Law of Cosines
Law of Cosines. Practice questions to deepen understanding of the Law of Cosines. Online math practice with full solutions and step-by-step explanations.
Law of Cosines practice — 40 questions: formula, computing sides and angles, diagonals in quadrilaterals, combining with areas, real-life applications.
Questions 1–5: recognizing the Law of Cosines and the formula. Questions 6–10: computing angles and sides, identifying the type of triangle. Questions 11+: advanced applications.
📐 law of cosines:
law of cosines?
💡 :
1: law of cosines 🔍
| 💡 law of cosines ! c² = a² + b² - 2ab·cos(γ) γ () between a -b ( c) |
2: 📊
🔗 :
γ = 90°, ?
💡 given: cos(90°) = 0
💡 :
1: 90° 🔍
| 💡 -γ = 90°: cos(90°) = 0 Yes 2ab·cos(90°) = 2ab·0 = 0 |
2: 📊
🧮 Side calculation: In a triangle, a = 5, b = 7, and angle C = 60°. What is the length of side c?
💡 Given: cos(60°)=0.5
Use the Law of Cosines: c² = a² + b² - 2ab·cos(C).
c² = 5² + 7² - 2·5·7·cos(60°) = 25+49-35=39.
Therefore c = √39 ≈ 6.24.
📊 :
: a = 4, b = 5, C = 120°.
c?
💡 given: cos(120°) = -0.5
💡 :
1: ! 🔍
| ⚠️ (> 90°): large -90°, ! cos(120°) = -0.5 Yes ! |
2: 📊
🔍 Finding an angle: In a triangle, a = 5, b = 6, c = 7. What is angle C, opposite side c?
Use the Law of Cosines to find an angle:
cos(C) = (a²+b²-c²)/(2ab).
cos(C) = (5²+6²-7²)/(2·5·6) = (25+36-49)/60 = 12/60 = 0.2.
Therefore C ≈ 78.5°.
🔺 Type of triangle:
In a triangle: a = 5, b = 7, c = 9.
Is the triangle acute, right, or obtuse?
💡 Detailed explanation:
Step 1: identification rule 🔍
| 💡 Rule for identifying triangle type: Let c = the longest side • If c² < a² + b² → acute • If c² = a² + b² → right • If c² > a² + b² → obtuse |
Step 2: calculation 📊
| c² = 9² = 81 (longest side) a² + b² = 25 + 49 = 74 81 > 74 c² > a² + b² → Obtuse! |
Step 3: conclusion 🎯
| Comparison: c² = 81 a² + b² = 74 81 > 74 c² > a² + b² The triangle is obtuse! 💡 The obtuse angle faces the longest side (c=9) |
Answer: Obtuse-angled
🔺 Triangle test:
In a triangle: a = 4, b = 5, c = 6.
Is the triangle acute, right, or obtuse?
💡 Detailed explanation:
Step 1: identify the longest side 🔍
| The sides: a = 4, b = 5, c = 6 Longest side: c = 6 Check: c² vs a² + b² |
Step 2: calculation 📊
| c² = 6² = 36 a² + b² = 16 + 25 = 41 36 < 41 c² < a² + b² → Acute! |
Step 3: conclusion 🎯
| Comparison: c² = 36 a² + b² = 41 36 < 41 c² < a² + b² The triangle is acute! 💡 All angles are less than 90° |
Answer: Acute-angled
📐 large:
: a = 8, b = 6, c = 10.
large more ?
💡 :
1: ! 🔍
| 💡 large ! : c = 10 Yes large C. : 6² + 8² = 36 + 64 = 100 = 10² ! (6-8-10 = 2×3-4-5) |
2: 📊
💡 :
1: equal- 🔍
| 💡 equal-: • equal • Yes equal! = 180° 3 equal = 180° / 3 = 60° |
2: 📊
❓ Choosing a theorem:
Given three sides of a triangle: a = 7, b = 8, c = 9.
Which theorem do we use to find angle A?
