Question 1: Calculate the length of side a in a triangle where sides b = 7 cm, c = 10 cm, and the included angle A = 60°.

Answer: 9.22

Question 2: A triangle has sides b = 8 m, c = 6 m, and an included angle A = 45°. Find side a, correct to two decimal places.

Answer: 9.01

Question 3: In triangle PQR, side PQ = 5 cm, side PR = 7 cm, and the angle at P is 120°. Calculate the length of side QR.

Answer: 9.89

Question 4: A swimmer swims from point A to point B, 12 km apart across a river. The swimmer maintains a constant heading, forming an angle of 30° with the bank. Find the shortest distance the swimmer swims directly across if the angle is measured at point A.

Answer: 10.39

Question 5: Construct a triangle with sides b = 9 cm, c = 12 cm, and included angle A = 70°. Use the grid to draw an accurate triangle.

Answer: Student draws accurately on grid

Question 6: Given a triangle with sides a = 10 cm, b = 8 cm, and angle A = 60°, verify whether side c calculated from the cosine rule matches the given length of 9.5 cm. Show your working.

Answer: Yes, approximately 9.49 cm

Question 7: A triangle has sides b = 5 cm, c = 9 cm, and side a = 8 cm. The included angle A is unknown. Use the cosine rule to find angle A, correct to one decimal place.

Answer: 53.1°

Question 8: In triangle XYZ, side XY = 6 cm, side XZ = 8 cm, and angle X = 135°. Find the length of side YZ.

Answer: 10.20

Question 9: Error analysis: A student attempts to find side a using the cosine rule but forgets to square the sides before subtracting the cosine term. Identify the mistake and suggest the correction.

Answer: They forgot to square b and c; correction: a² = b² + c² - 2bc cosA

Question 10: Challenge: In triangle DEF, sides b = 7.5 cm, c = 9.2 cm, and the angle A = 40°. Calculate side a accurately, then verify whether the sum of sides b and c exceeds side a.

Answer: a ≈ 12.47 cm; Yes, b + c > a