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Now that we've gone over how to calculate the angles and side lengths of both right- and non-right triangles, we can discuss some real-world applications.
Discussion
Imagine a person in the lighthouse and a person on the ship are staring at each other. Each wants to know how many miles the ship is from the cliffs. They'll need to use trigonometric ratios to do so, calculating with the use of the angles marked in blue on the diagram.
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Notice that the angle that the person on the ship is looking "up" from is the same as the angle that the person on the lighthouse is looking "down" from. Why is this the case? Will this always be true?
Since the water surface and the horizontal line through the lighthouse are parallel, the line of sight from one observer to the cliff top is a transversal cutting two parallel lines. Hence the blue angles are alternate‐interior angles and so are equal.
Equivalently, if the cliff height is h and its horizontal distance from the ship is d, then for the ship’s angle θ and the lighthouse’s angle ϕ we have
so
This will always be true whenever two parallel lines are cut by a transversal.
We call the angles that the boat and lighthouse observers are watching from the angles of elevation and depression respectively.
Angles of elevation & depression
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.
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(make this a discussion?)
This alternate interior angle relationship is a crucial one to keep in mind. Another way of thinking of it is this: Imagine two people are standing next to each other in a forest and looking at a tall pine tree. One of the people is quite tall, and the other is quite short. Each knows their own height as well as their distance from the tree and the angle of elevation from their sightline to the top of the tree.
(image)
Clearly, the respective heights of the pair do not affect the height of the tree itself. Further, neither can directly calculate the angle of depression from the top of the tree to their own line of sight. In order to calculate the tree's height, then, they can use their respective calculations of the angle of elevation as the angle of depression, and add the value they calculate as the height from the top of their heads to the top of the tree to their own height in order to find the tree's total height.
Exercise
An observer stands such that the horizontal distance to a lighthouse is 60m and the slant (line-of-sight) distance to the top is 80m. The angle of elevation to the top of the lighthouse is θ.
Find θ to the nearest degree.
Select the correct option
(exam style problem)
Triangle geometry can also be applied to find angle bearings.
True bearings
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.
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Example
A ship leaves port P at bearing of 075° and sails 4km to a lighthouse L. It then turns 55° away from north, and sails 3km to a dolphin watching spot D. Find
The bearing the ship must take to get back to L.
The distance P from the port.
The bearing the ship must take to return to port.
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The ship starts at a 075° bearing, and then turns 55° away from north, giving 130°. If the ship were to make a U-turn, it would have a bearing of 130°+180°=310° coming back to L.
Since ∠NPL=75°, ∠NLP=180°−75°=105°. Thus the angle ∠PLN=360°−130°−105°=125°. Using the cosine rule:
Now using the sine rule:
So ∠PDL=sin−1(0.526)=31.8°. Finally, we recall that the bearing from D→L was 310°, so the bearing from D→P is 310°−∠PDL=278°.
Exercise
A ship leaves port P and sails at a bearing of 44°N until it reaches an island I. It then turns and sails at a bearing of 133°N until it reaches a lighthouse L.
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Find the angle ∠PIL.
Find the bearing the ship would take to sail from L back to I.
Select the correct option
!
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