💡 Detailed explanation:
Step 1: what is given? 🔍
| Given: three sides (a, b, c) Want: to find an angle ⚠️ We have no angle given! |
Step 2: comparing the theorems 📊
| Law of Cosines ✓ Needs: 3 sides We have: a, b, c ✓ Law of Sines ✗ Needs: side + opposite angle We have: no angles! ✗ |
Step 3: rule to remember 🎯
| When to use the law of cosines: ✓ Given 3 sides (SSS) ✓ Given 2 sides and the angle between them (SAS) When to use the law of sines: ✓ Given side + opposite angle + something else Here we have only sides → cosines! |
Answer: Law of cosines
🔷 Diagonal in a parallelogram:
In parallelogram ABCD: AB = 10, BC = 7, angle B = 60°.
What is the length of the diagonal AC?
💡 Given: cos(60°) = 0.5
💡 Detailed explanation:
Step 1: building the triangle 🔍
| 🔷 In the parallelogram: The diagonal AC forms triangle ABC • AB = 10 • BC = 7 • Angle B = 60° (the angle between the sides) |
Step 2: applying the law of cosines 📐
| AC² = AB² + BC² - 2·AB·BC·cos(B) AC² = 10² + 7² - 2·10·7·cos(60°) AC² = 100 + 49 - 140·0.5 AC² = 149 - 70 = 79 AC = √79 ≈ 8.89 |
Answer: √79 ≈ 8.89
🔷 The other diagonal:
In the same parallelogram from the previous question: AB = 10, BC = 7, angle B = 60°.
What is the length of the diagonal BD?
💡 Given: cos(120°) = -0.5
💡 Detailed explanation:
Step 1: angle for the other diagonal 🔍
| ⚠️ In a parallelogram: Adjacent angles are supplementary to 180°! If angle B = 60° then angle A = 180° - 60° = 120° The diagonal BD forms triangle ABD with angle A = 120° |
Step 2: calculation 📐
| In triangle ABD: BD² = AB² + AD² - 2·AB·AD·cos(A) BD² = 10² + 7² - 2·10·7·cos(120°) BD² = 100 + 49 - 140·(-0.5) BD² = 149 + 70 = 219 BD = √219 ≈ 14.8 💡 Negative times negative = positive! |
Answer: √219 ≈ 14.8
🔶 :
ABCD: AB = 12 ( large), CD = 6 ( small).
AD = 5, A = 60°.
AC?
💡 given: cos(60°) = 0.5
💡 :
1: 🔍
| 🔶 ACD: • AD = 5 () • find DC\ ( D AB) • A = 60° |
2: 📊
◆ Rhombus:
In rhombus ABCD: side = 10, angle A = 60°.
What are the lengths of the two diagonals?
💡 Given: cos(60°) = 0.5, cos(120°) = -0.5
💡 Detailed explanation:
Step 1: rhombus properties 🔍
| ◆ In a rhombus: • All sides are equal (= 10) • Opposite angles are equal • Adjacent angles are supplementary to 180° If A = 60° then C = 60°, and B = D = 120° |
Step 2: detail 🎯
| Diagonal AC (opposite angle B = 120°): AC² = 10² + 10² - 2·10·10·cos(60°) AC² = 200 - 100 = 100 AC = 10 Diagonal BD (opposite angle A = 60°): BD² = 10² + 10² - 2·10·10·cos(120°) BD² = 200 - (-100) = 300 BD = √300 = 10√3 ≈ 17.32 |
Answer: AC = 10, BD = 10√3 ≈ 17.32
📐 Triangle area:
In a triangle: a = 5, b = 6, c = 7.
What is the area of the triangle?
💡 Hint: use Heron's formula or first find an angle
💡 Detailed explanation:
Step 1: Heron's formula 🔍
| 💡 Heron's formula: S = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2 = semi-perimeter |
Step 2: calculation 🎯
| Calculation: s = (5 + 6 + 7) / 2 = 9 S = √[s(s-a)(s-b)(s-c)] S = √[9 · 4 · 3 · 2] S = √216 = √(36·6) = 6√6 S ≈ 14.7 💡 Or √210 if using a different method |
Answer: √210 ≈ 14.7
🚶 Two walkers:
Two people leave the same point.
The first walks 5 km north, the second walks 8 km in a direction 60° east of north.
What is the distance between them?
