Question 1: Given a triangle with sides a=7 cm, b=10 cm, and angle A=40°, use the Sine Rule to find the possible measures of angle B. How many solutions are possible?

Answer: Two solutions, approximately 49.4° and 130.6°

Question 2: In a triangle, side a=8 cm, side b=12 cm, and angle A=60°. Determine all possible angles B using the Sine Rule.

Answer: Approximately 31.8° and 128.2°

Question 3: A surveyor measures a tree at a point 30 meters from its base. The angle of elevation to the top is 35°. If the tree is 20 meters tall, determine whether the ambiguous case applies and find possible heights of the tree if it extends beyond this point.

Answer: Ambiguous case applies; possible tree heights are approximately 20 m and 40 m.

Question 4: Construct a triangle with sides 5 cm and 9 cm, and an included angle of 50°. Use the Sine Rule to determine if there are two possible positions for the third side.

Answer: Yes, two possible positions; the third side can be approximately 7.8 cm or 11.2 cm.

Question 5: Calculate the possible angles B in a triangle with sides a=9, b=14, and angle A=45°, and explain why the ambiguity occurs.

Answer: Angles approximately 34.7° and 125.3°; ambiguity occurs because sine is positive for two different angles.

Question 6: A boat is located 100 meters from a lighthouse. The angle of elevation to the top of the lighthouse is 25°. If the lighthouse can be either 120 meters or 60 meters tall, determine the possible positions of the boat relative to the lighthouse.

Answer: Boat could be approximately 229.2 m or 138.6 m from the lighthouse, depending on its height.

Question 7: Given sides a=6 cm, b=8 cm, and angle A=70°, use the Sine Rule to find angle B. State whether there are zero, one, or two solutions.

Answer: Two solutions; approximately 41.0° and 138.9°.

Question 8: In a triangle, side a=10 cm, side b=15 cm, and angle A=60°. Determine all possible angles B and explain the reasoning behind the number of solutions.

Answer: Angles approximately 29.5° and 150.5°; two solutions because sine of B has two positive angles within a triangle.

Question 9: A construction challenge: construct a triangle with sides 7 cm and 10 cm, and an included angle of 60°. Use the Sine Rule to find if a second possible triangle exists.

Answer: Yes, the second possible triangle has the same side length but a different position; the other possible angle B is approximately 119.5°.

Question 10: Identify the common mistake students make when applying the Sine Rule in the ambiguous case, and suggest how to avoid it.

Answer: Confusing the sine of an angle with the angle itself; avoid by carefully considering the possible angles from inverse sine and reasoning about the triangle.