# 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.