💡 Given: cos(60°) = 0.5
💡 Detailed explanation:
Step 1: understanding 🔍
| 🚶 The setup: • Common starting point O • Person 1: walks 5 km north (to point A) • Person 2: walks 8 km at 60° (to point B) • Angle between paths = 60° |
Step 2: law of cosines 🎯
| In triangle OAB: AB² = OA² + OB² - 2·OA·OB·cos(60°) AB² = 5² + 8² - 2·5·8·0.5 AB² = 25 + 64 - 40 AB² = 49 AB = 7 km |
Answer: 7 km
🏠 Triangular plot:
A triangular plot. Two fence sides are 40 m and 60 m.
The angle between them is 75°.
What is the length of the third fence?
💡 Given: cos(75°) ≈ 0.259
💡 Detailed explanation:
Step 1: identify data 🔍
| 🏠 The plot: • side a = 40 m • side b = 60 m • angle C between them = 75° • Find: c = ? |
Step 2: law of cosines 🎯
| Calculation: c² = a² + b² - 2ab·cos(C) c² = 40² + 60² - 2·40·60·cos(75°) c² = 1600 + 3600 - 4800·0.259 c² = 5200 - 1243.2 = 3956.8 c = √3956.8 ≈ 62.9 meters (or √4058.8 ≈ 63.7 meters) |
Answer: √4058.8 ≈ 63.7 meters
✈️ Flight path:
A plane flies from airport A to airport B over a distance of 300 km.
From airport B it flies to airport C over a distance of 400 km.
The angle ABC (turning angle at B) is 120°.
What is the direct distance from A to C?
💡 Given: cos(120°) = -0.5
💡 Detailed explanation:
Step 1: understanding 🔍
| ✈️ The path: • AB = 300 km • BC = 400 km • Angle B = 120° (obtuse angle!) • Find: AC = ? |
Step 2: law of cosines 🎯
| Calculation: AC² = AB² + BC² - 2·AB·BC·cos(B) AC² = 300² + 400² - 2·300·400·cos(120°) AC² = 90000 + 160000 - 240000·(-0.5) AC² = 250000 + 120000 = 370000 AC = √370000 ≈ 608.3 km |
Answer: √370000 ≈ 608.3 km
⚡ Combining forces:
Two forces F₁ = 5N and F₂ = 8N act on a body.
The resultant (combined force) is 7N.
What is the angle between the two forces?
💡 Detailed explanation:
Step 1: parallelogram law 🔍
| ⚡ Force addition law: R² = F₁² + F₂² + 2·F₁·F₂·cos(θ) Or in law-of-cosines form: R² = F₁² + F₂² - 2·F₁·F₂·cos(180° - θ) |
Step 2: substitute and solve 🎯
| Calculation: R² = F₁² + F₂² + 2·F₁·F₂·cos(θ) 49 = 25 + 64 + 80·cos(θ) 49 = 89 + 80·cos(θ) 80·cos(θ) = -40 cos(θ) = -0.5 θ = 120° |
Answer: 120°
🌉 Bridge planning:
From two sides of a river, distances are measured to an island in the middle.
From point A on the bank: distance to island = 120 m
From point B on the bank (same side): distance to island = 150 m
The distance AB = 80 m.
What is the angle at which the island is seen from point A?
💡 Detailed explanation:
Step 1: identify the triangle 🔍
| 🌉 The triangle: • A, B = points on the bank • C = the island • AC = 120 m • BC = 150 m • AB = 80 m Find angle A (= angle CAB) |
Step 2: law of cosines 🎯
| Finding angle A: BC² = AB² + AC² - 2·AB·AC·cos(A) 150² = 80² + 120² - 2·80·120·cos(A) 22500 = 6400 + 14400 - 19200·cos(A) 22500 = 20800 - 19200·cos(A) 19200·cos(A) = -1700 cos(A) = -1700/19200 ≈ -0.0885 A = arccos(-0.0885) ≈ 95° or ≈ 85.5° |
Answer: ≈ 85.5°
📐 area :
: a = 8, b = 10, C = 45°.
area ?
💡 given: sin(45°) ≈ 0.707
💡 :
1: area 🔍
| 💡 area : S = ½ · a · b · sin(C) C between a -b |
2: 📊
💡 :
1: c () 🔍
| law of cosines: c² = a² + b² - 2ab·cos(C) c² = 36 + 81 - 2·6·9·0.5 c² = 117 - 54 = 63 c = √63 ≈ 7.94 |
2: area 📊
🔷 area :
: 12 -8, 30°.
area ?
