Trigonometry Review

Using the tangent function: height = distance * tan(angle). Here, height = 50 * tan(30°) = 50 * (1/√3) ≈ 28.87 metres. Thus, the height of the building is approximately 28.87 metres.

The Sine Law is appropriate here because we have two sides and the angle opposite one of them. It allows us to find the unknown angle opposite side b using the known angle and the sides.

To find the angle opposite side c in a triangle with known sides, the Cosine Law is appropriate. It relates the lengths of the sides to the cosine of one angle, making it suitable for this scenario.

Using the cosine function: cos(60°) = adjacent/hypotenuse. Here, adjacent = 3m (distance from wall), so hypotenuse (ladder length) = 3m/cos(60°) = 3m/(1/2) = 6m. Thus, the ladder is 6 metres long.

Using the Law of Cosines: c² = a² + b² - 2ab * cos(C). Plugging in values: c² = 10² + 12² - 2*10*12*cos(30°). This simplifies to c² = 100 + 144 - 120*√3/2. Calculating gives c ≈ 6.01 cm, the correct answer.

To find the altitude of point B, use the formula: altitude = distance * sin(angle). Here, altitude = 200 km * sin(20°) ≈ 68.40 km. Thus, the correct answer is 68.40 km.

To find the angle opposite side a in a triangle with sides a, b, and c, the Cosine Law is appropriate. It relates the lengths of the sides to the cosine of one of the angles, making it suitable for this scenario.

Using the Law of Cosines: c² = a² + b² - 2ab * cos(C). Plugging in values: c² = 9² + 12² - 2*9*12*cos(45°). This simplifies to c² = 81 + 144 - 108√2. Calculating gives c ≈ 8.50 cm.

To find the straight-line distance, use the Pythagorean theorem: distance = √(100² + 150²) = √(10000 + 22500) = √32500 = 180 km. Thus, the correct answer is 180 km.

Using the Law of Cosines: c² = a² + b² - 2ab * cos(C). Plugging in the values: c² = 6² + 8² - 2*6*8*cos(60°) = 36 + 64 - 48 = 52. Thus, c = √52 ≈ 7.21 cm, which is the correct answer.