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Law of Sines & Law of Cosines — Easy Worksheet 1
Grade 12 · MathSheets Name: ____________________ Date: __________
Find the missing side length.
How to solve
When two sides and the included angle are given, use the Law of Cosines: c² = a² + b² - 2ab·cos C. Remember cos 60° = 12 and cos 120° = -12, then take the square root to find c.
Example
In a triangle, side a = 5, side b = 8, and the included angle C = 60°. Find side c (Law of Cosines).
Answer: 7
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 72 + 152 − 2(7)(15)cos 60°. cos 60° = 12, so 2(7)(15)(12) = 105 and c2 = 49 + 225 − 105 = 169. c = √169 = 13.
✓ Correct!
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 72 + 152 − 2(7)(15)cos 60°. cos 60° = 12, so 2(7)(15)(12) = 105 and c2 = 49 + 225 − 105 = 169. c = √169 = 13.
💡 How to solve: Law of Sines & Law of Cosines
When two sides and the included angle are given, use the Law of Cosines: c² = a² + b² - 2ab·cos C. Remember cos 60° = 12 and cos 120° = -12, then take the square root to find c.
Try a similar one
In a triangle, side a = 3, side b = 8, and the included angle C = 60°. Find side c (Law of Cosines).
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 52 + 162 − 2(5)(16)cos 120°. cos 120° = -12, so 2(5)(16)(-12) = -80 and c2 = 25 + 256 + 80 = 361. c = √361 = 19.
✓ Correct!
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 52 + 162 − 2(5)(16)cos 120°. cos 120° = -12, so 2(5)(16)(-12) = -80 and c2 = 25 + 256 + 80 = 361. c = √361 = 19.
💡 How to solve: Law of Sines & Law of Cosines
When two sides and the included angle are given, use the Law of Cosines: c² = a² + b² - 2ab·cos C. Remember cos 60° = 12 and cos 120° = -12, then take the square root to find c.
Try a similar one
In a triangle, side a = 5, side b = 8, and the included angle C = 60°. Find side c (Law of Cosines).
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 52 + 82 − 2(5)(8)cos 60°. cos 60° = 12, so 2(5)(8)(12) = 40 and c2 = 25 + 64 − 40 = 49. c = √49 = 7.
✓ Correct!
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 52 + 82 − 2(5)(8)cos 60°. cos 60° = 12, so 2(5)(8)(12) = 40 and c2 = 25 + 64 − 40 = 49. c = √49 = 7.
💡 How to solve: Law of Sines & Law of Cosines
When two sides and the included angle are given, use the Law of Cosines: c² = a² + b² - 2ab·cos C. Remember cos 60° = 12 and cos 120° = -12, then take the square root to find c.
Try a similar one
In a triangle, side a = 7, side b = 15, and the included angle C = 60°. Find side c (Law of Cosines).
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 72 + 82 − 2(7)(8)cos 120°. cos 120° = -12, so 2(7)(8)(-12) = -56 and c2 = 49 + 64 + 56 = 169. c = √169 = 13.
✓ Correct!
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 72 + 82 − 2(7)(8)cos 120°. cos 120° = -12, so 2(7)(8)(-12) = -56 and c2 = 49 + 64 + 56 = 169. c = √169 = 13.
💡 How to solve: Law of Sines & Law of Cosines
When two sides and the included angle are given, use the Law of Cosines: c² = a² + b² - 2ab·cos C. Remember cos 60° = 12 and cos 120° = -12, then take the square root to find c.
Try a similar one
In a triangle, side a = 3, side b = 8, and the included angle C = 60°. Find side c (Law of Cosines).
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 32 + 82 − 2(3)(8)cos 60°. cos 60° = 12, so 2(3)(8)(12) = 24 and c2 = 9 + 64 − 24 = 49. c = √49 = 7.
✓ Correct!
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 32 + 82 − 2(3)(8)cos 60°. cos 60° = 12, so 2(3)(8)(12) = 24 and c2 = 9 + 64 − 24 = 49. c = √49 = 7.
💡 How to solve: Law of Sines & Law of Cosines
When two sides and the included angle are given, use the Law of Cosines: c² = a² + b² - 2ab·cos C. Remember cos 60° = 12 and cos 120° = -12, then take the square root to find c.
Try a similar one
In a triangle, side a = 3, side b = 5, and the included angle C = 120°. Find side c (Law of Cosines).
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 32 + 52 − 2(3)(5)cos 120°. cos 120° = -12, so 2(3)(5)(-12) = -15 and c2 = 9 + 25 + 15 = 49. c = √49 = 7.
✓ Correct!
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 32 + 52 − 2(3)(5)cos 120°. cos 120° = -12, so 2(3)(5)(-12) = -15 and c2 = 9 + 25 + 15 = 49. c = √49 = 7.
💡 How to solve: Law of Sines & Law of Cosines
When two sides and the included angle are given, use the Law of Cosines: c² = a² + b² - 2ab·cos C. Remember cos 60° = 12 and cos 120° = -12, then take the square root to find c.
Try a similar one
In a triangle, side a = 3, side b = 8, and the included angle C = 60°. Find side c (Law of Cosines).
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 32 + 82 − 2(3)(8)cos 60°. cos 60° = 12, so 2(3)(8)(12) = 24 and c2 = 9 + 64 − 24 = 49. c = √49 = 7.
✓ Correct!
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 32 + 82 − 2(3)(8)cos 60°. cos 60° = 12, so 2(3)(8)(12) = 24 and c2 = 9 + 64 − 24 = 49. c = √49 = 7.
💡 How to solve: Law of Sines & Law of Cosines
When two sides and the included angle are given, use the Law of Cosines: c² = a² + b² - 2ab·cos C. Remember cos 60° = 12 and cos 120° = -12, then take the square root to find c.
Try a similar one
In a triangle, side a = 5, side b = 16, and the included angle C = 120°. Find side c (Law of Cosines).
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 72 + 82 − 2(7)(8)cos 120°. cos 120° = -12, so 2(7)(8)(-12) = -56 and c2 = 49 + 64 + 56 = 169. c = √169 = 13.
✓ Correct!
Not quite. The answer is
💡 Law of Cosines: c2 = a2 + b2 − 2ab cos C = 72 + 82 − 2(7)(8)cos 120°. cos 120° = -12, so 2(7)(8)(-12) = -56 and c2 = 49 + 64 + 56 = 169. c = √169 = 13.
💡 How to solve: Law of Sines & Law of Cosines
When two sides and the included angle are given, use the Law of Cosines: c² = a² + b² - 2ab·cos C. Remember cos 60° = 12 and cos 120° = -12, then take the square root to find c.
Try a similar one
In a triangle, side a = 5, side b = 8, and the included angle C = 60°. Find side c (Law of Cosines).
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
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Law of Sines & Law of Cosines — Easy Worksheet 1
Find the missing side length.