7431 = 28.61 cm² (2 d.p.).
JUSTIFYYou know two sides and a non-included angle. Which rule do you use and what must you watch for?
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Model Answer
Use the sine rule since you have a side-angle pair. You must watch for the ambiguous case: when finding an angle, check whether the supplementary angle (180° − θ) also gives a valid triangle. If both angles produce a sum less than 180° with the known angle, both solutions must be stated.
PROVEExplain why the cosine rule is a generalisation of Pythagoras’ theorem.
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Model Answer
a² = b² + c² − 2bc cos A. When A = 90°, cos A = 0, so a² = b² + c², which is exactly Pythagoras’ theorem. The cosine rule adds the correction term −2bc cos A, which accounts for the angle not being 90°. For acute A, cos A > 0 so a² < b² + c²; for obtuse A, cos A < 0 so a² > b² + c².
Mathematics · Review
Flashcard Review
Tap each card to reveal the answer. Try to answer from memory first.
State the sine rule.
a/sin A = b/sin B = c/sin C. Each side divided by the sine of its opposite angle.
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State the cosine rule (find a side).
a² = b² + c² − 2bc cos A. Use with SAS (two sides and included angle).
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State the cosine rule (find an angle).
cos A = (b² + c² − a²) / 2bc. Use with SSS (three sides known).
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Area of any triangle formula?
Area = ½ ab sin C. Requires two sides and the included angle.
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When do you use the sine rule?
When you know a side and its opposite angle plus one more piece (AAS, ASA, or SSA).
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When do you use the cosine rule?
When you have SAS (two sides + included angle) or SSS (three sides). The sine rule won’t work in these cases.
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What is the ambiguous case?
When using sine rule to find an angle (SSA), sin θ = sin(180° − θ) gives two possible angles. Both must be checked for validity.
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Why does the cosine rule reduce to Pythagoras?
When A = 90°, cos 90° = 0, so a² = b² + c² − 0 = b² + c².
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What does a negative cosine value tell you about an angle?
The angle is obtuse (greater than 90°). cos A < 0 when 90° < A < 180°.
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How do you handle bearings in trig problems?
Bearings are measured clockwise from north as three-figure angles. Sketch the diagram, find the included angle, then apply the appropriate rule.
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What is an “included angle”?
The angle formed between two known sides. It is “included” (sandwiched) by those sides.
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If you know SSS but no angles, what do you do first?
Use the cosine rule rearranged to find an angle: cos A = (b² + c² − a²) / 2bc. Then you can find the area or other angles.
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Can you use SOHCAHTOA for non-right triangles?
No. SOHCAHTOA only works for right-angled triangles. For non-right triangles, use the sine rule or cosine rule.
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Third angle in a triangle?
C = 180° − A − B. The angles in any triangle always sum to 180°.
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How to check if the ambiguous case applies?
It applies when using the sine rule in SSA configuration and the known angle is acute. Check: does 180° − θ also give a valid triangle (angle sum < 180°)?