The angle of elevation of an aircraft from a point K on the horizontal ground 30$\alpha$. If the aircraft is 800m above the ground, how far is it from K?

Analyze the question: The question text is garbled: "30α" most likely means the angle of elevation is 30°. The aircraft is 800 m above the ground, and we need the straight-line distance from K to the aircraft. That distance is the hypotenuse of a right-angled triangle.

Concept: In a right-angled triangle, with θ the angle of elevation:

• Opposite side = height of the aircraft = 800 m
• Hypotenuse = distance from K to the aircraft

sin θ = opposite / hypotenuse

Solution:

sin 30° = 800 / d

1/2 = 800 / d

d = 800 × 2 = 1,600 m

Check against the options: The other options come from the wrong trig ratios.

• 800 / tan 30° = 800√3 ≈ 1,385.6 m, which is the horizontal distance and is not an option.
• 800 × tan 30° ≈ 461.9 m, which is not an option either.
• 692.82 = 800 × (√3/2) = 800 cos 30°, which is a misuse of the ratio.
• 400 = 800 × sin 30°, which multiplies instead of dividing.

Tip: Remember SOH-CAH-TOA. When the height (opposite) and the "direct distance" (hypotenuse) are involved, use sine. Since sin 30° = ½, the hypotenuse is always twice the opposite side, which is a quick check for any 30° elevation question.