Spherical Triangle

Spherical triangle ABC is on the surface of a sphere as shown in the figures.

Sides a, b, c (which are arcs of great circles) are measured by their angles subtended at center O of the sphere.
A, B, C are the angles opposite sides a, b, c respectively.

Area of the spherical triangle $ABC = (A + B + C - \pi)R^2$
where R is the radius of the sphere.

Relationships Between Sides and Angles of a Spherical Triangle

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

Law of Cosines
cos a = cos b ⋅ cos c + sin b ⋅ sin c ⋅ cos A
cos A = - cos B ⋅ cos C + sin B ⋅ sin C ⋅ cos a
with similar results involving other sides and angles.

Law of Tangents
$\frac{\tan(\frac{A + B}{2})}{\tan(\frac{A - B}{2})}=\frac{\tan(\frac{a + b}{2})}{\tan(\frac{a - b}{2})}$

with similar results involving other sides and angles.
$\cos\frac{A}{2}=\sqrt{\frac{\sin s \ \sin(s - c)}{\sin b \ \sin c}}$

where $s = \frac{a + b + c}{2}$.

Similar results hold for other sides and angles.
$\cos\frac{a}{2}=\sqrt{\frac{\cos(S - B) \cos(S - C)}{\sin B \ \sin C}}$

where $S = \frac{A + B + C}{2}$.
Similar results hold for other sides and angles.

Napier's Rules for Right Angled Spherical Triangles

Except for right angle C, there are five parts of spherical triangle ABC if arranged in other as given in Fig.5-19 would be a, b, A, c, B.

Suppose these quantities are arranged in a circle as in Fig. 5 - 20 where we attach the prefix co (indicating complement) to hypotenuse c and angles A and B.

Any one of the parts of this circle is called a middle part, the two neighbouring parts are called adjacent parts and the two remaining parts are called opposite parts.

The Napier's rules are

The sine of any middle part equals the product of the tangents of the adjacent parts.

The sine of any middle part equals the product of the cosines of the opposite parts.

Example:
Since co-A = 90° - A, co-B = 90° - B, we have
sin a = tan b ⋅ tan(co-B) or sin a = tan b ⋅ cot B
sin(co-A) = cos a ⋅ cos(co-B) or cos A = cos a ⋅ sin B.


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