Trigonometry & Heights / Distances • Topic 4 of 4

Heights & Distances

Model the situation as a right triangle: the angle of elevation (looking up) or depression (looking down) sits at the observer, the height is the opposite side and the ground distance the adjacent side, so tan(angle) = height/distance. Standard angles 30, 45, 60 give clean answers. For two observations, set up two equations and subtract.

✅ Solved examples

1. The angle of elevation of a tower top from 30 m away is 45 degrees. Tower height?
tan 45 = h/30 = 1 -> h = 30 m.
2. From 20 m, the elevation of a pole top is 60 degrees. Height?
tan 60 = h/20 = root3 -> h = 20 root3 m.
3. A 15 m ladder leans at 60 degrees to the ground. Height it reaches?
sin 60 = h/15 -> h = 15 x root3/2 = 7.5 root3 m.
4. Elevation of a tower top is 30 degrees from 90 m. Height (use tan 30 = 1/root3)?
h = 90/root3 = 30 root3 m.

✏️ Practice — try these, take hints as needed

1. Elevation 45 at 50 m. Height?
tan 45 = h/50.
—
—
50 m
2. Elevation 60 at 10 m. Height?
tan 60 = h/10.
10 root3.
—
10 root3 m
3. A 10 m ladder at 30 degrees reaches what height?
sin 30 = h/10.
10 x 1/2.
—
5 m
4. Elevation 30 at 60 m, height?
tan 30 = h/60.
60/root3.
—
20 root3 m
5. Shadow of a pole equals its height. Sun elevation?
tan = h/h = 1.
—
—
45 degrees

📝 Topic test — 8 questions

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