Question 1: Given a triangle where sides a=7 cm, side b=10 cm, and the angle opposite side a is 40°, determine whether the Sine Rule can be used to find angle B, and if so, find angle B.

Answer: Yes; approximately 63.4°

Question 2: Two sides of a triangle are 8 cm and 13 cm, with the included angle of 50°. Use the Sine Rule to determine if the third side can be uniquely found and state the possible number of solutions.

Answer: Yes; one or two solutions depending on the case

Question 3: A triangle has sides 9 cm and 12 cm, with an included angle of 70°. Can the Ambiguous Case occur here? Explain your reasoning briefly.

Answer: No; because the given data does not satisfy the criteria for ambiguity

Question 4: Construct a triangle with sides 5 cm and 9 cm, and an angle of 60° between them. Use the Sine Rule to find the possible lengths of the third side.

Answer: Approximately 4.2 cm and 13.9 cm

Question 1: A triangle has sides 15 cm and 20 cm with an included angle of 120°. Use the Sine Rule to determine whether there are zero, one, or two possible positions for the third side, and find its length if solutions exist.

Answer: Two solutions; third side approximately 10.4 cm and 27.6 cm

Question 2: Explain why the Ambiguous Case occurs only under certain conditions when using the Sine Rule for solving triangles.

Answer: Because the sine of the angle can correspond to two different angles between 0° and 180°, leading to two possible triangles when the given data satisfies certain criteria.

Question 3: In a scenario with sides 6 cm and 10 cm and an included angle of 80°, determine whether the Sine Rule will produce one solution, two solutions, or no solution. Justify your answer.

Answer: One solution; the height corresponding to the given data allows only one triangle.

Question 4: Given two possible triangles with sides 7 cm and 24 cm, and a common angle of 30°, explain how to determine which triangle is valid based on the Law of Sines.

Answer: Compare the calculated third side lengths for each possible angle; the valid triangle must satisfy the triangle inequality.

Question 1: A surveyor measures two sides of a triangular plot of land as 50 m and 70 m, with the included angle measured as 100°. Using the Sine Rule, identify how many possible positions the third corner could have and find the length of the third side for each case.

Answer: Two solutions; third side approximately 43.6 m and 109.9 m

Question 1: Construct a triangle where sides a=8 cm, b=15 cm, and the included angle is 100°. Use the Sine Rule to determine all possible configurations of the triangle. Discuss the conditions for each and calculate the third side length in each case.

Answer: Two solutions; third side approximately 17.4 cm and 8.6 cm

Question 2: Given a triangle with sides 9 cm and 14 cm, and an included angle of 90°, analyze whether the Ambiguous Case applies, and if not, justify why.

Answer: No; because the angle is 90°, so the data does not satisfy the criteria for ambiguity; the triangle is uniquely determined.

Question 1: A student attempts to use the Sine Rule with sides 10 cm, 15 cm, and an angle of 50°, claiming there are two possible triangles. Identify and explain the mistake in the student's reasoning.

Answer: The student assumes two solutions exist without analyzing the height or the sine of the angle; proper check shows only one valid triangle.

Question 2: List common errors students make when applying the Ambiguous Case and how to avoid them.

Answer: Incorrectly assuming two solutions without verifying the sine value, misapplying the Law of Sines, or ignoring the constraints of the sine function.