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I found from a book of mine the formula $\cos a=\cos b\cos c+\sin b\sin c\cos\alpha.$ Can this be true? If for example $a=1m,b=1m,c=1m,\alpha=1$, $m$ denotes by meter, then $\cos m=\cos^2 m+\sin^2m\cos 1.$ There seems to be some mistake in units if cosine meters are sum of cosine square meters and sine square meters.

Curious
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  • Are you sure that meters and square meters are present in trigonometry ? – Claude Leibovici Oct 30 '13 at 11:15
  • Meters and square meters are not part of the trigonometry but I'm just confused by the formula.

    I'm just reading a book "Mathematics Handbook for Science and Engineering". Here authors wrote: $a,b,c$=sides, $\alpha$=angle, $\cos a=\cos b \cos c + \sin b \sin c \cos \alpha.$ But as this book has only the formula, I'm not sure how to use it as there is no proof for the formula.

     – Curious Oct 30 '13 at 11:21

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Aloha Curious, This formula is the law of cosines for spherical triangles. If you stick three dots on a sphere, and connect the three dots with the shortest segments you can, you get a spherical triangle. The interior angles $\alpha$, $\beta$, and $\gamma$ are similar to the angles in a flat triangle; they're just the angles between the sides. The sides to a spherical triangle can be represented by distances, since they are arcs of a great circle on the Earth, so they can have units of meters, nautical miles, or any other distance unit. Since they are arcs, you could also represent them as angles. $a$ is probably the "side" between points labeled $B$ and $C$ in the triangle. If you imagine a line from Earth's center all the way to point $B$, and another line from Earth's center all the way to point $C$, then $a$ is the angle between those two lines where they touch at the center of the Earth.

So $\alpha$, $\beta$, and $\gamma$ are angles on the surface of the Earth, between the sides of the spherical triangle. And $a$, $b$, and $c$ are angles inside the Earth, or arc lengths on the surface of the Earth, the sides of the spherical triangle.

If you want to use this formula you posted, make sure you put $a$, $b$, and $c$ in either degrees or radians though. You can get the distance by multiplying $a$ in radians times the radius of the Earth. An even faster way is to multiply $a$ in degrees times 60 nautical miles per degree of a great circle arc. This is very close to how a nautical mile was originally defined.

Atomj
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