Proofs for the Spherical Laws of sines and Cosines

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I am looking for "classical" proofs for the spherical laws of sines and cosines. A proof that relies only on knowledge that was common to the ancient greek geometers, not containing analytical geometry and vectors and such.
Thanks!

geometry trigonometry euclidean-geometry spherical-geometry spherical-trigonometry

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edited Jul 23 at 7:45

asked Jul 23 at 7:28

Gouni Jacobi

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up vote
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I am looking for "classical" proofs for the spherical laws of sines and cosines. A proof that relies only on knowledge that was common to the ancient greek geometers, not containing analytical geometry and vectors and such.
Thanks!

geometry trigonometry euclidean-geometry spherical-geometry spherical-trigonometry

share|cite|improve this question

edited Jul 23 at 7:45

asked Jul 23 at 7:28

Gouni Jacobi

63

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

I am looking for "classical" proofs for the spherical laws of sines and cosines. A proof that relies only on knowledge that was common to the ancient greek geometers, not containing analytical geometry and vectors and such.
Thanks!

geometry trigonometry euclidean-geometry spherical-geometry spherical-trigonometry

share|cite|improve this question

edited Jul 23 at 7:45

asked Jul 23 at 7:28

Gouni Jacobi

63

I am looking for "classical" proofs for the spherical laws of sines and cosines. A proof that relies only on knowledge that was common to the ancient greek geometers, not containing analytical geometry and vectors and such.
Thanks!

geometry trigonometry euclidean-geometry spherical-geometry spherical-trigonometry

share|cite|improve this question

edited Jul 23 at 7:45

asked Jul 23 at 7:28

Gouni Jacobi

63

share|cite|improve this question

share|cite|improve this question

edited Jul 23 at 7:45

edited Jul 23 at 7:45

edited Jul 23 at 7:45

asked Jul 23 at 7:28

Gouni Jacobi

63

asked Jul 23 at 7:28

Gouni Jacobi

63

asked Jul 23 at 7:28

Gouni Jacobi

63

63

add a comment | 

add a comment | 

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Let $A$, $B$, $C$ the vertices of a spherical triangle on a sphere of unit radius and center $O$ (see diagram below). Spherical angle $angle ACB=gamma$ is by definition angle $angle ECF$ formed by tangent lines at $C$.

In right triangles $OCE$ and $OCF$ we have then $EC=tan b$ and $FC=tan a$, so by the cosine rule in triangle $CEF$:

$$
EF^2=tan^2 a+tan^2 b-2tan atan bcosgamma.
$$
On the other hand we also have $OE=sec b$ and $OF=sec a$, so by the cosine rule in triangle $OEF$:
$$
EF^2=sec^2 a+sec^2 b-2sec asec bcos c.
$$
Equating the right hand sides of both formulas leads, after some simplifications, to:
$$
cos c=cos acos b+sin a sin bcosgamma,
$$
which is the spherical cosine rule.

This proof works as long as $a$ and $b$ are acute angles, but I think it can also be extended, with some modifications, to the other cases.

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edited Jul 23 at 14:19

answered Jul 23 at 14:11

Aretino

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Let $A$, $B$, $C$ the vertices of a spherical triangle on a sphere of unit radius and center $O$ (see diagram below). Spherical angle $angle ACB=gamma$ is by definition angle $angle ECF$ formed by tangent lines at $C$.

In right triangles $OCE$ and $OCF$ we have then $EC=tan b$ and $FC=tan a$, so by the cosine rule in triangle $CEF$:

$$
EF^2=tan^2 a+tan^2 b-2tan atan bcosgamma.
$$
On the other hand we also have $OE=sec b$ and $OF=sec a$, so by the cosine rule in triangle $OEF$:
$$
EF^2=sec^2 a+sec^2 b-2sec asec bcos c.
$$
Equating the right hand sides of both formulas leads, after some simplifications, to:
$$
cos c=cos acos b+sin a sin bcosgamma,
$$
which is the spherical cosine rule.

