Question 1: Rearrange the Cosine Rule formula to explicitly solve for the angle C in triangle ABC where side a is opposite angle A, side b is opposite B, and side c is opposite C.

Answer: cos C = (a^2 + b^2 - c^2) / (2ab)

Question 2: Using the rearranged formula, find the measure of angle C if a = 7 cm, b = 9 cm, and c = 10 cm. Show your working.

Answer: cos C = (7^2 + 9^2 - 10^2) / (2 * 7 * 9) = (49 + 81 - 100) / 126 = 30/126 = 0.2381; C ≈ 76.2°

Question 3: Rearrange the Cosine Rule to solve for side c when angles A and B and sides a and b are known.

Answer: c^2 = a^2 + b^2 - 2ab * cos C

Question 4: In triangle PQR, sides p=8 cm, q=6 cm, and the included angle R is 60°. Use the Cosine Rule to find side r. Show your working.

Answer: r^2 = 8^2 + 6^2 - 2 * 8 * 6 * cos 60° = 64 + 36 - 96 * 0.5 = 100 - 48 = 52; r ≈ 7.21 cm

Question 5: Explain why incorrectly rearranging the Cosine Rule might lead to an incorrect calculation of the angle.

Answer: Common mistakes include misplacing terms, forgetting to square sides, or misapplying the formula, leading to wrong cosine values and angles.

Question 6: Given sides a=5, b=7, and an angle C of 60°, rearranged formula to find side c. Calculate c.

Answer: c^2 = 5^2 + 7^2 - 2 * 5 * 7 * cos 60° = 25 + 49 - 70 * 0.5 = 74 - 35 = 39; c ≈ 6.24 cm

Question 7: Construct a triangle with sides 8 cm, 10 cm, and an included angle of 45°. Use the rearranged Cosine Rule to find the third side.

Answer: c^2 = 8^2 + 10^2 - 2 * 8 * 10 * cos 45° ≈ 64 + 100 - 160 * 0.7071 ≈ 164 - 113.14 ≈ 50.86; c ≈ 7.13 cm

Question 8: A triangle has sides a=6, b=9, and an angle C of 120°. Use the formula to find side c.

Answer: c^2 = 6^2 + 9^2 - 2 * 6 * 9 * cos 120° = 36 + 81 - 108 * (-0.5) = 117 + 54 = 171; c ≈ 13.07 cm

Question 9: Identify the common mistake students make when rearranging the Cosine Rule to find an angle, and explain how to avoid it.

Answer: A common mistake is forgetting to take the inverse cosine after calculating the cosine value. To avoid this, always remember to apply cos⁻¹ to find the angle.

Question 10: Using the rearranged formula, determine whether the following set of sides could form a triangle: a=4, b=7, c=10. Justify your answer.

Answer: Check the triangle inequality: 4 + 7 = 11 > 10 (OK), 4 + 10 = 14 > 7 (OK), 7 + 10 = 17 > 4 (OK). Yes, they can form a triangle.