The Law of Cosines Questions

Liam and Abigail are planning a treasure hunt and need to calculate the distance between two points on their map. They know the lengths of two paths and the angle between them. Can you help them find the formula to calculate the missing side using the Law of Cosines?

Answer explanation

The Law of Cosines states that for a triangle with sides a, b, c and included angle A, the correct formula is a^2 = b^2 + c^2 - 2bc * cos(A). This allows calculation of side a when b, c, and angle A are known.

In triangle ABC, if a=5, b=7, and C=60°, use the Law of Cosines to find c.

Answer explanation

Use the Law of Cosines: c² = a² + b² - 2ab * cos(C). Plugging in values: c² = 5² + 7² - 2*5*7*cos(60°). This simplifies to c² = 25 + 49 - 35 = 39. Thus, c = √39, approximately 6.24.

Elijah and Zoe are on a treasure hunt and they come across a triangular map marked as triangle ABC. If side a = 8 meters, side b = 6 meters, and side c = 10 meters, use the Law of Cosines to help them find the angle at point C so they can continue their adventure!

• 60°

• 45°

• 90°

• 75°

Answer explanation

Using the Law of Cosines: c² = a² + b² - 2ab * cos(C). Plugging in values: 10² = 8² + 6² - 2*8*6*cos(C). Solving gives cos(C) = 0, thus C = 90°. Therefore, the correct answer is 90°.

Abigail and Mason are on a treasure hunt and they need to find the length of a mysterious path to reach the treasure. In triangle ABC, where the path is side c, they know that a=12, b=15, and the angle C=120°. Can you help them find the length of side c?

• The length of side c is approximately 23.4.

• The length of side c is approximately 20.5.

• The length of side c is approximately 22.3.

• The length of side c is approximately 25.1.

Answer explanation

Using the Law of Cosines: c² = a² + b² - 2ab * cos(C). Plugging in values: c² = 12² + 15² - 2*12*15*cos(120°). This simplifies to c² ≈ 549, giving c ≈ 23.4. Thus, the correct answer is approximately 23.4.

Mia, Emma, and Liam are on a treasure hunt and they come across a mysterious triangular map. The sides of the triangle are marked as a=7, b=9, and c=11. Help them find the angle C to unlock the next clue!

• angle C = 30°

• angle C = 45°

• angle C = 85.9°

• angle C = 90°

Answer explanation

Using the Law of Cosines: c² = a² + b² - 2ab*cos(C). Plugging in values: 11² = 7² + 9² - 2*7*9*cos(C). Solving gives cos(C) = 0.0714, thus angle C = 85.9°.

Elijah and Abigail are on a treasure hunt and they need to find the length of a mysterious path to reach the treasure. In triangle ABC, where the path is side c, they know that a=8, b=6, and angle C=120°. Use the Law of Cosines to help them find the length of the path c. (Your answer in 3 significant figures.)

(a)  

Answer explanation

Using the Law of Cosines: c² = a² + b² - 2ab * cos(C). Plugging in values: c² = 8² + 6² - 2*8*6*(-0.5). This simplifies to c² = 64 + 36 + 48 = 148. Thus, c = √148, which is approximately 12.2, not 0.

Liam and Nora are exploring a mysterious triangular island. They discover that angle C of the triangle is 120°, and the sides leading to it measure 9 meters and 10 meters. Can you help them find the length of the third side, c, so they can complete their map?

• 7.5

• 12.5

• 15.5

• 16.5

Answer explanation

Using the Law of Cosines: c² = a² + b² - 2ab*cos(C). Plugging in the values: c² = 9² + 10² - 2*9*10*(-0.5). This simplifies to c² = 81 + 100 + 90 = 271, so c = √271, which is approximately 16.5. Thus, the correct answer is 16.5.

Noah and Ava are on a treasure hunt and they find a mysterious triangular map. The map shows two sides of the triangle, a=4 and b=5, with an angle C=45° between them. They need to use the Law of Cosines to find the length of the third side, c, to locate the treasure. Can you help them find side c?

• c = 3.57

• c = 5.12

• c = 6.78

• c = 4.47

Answer explanation

Using the Law of Cosines: c² = a² + b² - 2ab * cos(C). Plugging in values: c² = 4² + 5² - 2*4*5*cos(45°). This simplifies to c² = 16 + 25 - 20*0.7071, resulting in c ≈ 3.57.

Abigail and Liam are on a treasure hunt and they come across a triangular map with sides a=5, b=8, and c=10. To find the next clue, they need to determine the measure of angle C in the triangle. Can you help them solve this mystery?

• 36.87°

• 53.13°

• 97.9°

• 60°

Answer explanation

Using the Cosine Law equals 97.9°.

In triangle ABC, if a=5, b=7, and C=60°, use the Law of Cosines to find the missing side.

• 6.24

• 5

• 8.19

• 10.2

Answer explanation

The answer should be approximately 6.24.