Question 1: Given a triangle where side a = 8 cm, side b = 10 cm, and the included angle between them is 30°, use the sine rule to find the possible measures of the opposite angle to side b. State whether the ambiguous case applies.

Answer: The possible angles are 45.6° and 134.4°. The ambiguous case applies because sin B has two possible solutions.

Question 2: Calculate the length of side c in a triangle where side a = 7 cm, side b = 9 cm, angle A = 50°, and angle B is unknown. Use the sine rule and consider the ambiguous case.

Answer: c ≈ 11.4 cm

Question 3: A surveyor measures a triangle where two sides are 15 m and 20 m, with an included angle of 60°. Using the sine rule, find the possible measures of the third side and comment on the ambiguity.

Answer: The third side can be approximately 19.1 m or 31.4 m. Ambiguity occurs because two solutions are possible.

Question 4: Construct a triangle with sides 6 cm, 8 cm, and an angle of 40° between the sides of length 6 cm and 8 cm. Use the sine rule to find the possible angles opposite the side of length 8 cm.

Answer: Angles are approximately 59.0° and 121.0°.

Question 5: In a triangle, side a = 12 m, side b = 14 m, and the angle between them is 20°. Determine if the ambiguous case occurs and find the possible angles opposite side b.

Answer: Yes, the ambiguous case occurs; angles are approximately 18.0° and 162.0°.

Question 6: A boat is sailing to a point where the angle between its current and a landmark is measured as 70°. The boat's speed and the distance to the landmark are known, but the exact position is uncertain due to the ambiguous case. Explain how the ambiguous case affects navigation decisions.

Answer: The ambiguous case means there are two possible positions of the boat relative to the landmark, which could lead to different navigation routes. Additional information is needed to determine the correct position.

Question 7: A tower casts a shadow measuring 25 m. The angle of elevation from the tip of the shadow to the top of the tower is 35°. Using the sine rule, estimate the height of the tower, considering the possibility of an ambiguous solution.

Answer: Tower height ≈ 27.2 m; ambiguity is unlikely due to the context, but theoretically possible if other conditions match.

Question 8: Explain what conditions lead to the ambiguous case in the sine rule and how students can identify when it applies.

Answer: The ambiguous case occurs when two sides and a non-included angle are known, and sin of the unknown angle yields two possible solutions. It applies when the given data satisfies the criteria for two triangles: sin B less than 1, and side lengths and angles are compatible.

Question 9: A triangle has sides of 9 cm and 12 cm with an included angle of 70°. Find all possible measures for the third side, considering the ambiguous case.

Answer: Possible third sides are approximately 5.0 cm or 19.4 cm.

Question 10: Identify the errors in the following statement: 'Since sin B can only have one value between 0° and 180°, the ambiguous case cannot occur when using the sine rule.'

Answer: Incorrect; sine B can have two solutions when sin B is between 0 and 1, leading to the ambiguous case.