This proof works as long as $a$ and $b$ are acute angles, but I think it can also be extended, with some modifications, to the other cases.

share|cite|improve this answer

edited Jul 23 at 14:19

answered Jul 23 at 14:11

Aretino

21.7k21342

add a comment | 

up vote
0
down vote

Let $A$, $B$, $C$ the vertices of a spherical triangle on a sphere of unit radius and center $O$ (see diagram below). Spherical angle $angle ACB=gamma$ is by definition angle $angle ECF$ formed by tangent lines at $C$.

In right triangles $OCE$ and $OCF$ we have then $EC=tan b$ and $FC=tan a$, so by the cosine rule in triangle $CEF$:

$$
EF^2=tan^2 a+tan^2 b-2tan atan bcosgamma.
$$
On the other hand we also have $OE=sec b$ and $OF=sec a$, so by the cosine rule in triangle $OEF$:
$$
EF^2=sec^2 a+sec^2 b-2sec asec bcos c.
$$
Equating the right hand sides of both formulas leads, after some simplifications, to:
$$
cos c=cos acos b+sin a sin bcosgamma,
$$
which is the spherical cosine rule.

This proof works as long as $a$ and $b$ are acute angles, but I think it can also be extended, with some modifications, to the other cases.

share|cite|improve this answer

edited Jul 23 at 14:19

answered Jul 23 at 14:11

Aretino

21.7k21342

add a comment | 

up vote
0
down vote

up vote
0
down vote

Let $A$, $B$, $C$ the vertices of a spherical triangle on a sphere of unit radius and center $O$ (see diagram below). Spherical angle $angle ACB=gamma$ is by definition angle $angle ECF$ formed by tangent lines at $C$.

In right triangles $OCE$ and $OCF$ we have then $EC=tan b$ and $FC=tan a$, so by the cosine rule in triangle $CEF$:

$$
EF^2=tan^2 a+tan^2 b-2tan atan bcosgamma.
$$
On the other hand we also have $OE=sec b$ and $OF=sec a$, so by the cosine rule in triangle $OEF$:
$$
EF^2=sec^2 a+sec^2 b-2sec asec bcos c.
$$
Equating the right hand sides of both formulas leads, after some simplifications, to:
$$
cos c=cos acos b+sin a sin bcosgamma,
$$
which is the spherical cosine rule.

This proof works as long as $a$ and $b$ are acute angles, but I think it can also be extended, with some modifications, to the other cases.

share|cite|improve this answer

edited Jul 23 at 14:19

answered Jul 23 at 14:11

Aretino

21.7k21342

Let $A$, $B$, $C$ the vertices of a spherical triangle on a sphere of unit radius and center $O$ (see diagram below). Spherical angle $angle ACB=gamma$ is by definition angle $angle ECF$ formed by tangent lines at $C$.

In right triangles $OCE$ and $OCF$ we have then $EC=tan b$ and $FC=tan a$, so by the cosine rule in triangle $CEF$:

$$
EF^2=tan^2 a+tan^2 b-2tan atan bcosgamma.
$$
On the other hand we also have $OE=sec b$ and $OF=sec a$, so by the cosine rule in triangle $OEF$:
$$
EF^2=sec^2 a+sec^2 b-2sec asec bcos c.
$$
Equating the right hand sides of both formulas leads, after some simplifications, to:
$$
cos c=cos acos b+sin a sin bcosgamma,
$$
which is the spherical cosine rule.

This proof works as long as $a$ and $b$ are acute angles, but I think it can also be extended, with some modifications, to the other cases.

share|cite|improve this answer

edited Jul 23 at 14:19

answered Jul 23 at 14:11

Aretino

21.7k21342

share|cite|improve this answer

share|cite|improve this answer

edited Jul 23 at 14:19

edited Jul 23 at 14:19

edited Jul 23 at 14:19

answered Jul 23 at 14:11

Aretino

21.7k21342

answered Jul 23 at 14:11

Aretino

21.7k21342

answered Jul 23 at 14:11

Aretino

21.7k21342

21.7k21342

add a comment | 

add a comment | 

 

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