Applications of trigonometry
All the trigonometry of oriented angles that you have seen has a very concrete application: the Earth is (almost) a sphere, and its shape is described with two angles, latitude and longitude. We are going to use the cosine to work out how big a parallel really is —one of those horizontal lines you see on maps— and the distance between two cities that lie on the same one.
The radius of a parallel: why the cosine appears
A parallel is a circle made up of all the points on the Earth that are at the same latitude (the angle measured from the plane of the equator). The equator, at latitude $0º$, has the same radius as the whole Earth; any other parallel is smaller, because it "narrows" as it gets closer to the poles.
Look at the right-angled triangle formed by the centre of the Earth, the centre of the parallel and a point on the parallel: the hypotenuse is $R$ (the Earth's radius), and the side adjacent to the latitude angle $\alpha$ is, precisely, $r$ (the radius of the parallel).
For a school project, you need to calculate the length of the parallel of latitude $40^\circ$ N. Knowing that the Earth's radius is $R \approx 6\,371$ km, calculate: a)…
For a school project, you need to calculate the length of the parallel of latitude $40^\circ$ N. Knowing that the Earth's radius is $R \approx 6\,371$ km, calculate:
- a) The radius of the circle formed by this parallel (use the expression $r = R \cdot \cos(\text{latitud})$ and justify it from the right-angled triangle in the figure).
- b) The total length (circumference) of this parallel.
The length of a parallel
Once the radius $r$ of a parallel is known, calculating its length (its perimeter) uses the same formula as always for a circle: $2\pi r$. The difference from the equator is that here the radius is no longer $R$, but the $r$ reduced by the cosine of the latitude.
Distance between two points on the same parallel
If two cities are on the same parallel but at different longitudes (the east–west angle), the distance between them along the parallel is an arc of that smaller circle, not of the whole globe. You first need the radius of the parallel, and then the usual arc length formula with the angle that separates them.
Two cities, A and B, are on the same parallel at a latitude of $45^\circ \text{ N}$. City A has a longitude of $20^\circ \text{ E}$ and city B a longitude of…
Two cities, A and B, are on the same parallel at a latitude of $45^\circ \text{ N}$. City A has a longitude of $20^\circ \text{ E}$ and city B a longitude of $70^\circ \text{ E}$.
Given that the radius of the Earth is approximately $6371 \text{ km}$:
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Calculate the radius of the $45^\circ \text{ N}$ parallel (Remember that the radius of a parallel at a latitude $\alpha$ is calculated as $r = R \cdot \cos \alpha$; for $45^\circ$ use the exact value $\cos 45^\circ = \frac{\sqrt{2}}{2}$).
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Determine the distance in kilometres separating both cities along the arc of the parallel.