Prove that the length of any parallel of latitude is equal to the length of the equator times the cosine of the latitude angle.
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Assume as an approximation that the earth is a perfect sphere.
Call the radius of the sphere r.
Using the formula for the circumference of a circle, the length of the equator is `2pir`
Now, draw a triangle with a point at the centre of the earth A, a point on a particular latitude line B and a point horizontally across from that directly above the centre of the earth C. This is a right-angled triangle.
b
B_ _ _ C
\`theta` ) |
r \ |
\ |
_ _( `theta` \| A (centre of earth)
Call the length of the line BC 'b'. This is the radius of the circle round the earth at ``the latitude of B. The length of the line AB is the radius of the earth, r.
Call the latitude angle (the angle between the horizontal and the line AB) `theta` .
Using the z-angle rule of triangles, the angle between the lines AB and BC is also equal to `theta`
Now, `cos (theta) = b/r`
So the length of the equator `2pir` x `cos(theta)` is equal to
```2pir (b/r)= 2 pi b`
which is the length of the circumference of the earth at latitude `theta`
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