Question 1: Given a triangle where side a = 8 cm and angle A = 30°, and side b = 10 cm with angle B = 45°, verify whether the lengths and angles satisfy the sine rule. Calculate the missing side c if angle C = 105°.

Answer: Calculate c using a/sinA = c/sinC; c = (a * sinC) / sinA

Question 2: In triangle XYZ, side x = 7.5 cm, angle X = 40°, and angle Y = 60°. Find the length of side y.

Answer: Use a/sinA = b/sinB to find y

Question 3: A boat is anchored at a point where the angle of elevation to the top of a lighthouse is 25°. From a point 200 meters closer to the lighthouse, the angle of elevation is 35°. Calculate the height of the lighthouse.

Answer: Use tangent or sine rule reasoning, set up equations based on angles and distances

Question 4: Construct a triangle with sides of lengths 6 cm and 10 cm, and an included angle of 60°. Find the length of the third side using the sine rule.

Answer: Apply sine rule after constructing the triangle: c = √(a² + b² - 2ab cosC)

Question 5: In triangle PQR, side p = 9 cm, side q = 12 cm, and angle P = 50°. Calculate angle Q.

Answer: Use a/sinA = b/sinB to find angle Q

Question 6: A surveyor measures a tree's height indirectly. The angle of elevation from a point 30 meters away is 40°, and from 50 meters away is 50°. Find the height of the tree.

Answer: Set up right triangle equations or sine rule based on measurements

Question 7: Given triangle with sides 5 cm and 7 cm and included angle of 80°, find the length of the third side.

Answer: Use the Law of Cosines or sine rule where appropriate

Question 8: A triangle has sides 9 cm, 12 cm, and an unknown side x. If angles opposite sides 9 cm and 12 cm are 40° and 60°, respectively, find side x.

Answer: Apply sine rule to find x based on given angles and sides

Question 9: Determine whether the following set of sides and angles satisfy the sine rule: a= 13 cm, A= 35°, b= 15 cm, B= 60°, C= 85°.

Answer: Yes or No based on checking a/sinA and b/sinB ratios

Question 10: Identify and correct the mistake in the following calculation: Given a=10 cm, A=30°, b=8 cm, B=45°, the student claims a/sinA = b/sinB implies c=12 cm in a different triangle.

Answer: Students explain the mistake, e.g., incorrect application of the sine rule or misinterpretation of angles