Question 1: In a right-angled triangle, the side opposite the angle is 6 cm long and the hypotenuse is 10 cm. Calculate the sine of the angle.

Answer: 0.6

Question 2: A ladder leaning against a wall makes an angle of 30° with the ground. The ladder is 8 meters long. Find the height the ladder reaches on the wall.

Answer: 4

Question 3: Construct a right-angled triangle where the hypotenuse is 15 cm and the angle is 45°. Then, find the length of the side opposite the angle.

Answer: 10.6

Question 4: A satellite dish forms a 60° angle with the ground. The line from the dish to the satellite is 20 meters long. Using SOH, find the height of the satellite above ground level.

Answer: 17.3

Question 5: Identify and correct the mistake in this solution: 'Using sine = Opposite / Hypotenuse, Opposite = 8, Hypotenuse = 10, so sine = 8/10 = 0.8. Therefore, the angle is 36°.'

Answer: The mistake is that the inverse sine (arcsin) should be used to find the angle, not the sine value itself.

Question 6: In a right triangle, the angle is 25°, and the side opposite is 7 cm. Calculate the hypotenuse using SOH.

Answer: 15.2

Question 7: A tree casts a shadow 12 meters long when the angle of elevation of the sun is 45°. How tall is the tree?

Answer: 12

Question 8: Explain how you would find the length of the hypotenuse if you know the opposite side and the sine of the angle.

Answer: Use the formula hypotenuse = Opposite / sin(angle).

Question 9: A right triangle has an angle of 40° and the side adjacent to it is 9 cm. Find the length of the hypotenuse.

Answer: 13.9

Question 10: A plane flies at an altitude of 200 meters and the angle of depression to a ship on the sea is 10°. How far is the ship from the point directly below the plane?

Answer: 1149.8

Question 11: Challenging: The angle of elevation to the top of a tower is 35°. If the distance from the observer to the tower is 50 meters, find the height of the tower.

Answer: 34.2

Question 12: Identify the common mistake in this problem: 'Given Opposite = 5, Hypotenuse = 13, and the angle is 25°, the student calculates sine as 5/13 and finds the angle using arcsin. However, they forget to verify if the calculated angle makes sense in context.'

Answer: The mistake is not checking if the calculated angle fits the triangle's constraints; students should verify the context and reasoning.