Solving Height and Distance Problems with Trigonometry
When you need to find the height of something tall without measuring it directly, trigonometry comes to the rescue! In problem 7, we used the tangent ratio to find a traffic light's height. When the angle of elevation is 30° from 30 meters away, we simply use tan 30° = height/30 to solve for the height.
For situations involving airplanes, like problem 8, we use the angle of depression (the downward angle from horizontal). When a plane flies at 12,000 meters and sees an airport tower at a 37° depression angle, we calculate the horizontal distance using tangent again.
Problem 9 shows a practical application with a ladder. When a 12-meter ladder leans against a wall at a 15° angle, we use sine to find the wall's height: sin 15° = wall height/12. This gives us a wall height of 3 meters.
Quick Tip: Remember that angles of elevation look up from horizontal, while angles of depression look down. Both types of problems use the same trigonometric ratios!
In problem 10, we find the minimum takeoff angle for an airplane to clear a building. By using the sine ratio with the building's height and the distance, we can determine the plane needs a 45° angle of elevation to safely clear the 63-foot building.
