IB Maths AA SL Topic 3 — Geometry & Trig Paper 1 & 2 🎯 Skill ~3 min practice

AA SL Cosine Rule skills

The cosine rule covers the cases the sine rule can’t. Two versions in your booklet — one for finding a side (when you have two sides and the angle between them), one for finding an angle (when you have all three sides). Pick the right form and substitute carefully.

The Method

Form 1

Finding a side

a² = b² + c² − 2bc cos A

use when given SAS — two sides and the included angle

Form 2

Finding an angle

cos A = b² + c² − a²2bc

use when given SSS — all three sides

Label sides and angles correctly. Side a is opposite angle A. The angle in the formula is the one between the two named sides (b and c).
Pick the right form — finding a side use Form 1, finding an angle use Form 2.
Substitute and solve. For an angle, take cos⁻¹ at the end.

The labelling rule

In a² = b² + c² − 2bc cos A: side a is what you’re finding, angle A is the one between sides b and c.

When to use the cosine rule

✓ USE cosine rule when

You have three sides (SSS) or two sides and the angle between them (SAS).

No matching angle-side pair given.

✗ USE sine rule instead when

You have an angle and its opposite side already paired up.

Triangle types: AAS, ASA, SSA

Worked examples

WE 1 EASY

In triangle ABC, b = 7, c = 9, A = 60°. Find side a.

step 1 — pick Form 1 (SAS, find side) a² = b² + c² − 2bc cos Astep 2 — substitute a² = 7² + 9² − 2(7)(9) cos 60° a² = 49 + 81 − 126 × 0.5 a² = 130 − 63 = 67step 3 — square root a = √67a ≈ 8.19 (3 sf) don’t forget the square root at the end — a², not a!

WE 2 MEDIUM

In triangle PQR, p = 5, q = 8, r = 10. Find angle P.

step 1 — pick Form 2 (SSS, find angle) cos P = (q² + r² − p²) / (2qr)step 2 — substitute cos P = (64 + 100 − 25) / (2 × 8 × 10) cos P = 139 / 160 = 0.8688step 3 — take cos⁻¹ P = cos⁻¹(0.8688)P ≈ 29.6° (3 sf) side opposite the angle being found goes alone in the numerator (subtracted).

WE 3 HARD

In triangle XYZ, x = 14, y = 6, z = 9. Find angle X.

step 1 — Form 2 (SSS, find angle) cos X = (y² + z² − x²) / (2yz)step 2 — substitute cos X = (36 + 81 − 196) / (2 × 6 × 9) cos X = −79 / 108 cos X = −0.7315step 3 — negative → obtuse X = cos⁻¹(−0.7315)X ≈ 137° (3 sf) unlike sin⁻¹, cos⁻¹ gives the correct obtuse answer directly — no ambiguous case!

Practice questions

Try each one yourself first, then click the question to reveal the worked answer. Identify SAS or SSS before you start — that picks the form.

Q1 EASY In triangle ABC, b = 5, c = 8, A = 50°. Find a. Show answer ▼Hide answer ▲

a² = 25 + 64 − 80 cos 50° = 89 − 51.42 = 37.58 a ≈ 6.13 (3 sf)

Q2 EASY In triangle PQR, p = 4, q = 7, r = 6. Find angle P. Show answer ▼Hide answer ▲

cos P = (49 + 36 − 16) / 84 cos P = 69/84 = 0.8214 P ≈ 34.8° (3 sf)

Q3 MEDIUM In triangle ABC, a = 11, b = 14, C = 75°. Find c. Show answer ▼Hide answer ▲

find c, given a, b, C → SAS form c² = a² + b² − 2ab cos C = 121 + 196 − 308 cos 75° = 317 − 79.71 = 237.29 c ≈ 15.4 (3 sf) the formula adapts — swap letters to match what you’re finding.

Q4 MEDIUM In triangle XYZ, x = 12, y = 9, z = 7. Find angle Y. Show answer ▼Hide answer ▲

cos Y = (x² + z² − y²) / (2xz) = (144 + 49 − 81) / 168 = 112/168 = 0.6667 Y ≈ 48.2° (3 sf) y² stands alone — the side opposite the angle you want.

Q5 HARD In triangle ABC, a = 6, b = 8, c = 13. Find the largest angle. Show answer ▼Hide answer ▲

largest angle is opposite the longest side (c = 13) cos C = (a² + b² − c²) / (2ab) = (36 + 64 − 169) / 96 = −69/96 = −0.7188 C ≈ 135.9° (3 sf) negative cosine → obtuse angle. always check which side is longest first!

⚠ Common mistakes

Forgetting the square root. The formula gives a², not a. Always √ at the end when finding a side.
Using the wrong form. Form 1 finds a side; Form 2 finds an angle. Check what you’re solving for before substituting.
Wrong angle in the formula. The angle is always the one between the two named sides (the included angle).
Calculator in wrong mode. Degrees vs radians — set it before you start. cos 60° ≠ cos(60 rad).
Order of operations error. 2bc cos A is multiplied then subtracted — not 2bc and then cos separately. Use brackets.
Mismatching letters. If finding angle P, it’s p² that stands alone in the numerator — not a² (unless the triangle is labelled ABC).

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Want the theory?

Read the full Sine & Cosine Rules notes for the proof, the link to area = ½ ab sin C, and how cosine rule is just a generalisation of Pythagoras.

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