Question 1: A company is designing a triangular billboard where two sides measure 8 meters and 10 meters, and the included angle is unknown. If the angle between these sides is 60°, rearrange the Cosine Rule to find the angle in terms of the sides and cosine value.

Answer: cos(C) = (8^2 + 10^2 - c^2) / (2 * 8 * 10)

Question 2: Rearrange the Cosine Rule to solve for the angle when the sides are 5 meters, 7 meters, and 9 meters.

Answer: cos(C) = (5^2 + 7^2 - 9^2) / (2 * 5 * 7)

Question 3: In a business context, a triangular plot of land has sides measuring 50 meters, 80 meters, and an unknown side. The included angle between the two known sides is 45°. Rearrange the formula to find the unknown side length.

Answer: c = √(50^2 + 80^2 - 2 * 50 * 80 * cos(45°))

Question 4: A business is analyzing the distance between two cities. The roads form two sides of a triangle with lengths 120 km and 150 km, and the included angle is 70°. Rearrange the Cosine Rule to find the third side.

Answer: c = √(120^2 + 150^2 - 2 * 120 * 150 * cos(70°))

Question 5: Calculate the angle between two investments if their returns and the covariance are modeled as sides of a triangle with sides 12, 15, and 20 units. Rearrange the Cosine Rule to find the angle.

Answer: cos(θ) = (12^2 + 15^2 - 20^2) / (2 * 12 * 15)

Question 6: A company’s marketing campaign results in a triangle of data points where two sides are 60% and 80%. The included angle is unknown. Rearrange the Cosine Rule to express the angle in terms of the sides and the third side length.

Answer: cos(C) = (60^2 + 80^2 - c^2) / (2 * 60 * 80)

Question 7: A retailer is analyzing the relationship between sales in two regions forming a triangle with sides 30 and 40 units. The angle between these sales figures is unknown. Rearrange the Cosine Rule to find the angle.

Answer: cos(C) = (30^2 + 40^2 - c^2) / (2 * 30 * 40)

Question 8: A profit analysis involves a triangle with known sides of 100 and 150 units. Rearrange the Cosine Rule to find the unknown side if the angle between the known sides is 55°.

Answer: c = √(100^2 + 150^2 - 2 * 100 * 150 * cos(55°))

Question 9: A project involves three costs forming a triangle, with two costs being 200 and 300 units, and the included angle is unknown. Rearrange the Cosine Rule to express the third cost in terms of the sides and the cosine of the angle.

Answer: c = √(200^2 + 300^2 - 2 * 200 * 300 * cos(C))

Question 10: Identify and correct the mistake in the following rearranged formula: cos(C) = (a^2 + b^2 + c^2) / (2ab).

Answer: The sign before c^2 should be minus, not plus.