Height and Distance with a Theodolite
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flagWhat you'll discover
- arrow_forwardIdentify the angle of elevation and the angle of depression in a figure
- arrow_forwardUse tan θ = perpendicular ⁄ base to find a height or a distance
- arrow_forwardSolve problems in which the instrument has its own height above the ground
- arrow_forwardDraw a clean, labelled figure before substituting any numbers
- arrow_forwardRound the answer sensibly and state the unit
The two angles
The angle of elevation is the angle between the horizontal line through the observer's eye and the line of sight to an object above the horizontal. If you look up at the top of a tower, that is the angle of elevation.
The angle of depression is the angle between the horizontal through the observer and the line of sight to an object below. It is measured downwards. Because the two horizontal lines are parallel, the angle of depression from the top of a tower equals the angle of elevation from the foot — a fact that turns most "depression" problems into the same triangle.
The working rule
In the right-angled triangle, tan θ = perpendicular ⁄ base. So height = distance × tan θ, and distance = height ⁄ tan θ.
Use sin when the hypotenuse and the opposite side are involved, and cos when the hypotenuse and the adjacent side are involved — but in height-and-distance questions the known pair is almost always the base and the height, which means tan.
If the instrument is not at ground level, work in two steps: find the height above the instrument using tan, then add the instrument's own height. Forgetting that addition is the commonest lost mark in the whole question.
Standard values to memorise
tan 30° = 1/√3 ≈ 0.5774, tan 45° = 1, tan 60° = √3 ≈ 1.7321. Also sin 30° = 1/2, sin 60° = √3/2, cos 30° = √3/2 and cos 60° = 1/2.
SEE questions are written so that the arithmetic comes out clean with these values. If you find yourself reaching for a calculator for a 45° problem, you have probably set the triangle up wrongly. Always draw the figure first, mark the right angle and the angle of elevation, and write which side is known and which is wanted before substituting.