Question 1: A satellite is in orbit around Earth. The distance from the satellite to the Earth's surface is 300 km. The satellite forms a 60° angle with the Earth's surface at a certain point. Calculate the straight-line distance from the satellite to the surface using the cosine rule.

Answer: Calculate using a² = b² + c² - 2bc cosA

Question 2: A space probe measures 500 km between two planets and forms a 45° angle at a certain observation point. Calculate the distance from the observation point to the second planet.

Answer: Apply a² = b² + c² - 2bc cosA

Question 3: In a triangular configuration in space, side b is 200 km, side c is 250 km, and the included angle A is 70°. Find side a.

Answer: Use a² = b² + c² - 2bc cosA

Question 4: Construct a triangle representing a satellite, a planet, and a station, with the satellite 400 km from the planet and forming a 50° angle at the station. Calculate the distance from the station to the satellite.

Answer: Apply cosine rule accordingly

Question 1: A telescope observes two stars forming a triangle with the Earth at the vertex. The distances from Earth to Star A and Star B are 1500 km and 2000 km respectively. The angle between the two lines of sight is 40°. Calculate the distance between the two stars.

Answer: Use the cosine rule to find side between the stars

Question 2: Explain why the cosine rule is useful in calculating distances between celestial bodies when only angles and some distances are known.

Answer: It allows calculation of unknown sides in non-right-angled triangles using known angles and sides.

Question 1: A space explorer sights two planets forming a 75° angle at their spaceship. The distance from the spaceship to Planet X is 600 km, and to Planet Y is 800 km. Calculate the straight-line distance between the two planets.

Answer: Apply the cosine rule for the two sides and the included angle.

Question 2: During a mission, a satellite observes two planets forming a 60° angle. The distances from the satellite to each planet are 400 km and 500 km. Find the distance between the two planets.

Answer: Calculate using a² = b² + c² - 2bc cosA

Question 1: In a space triangle, side b is 350 km, side c is 400 km, and the included angle A is 85°. Calculate side a and discuss the potential sources of error in the measurement.

Answer: Use cosine rule; consider measurement inaccuracies.

Question 2: A spaceship forms a 90° angle with two planets, with known distances to each. Explain how the cosine rule simplifies calculations in this orthogonal case.

Answer: When angle is 90°, cosine rule reduces to Pythagoras' theorem.

Question 1: A triangle in space has sides 180 km and 220 km with an included angle of 55°. Calculate the third side.

Answer: Apply a² = b² + c² - 2bc cosA

Question 2: Identify and correct the mistake: Given two sides of length 300 km each and an included angle of 60°, the student uses the formula a² = b² + c² + 2bc cosA to find side a.

Answer: Incorrect sign in formula; should be minus, not plus.