Spherical trigonometry

The spherical trigonometry is the branch of spherical geometry which deals with spherical triangles defined by great circles on the sphere.

It allows us to calculate the trigonometric functions of the sides and angles of these spherical polygons.

A great circle on the sphere is any circle having its center as the center of the sphere.

The spherical polygon is a polygon on the surface of the sphere defined by great-circle arcs.

The angles of proper spherical triangle are less than $\pi$

This means $\pi<A+B+C<3\pi$

The same for the sides:

The sides of proper spherical triangle are less than $\pi$

This means $\pi<a+ b+ c<3\pi$

Spherical law of cosines

Cosine rule for sides:

$\cos c=\cos a\cos b+ \sin a \sin b \cos C$

When $C=\frac{\pi}{2}$

We get:

$\cos C=0$

The formula becomes

$\cos c=\cos a\cos b$

Cosine rule for angles:

$\cos C=-\cos A\cos B+ \sin A \sin B \cos c$

Both of these formulae can be rearranged to get $\cos C$ or $\cos c$

Spherical Law of Sine:

$\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}$

Half-side Formula:

In a spherical triangle:

$\tan {(\frac{a}{2})}=R\cos{(S-A)}$

$\tan {(\frac{b}{2})}=R\cos{(S-B)}$

$\tan {(\frac{c}{2})}=R\cos{(S-C)}$

$S$ is half the sum of the angles:

$S=\frac{1}{2}(A+B+C)$

Also:

$R=\sqrt{\frac{-\cos S}{\cos(S-A)\cos(S-B)\cos(S-C)}}$

Cotangents formula:(4 consecutive elements)

If we write:

$\cos C=\frac{\cos c-\cos a \cos b}{\sin a \sin b}$

Dividing by $\sin B$ and using sine rule we get:

$\sin c \cot b=\sin A \cot B +\cos A \cos c$

Formula involving half-angles and half-sides:

If we let:

$s=\frac{1}{2}(a+b+c)$ the semi-perimeter of the triangle.

$\tan^{2}{\frac{C}{2}}=\frac{\sin(s-a) \sin (s-b)}{\sin s \sin(s-c)}$

If $\alpha=\frac{1}{2}(A+B+C)$

$\tan^{2}{\frac{c}{2}}=-\frac{\cos(\alpha) \cos (\alpha-C)}{\cos {(\alpha-A} \cos{(\alpha-B)}}$

Gauss Formulas:

$\displaystyle{\frac{\cos {\frac{a+b}{2}}}{\cos {\frac{c}{2}}}}=\frac{\cos {\frac{A+B}{2}}}{\cos {\frac{C}{2}}}$

AND

$\displaystyle{\frac{\sin {\frac{a+b}{2}}}{\sin {\frac{c}{2}}}}=\frac{\cos {\frac{A-B}{2}}}{\sin {\frac{C}{2}}}$

OR:

$\displaystyle{\frac{\cos {\frac{a-b}{2}}}{\cos {\frac{c}{2}}}}=\frac{\sin {\frac{A+B}{2}}}{\cos {\frac{C}{2}}}$

AND

$\displaystyle{\frac{\sin {\frac{a-b}{2}}}{\sin {\frac{c}{2}}}}=\frac{\sin {\frac{A-B}{2}}}{\cos {\frac{C}{2}}}$

Leading to:

$\displaystyle{\frac{\tan {\frac{a-b}{2}}}{\tan {\frac{a+b}{2}}}}=\frac{\tan {\frac{A-B}{2}}}{\tan {\frac{A+B}{2}}}$

Napier Formulas:

Obtained from Gauss’s Formulas:
$\tan {\frac{c}{2}}\cos {\frac{A-B}{2}}=\tan {\frac{a+b}{2}}\cos {\frac{A+B}{2}}$
$\tan {\frac{c}{2}}\sin {\frac{A-B}{2}}=\tan {\frac{a-b}{2}}\sin {\frac{A+B}{2}}$

$\cot {\frac{C}{2}}\cos {\frac{a-b}{2}}=\tan {\frac{A+B}{2}}\cos {\frac{a+b}{2}}$
$\cot {\frac{C}{2}}\sin {\frac{a-b}{2}}=\tan {\frac{A-B}{2}}\sin {\frac{a+b}{2}}$

Spherical triangle area:

This is called the GIRARD formula. Using the difference of the sum of the angles of the triangle and $\pi$
If $R$ is the Radius of the sphere, the Area in STERADIANS (From angles in RADIANS) is:
$S=(\hat{A}+\hat{B}+\hat{C}-\pi)R^{2}=R^{2}\epsilon$

Solving Spherical Triangles:

It should be known that the SPHERICAL EXCESS $E=S/R^{2}$ where $S$ is the Area of the triangle and $R$ is the radius of the Sphere.

Given the 3 sides $a$ , $b$ and $c$ :

For angle $A$
$\displaystyle{A}=\arccos(\frac{\cos {a}-\cos {b}\cos {c}}{\sin {b}\sin {c}})$

For angle $B$
$\displaystyle{B}=\arccos(\frac{\cos {b}-\cos {a}\cos {c}}{\sin {a}\sin {c}})$

For angle $C$
$\displaystyle{C}=\arccos \left(\frac{\cos {c}-\cos {a}\cos {b}}{\sin {a}\sin {b}}\right)$
$\displaystyle{E}=4\arctan {\sqrt{\tan {\frac{p}{2}} \tan {\frac{p-a}{2}} \tan {\frac{p-b}{2}} \tan {\frac{p-c}{2}}} }$
With:
$p=\frac{1}{2}(a+b+c)$

Given one angle with 2 adjacent sides:

Example, Given $a$ , $b$ and $C$
The side $c$ :
$c=\arccos(\cos {a} \cos {b}+\sin {a}\sin {b}\cos {C})$

For Angle $A$ we use Napier’s Analogies:
$\displaystyle{A}=\arctan {\left(\frac{2\sin {a}}{\tan {\frac{C}{2}} \sin {(b+a)}+\cot {\frac{C}{2}} \sin {(b-a)}}\right)}$

For Angle $B$ we use Napier’s Analogies:
$\displaystyle{B}=\arctan {\left(\frac{2\sin {b}}{\tan {\frac{C}{2}} \sin {(a+b)}+\cot {\frac{C}{2}} \sin {(a-b)}}\right)}$

Same goes for the Excess $E$ :
$\displaystyle{E}=C+2\arctan {\left(\cot {(\frac{C}{2})\frac{\cos {\frac{1}{2}(a-b)}}{\cos {\frac{1}{2}(a+b)}}}\right)}-\pi$