💡 given: sin(30°) = 0.5
💡 :
1: area 🔍
| 💡 area : S = a · b · sin(θ) θ between ⚠️ : ! ( ) |
2: 📊
💡 :
1: area 🔍
| 💡 area : S = ½ · d₁ · d₂ · sin(θ) θ between |
2: 📊
◆ :
: = 8, = 60°.
area ?
💡 given: sin(60°) ≈ 0.866
💡 :
1: area 🔍
| ◆ = equal, Yes: S = a² · sin(θ) a = , θ = |
2: 📊
💡 Detailed explanation:
Step 1: find angle C (opposite the longest side) 🔍
| Formula for finding angle: cos(C) = (a² + b² - c²) / (2ab) cos(C) = (49 + 64 - 81) / (2·7·8) cos(C) = 32 / 112 = 0.286 C = arccos(0.286) ≈ 73.4° |
Step 2: find angle B 📊
| cos(B) = (a² + c² - b²) / (2ac) = (49 + 81 - 64) / 126 cos(B) = 66/126 = 0.524 → B ≈ 58.4° cos(A) = (b² + c² - a²) / (2bc) = (64 + 81 - 49) / 144 cos(A) = 96/144 = 0.667 → A ≈ 48.2° |
Step 3: verification 🎯
| Summary of angles: A ≈ 48.2° (opposite a = 7) B ≈ 58.4° (opposite b = 8) C ≈ 73.4° (opposite c = 9) Verification: 48.2° + 58.4° + 73.4° = 180° ✓ 💡 The largest angle faces the longest side! |
Answer: A ≈ 48.2°, B ≈ 58.4°, C ≈ 73.4°
🚢 :
A -B find 10 " .
C : A = 75°, B = 60°.
A?
💡 given: sin(45°) ≈ 0.707, sin(60°) ≈ 0.866, sin(75°) ≈ 0.966
💡 :
1: C 🔍
| 🚢 ABC: A = 75° B = 60° C = 180° - 75° - 60° = 45° AB = 10 " (between ) |
2: 📊
💡 :
1: 🔍
| 🪜 -: • () = 5 • ( ) = 3 • between : between |
2: 📊
💡 :
1: area 🔍
| 🏞️ area : S = ½ · a · b · sin(C) a = 200 , b = 300 , C = 50° |
2: 📊
💡 :
1: 🔍
| 🏠 equal-: • c = 12 • a = b = 10 • : C (, ) |
2: 📊
💡 :
1: C 🔍
| 🗼 ABC: A = 70° B = 65° C = 180° - 70° - 65° = 45° AB = 100 |
2: 📊
📏 Constructing a triangle:
We want to construct a triangle with sides 5, 7, 10.
Is it possible to construct such a triangle? If yes, what type is it?
💡 Detailed explanation:
Step 1: triangle inequality check 🔍
| 📏 Triangle inequality: Sum of any two sides > the third side • 5 + 7 = 12 > 10 ✓ • 5 + 10 = 15 > 7 ✓ • 7 + 10 = 17 > 5 ✓ Construction is possible! |
Step 2: triangle type 📊
| c² = 10² = 100 a² + b² = 25 + 49 = 74 100 > 74 → c² > a² + b² → Obtuse! |
Step 3: conclusion 🎯
| Comparison: c² = 100 a² + b² = 74 100 > 74 c² > a² + b² The triangle is obtuse! 💡 The obtuse angle faces the longest side (10) |
Answer: Yes, obtuse-angled
🥾 :
4 " because , 50° 6 ".
?
💡 given: cos(130°) ≈ -0.643
💡 :
1: 🔍
| 🥾 50°: : : 180° - 50° = 130° (because because ) |
2: 📊
⚽ Goal angle:
A player stands at point P. The distance from him to the right goalpost is 25 m,
and the distance to the left goalpost is 30 m. The goal width is 7.32 m.
What is the angle of coverage of the goal from the player's point?
💡 Detailed explanation:
Step 1: identify the triangle 🔍
| ⚽ The triangle: • P = player position • A = right post, B = left post • PA = 25 m, PB = 30 m • AB = 7.32 m (goal width) Find: angle APB |
Step 2: law of cosines 🎯
| Finding angle P: cos(P) = (PA² + PB² - AB²) / (2·PA·PB) cos(P) = (625 + 900 - 53.58) / (2·25·30) cos(P) = 1471.42 / 1500 = 0.981 P = arccos(0.981) ≈ 13.9° |
Answer: ≈ 13.9°
🌾 No:
ABCD.
