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Angle of elevation and depression

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Angle of Elevation and Depression

The angle of elevation is measured upward from the horizontal to a line of sight; the angle of depression is measured downward. Learn why the two are equal as alternate angles, and how to solve height and distance problems using the tangent, sine, and cosine ratios, with a worked example.

Elevation and depression

The angle of elevation is the angle measured upward from a horizontal line to your line of sight when you look up at an object. The angle of depression is the angle measured downward from the horizontal when you look down. Both are always measured from the horizontal, not from the vertical.

Angle of elevation and angle of depression From an observer's eye, a horizontal dashed reference line runs to the right. A line of sight goes up to a tall object, making the angle of elevation of 25 degrees with the horizontal. Another line of sight goes down to a low object, making the angle of depression of 20 degrees below the horizontal. horizontal observer angle of elevation angle of depression object object
The angle of elevation is measured up from the horizontal; the angle of depression is measured down.

Why the two angles are equal

When you look down at an object and it looks back up at you, the angle of depression from your eye equals the angle of elevation from the object. The two horizontals are parallel, so these are alternate interior angles and therefore equal. This lets you move the angle to whichever triangle is easier to solve.

Solving with trigonometric ratios

These problems become right triangles, so you solve them with the trig ratios. The tangent ratio is the most common because it links the opposite and adjacent sides — usually a height and a ground distance. The sine ratio and cosine ratio are used when the hypotenuse (a line of sight distance) is involved.

Worked example

From a point 40 m from the base of a tower, the angle of elevation to the top is 35°. The height is h = 40 × tan(35°) ≈ 28 m. Because tangent relates the opposite side (height) to the adjacent side (distance), one ratio solves it.

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