GCSE · Maths · AQA · Spec 8300 · Higher
Sine rule and cosine rule (Higher)
Pythagoras and SOH CAH TOA need a right angle. Most triangles don't have one. Give two rules enough sides and angles to start from, and they crack those too, both built on one idea: every angle has a partner side.
Maths · Triangles
Opposite pairs: a with A · b with B · c with C
a ÷ sin A = b ÷ sin B = c ÷ sin C — in every triangle, whatever its shape.
Drag C sideways, or slide B along the base. Keep an eye on two things: which angle is the biggest, and which side is the longest.
Sine rule: finding a side
Problem
Triangle ABC has angle A = 40°, angle B = 65° and side a = 8.3 cm. Find side b.
Maths · Algebra
The cosine rule, line by line
No complete pair here, so the sine rule has nothing to start from. The cosine rule links all three sides with one angle: a² = b² + c² − 2bc cos A.
A is the angle between b and c, and a is the side opposite it — the side we want.
Step 1 of 6
A is the angle between b and c, and a is the side opposite it — the side we want.
Choose the rule
Sine rule or cosine rule?
For each problem, pick the rule you'd use. Start by asking: is any side known together with its opposite angle? Watch the last one — both rules can do it, so pick the quicker.
Still to sort
Sine rule (0)
A complete opposite pair is known, and you want part of another pair.
Where the line is: The sine rule needs at least one side AND its opposite angle both known. Without a complete pair it has nothing to start from.
Cosine rule (0)
Three sides and one angle are involved: two sides and the angle between them, or all three sides.
Where the line is: No complete pair, but you know two sides and the angle they make — or all three sides. It can also be the quicker choice when the sine rule would need several steps.
No numbers needed. Look at what's known, find the complete opposite pairs, and decide.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
What you need to know
- Label a triangle so each side sits opposite the angle with the same letter: a is opposite A, b is opposite B, c is opposite C.
- The sine rule links opposite pairs; the cosine rule links three sides with one angle.
- Knowing which pairs are complete tells you which rule to use.
The big picture
In any triangle, each angle is paired with the side opposite it, and the biggest angle always faces the longest side. The sine rule, a / sin A = b / sin B = c / sin C, says each side divided by the sine of its opposite angle gives the same value, so it links one complete pair to another. The cosine rule, a² = b² + c² − 2bc cos A, links all three sides with one angle. Spot which pairs you know, choose the rule, then substitute carefully.
Key points
Worked example
Problem
In triangle ABC, angle A = 38°, angle B = 76° and side c = 12 cm. Find side a, correct to 3 significant figures.
⚠ Watch out
Pairing an angle with a side that touches it. The side that goes with angle A is the one that doesn't touch A at all — it's across the triangle. Get a pair wrong and both rules still give you a number, just the wrong one.
Memory hook
Pair up, then count. A complete pair plus half of another? Sine rule. No complete pair, but three sides and an angle? Cosine rule.
Check yourself
In triangle XYZ, x = 9 cm, y = 12 cm and Z = 58°. Which rule gives z, and what is it? (Cosine rule: Z is between x and y. z ≈ 10.5 cm.)
Flashcards
(12)Which side pairs with angle A?
State the sine rule.
What does the sine rule say in words?
State the cosine rule for side a.
In the cosine rule a² = b² + c² − 2bc cos A, where is angle A?
The cosine rule rearranged to find angle A
When is the sine rule useful?
When is the cosine rule useful?
Where is the longest side of a triangle?
Finding an angle with the sine rule: which way up?
In b² + c² − 2bc cos A, what does cos A multiply?
Is the sine rule always the quickest route?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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