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Applied triangle geometry
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Angles of elevation and depression, true bearings, projections and more
Want a deeper conceptual understanding? Try our interactive lesson!
Angles of elevation & depression
SL Core 3.3
Angles of elevation and depression describe how far upward or downward you look relative to a horizontal line when observing an object.
The angle of elevation is the angle formed by looking upward from the horizontal line to an object above your line of sight.
The angle of depression is the angle formed by looking downward from the horizontal line to an object below your line of sight.
These angles are always measured relative to a horizontal line, never vertical. Because the lines of sight form alternate interior angles with horizontal lines, the angle of elevation from one viewpoint equals the angle of depression from the other viewpoint.
True bearings
SL Core 3.3
A true bearing describes a direction measured clockwise from the north direction (0°) around a full circle up to 360°. Bearings are always given using three digits (e.g. 045°,120°,270°) to avoid confusion.
A bearing of 000∘ points directly north.
090∘ points east.
180∘ points south.
270∘ points west.
When working with true bearings, clearly draw a compass rose to visualize directions and measure angles clockwise from the north line.
One very useful fact about bearings is that returning in the direction something came from means adding or subtracting 180° from the original bearing - since it is doing a U-turn.
Whether you add or subtract depends on whether the original bearing is smaller or bigger than 180°, since the resulting bearing must be between 0 and 360°.
In the real world, bearings are a navigational tool, often used in the context of sea navigation. The bearing of an object describes a direction measured clockwise from due north, and is useful as a standardization of direction, instead of other settings (like angles of depression or elevation) where the reported angle is subjectively based upon the observer's position.
The diagram below shows a kind of compass rose situated on the Cartesian plane. Try dragging around the point at the circle's exterior to build a visual understanding of the information that different bearings convey.
Angles in 3D
SL Core 3.1
By carefully imagining right triangles in 3D diagrams, we can find angles in space.
In the triangular prism below, we can find the length CB using Pythagoras. Then, since there is a right angle at C, we can write
tanθ=4√29⇒θ≈53.4∘
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