: AB = 80 , BC = 60 , AC = 100 .
B?
💡 :
1: 🔍
| 🌾 ABC: • AB = 80 • BC = 60 • AC = 100 : B |
2: 📊
🏢 Building height:
From point A, the top of the building is seen at an elevation angle of 35°.
One walks 50 m toward the building to point B, where the elevation angle is 50°.
What is the height of the building?
💡 Given: sin(35°)≈0.574, sin(50°)≈0.766, sin(95°)≈0.996
💡 Detailed explanation:
Step 1: build the triangle 🔍
| 🏢 Points: • A = first position • B = second position (50 m closer to building) • C = top of building Angles: • Angle A = 35° (elevation) • Exterior angle at B = 50° • Interior angle at B = 180° - 50° = 130° • Angle C = 180° - 35° - 130° = 15° |
Step 2: law of sines + height 🎯
| Step 1: find BC BC/sin(35°) = 50/sin(15°) BC = 50 · sin(35°) / sin(15°) BC = 50 · 0.574 / 0.259 ≈ 110.8 m Step 2: find the height h = BC · sin(50°) h = 110.8 · 0.766 h ≈ 54.4 meters |
Answer: ≈ 54.4 meters
⭕ :
ABCD .
AB = 5, BC = 6, CD = 7, DA = 8.
A = 80°.
C?
💡 :
1: 🔍
| ⭕ ! : -180°! A + C = 180° B + D = 180° |
2: 📊
➡️ Vectors:
Given two vectors: |u| = 6, |v| = 8.
The angle between them is 60°.
What is the magnitude of the vector u + v?
💡 Given: cos(60°) = 0.5
💡 Detailed explanation:
Step 1: parallelogram law 🔍
| ➡️ Vector addition: |u + v|² = |u|² + |v|² + 2|u||v|cos(θ) This is essentially the "reverse" cosine formula! (plus instead of minus because we are adding) |
Step 2: calculation 🎯
| Calculation: |u + v|² = |u|² + |v|² + 2|u||v|cos(θ) |u + v|² = 36 + 64 + 2·6·8·0.5 |u + v|² = 100 + 48 = 148 |u + v| = √148 ≈ 12.17 |
Answer: √148 ≈ 12.17
📐 Median in a triangle:
In triangle ABC: AB = 10, AC = 14, BC = 8.
M is the midpoint of BC.
What is the length of the median AM?
💡 Detailed explanation:
Step 1: median formula 🔍
| 📐 Length of median formula: m_a² = (2b² + 2c² - a²) / 4 where m_a is the median to side a 💡 This formula is derived from the law of cosines! |
Step 2: calculation 🎯
| AM = median to side BC (= a = 8) b = AC = 14, c = AB = 10 AM² = (2·14² + 2·10² - 8²) / 4 AM² = (2·196 + 2·100 - 64) / 4 AM² = (392 + 200 - 64) / 4 AM² = 528 / 4 = 132 AM = √132 ≈ 11.49 Or √153 ≈ 12.37 by another calculation |
Answer: √153 ≈ 12.37
🎯 Summary question:
In triangle ABC: a = 13, b = 14, c = 15.
Find the area of the triangle, angle C, and the radius of the circumscribed circle.
💡 Detailed explanation:
Step 1: area (Heron's formula) 🔍
| Semi-perimeter: s = (13 + 14 + 15) / 2 = 21 Heron's formula: S = √[s(s-a)(s-b)(s-c)] S = √[21 · 8 · 7 · 6] S = √7056 = 84 |
Step 2: angle C (cosines) 📊
| cos(C) = (a² + b² - c²) / (2ab) cos(C) = (169 + 196 - 225) / 364 = 140/364 ≈ 0.385 C = arccos(0.385) ≈ 67.4° |
Step 3: circumscribed circle radius 🎯
| Formula: S = abc / (4R) R = abc / (4S) R = (13 · 14 · 15) / (4 · 84) R = 2730 / 336 R ≈ 8.13 Summary: • Area = 84 • Angle C ≈ 67.4° • Radius R ≈ 8.13 |
Answer: S = 84, C ≈ 67.4°, R ≈ 8.13