Spherical Trigonometry

SPHERICAL TRIANGLE

B

AC

O

b

a

O c

B

A

C

A spherical triangle is that part of the surface of a sphere bounded by three arcs of great circles. The bounding arcs are called the sides of the spherical triangle and the intersection of these arcs are called the vertices. The angle formed by two intersecting arcs is called a spherical angle.

DEFINITION OF TERMS:• Spherical Trigonometry – is a branch of trigonometry that concerns with

triangles extracted from the surface of the sphere.• Great Circle – is a circle obtained by passing a section through the center of

the sphere.• Spherical Triangle – is a spherical surface bounded by the area of three

great circles. The magnitude of the side of a spherical triangle is the angle subtended by it at the Centre of the sphere and is expressed in degrees and minutes of arc. The maximum value of an angle of a spherical triangle is 1800. The sum of the three sides of a spherical triangle is less than 3600.

• Right Spherical Triangle – is a spherical triangle having a right angle.• Quadrantal - spherical triangle is one in which one side equals to 900.• Pole – this axis intersects the sphere at two points, of the great circle.• Axis of the circle – the through the center of the sphere perpendicular to

the plane of the circle.• Polar triangle – the vertices of a spherical triangle ABC as poles, construct

three great circles.

RIGHT SPHERICAL TRIANGLE

Rule 1: The sine of any middle part is equal to the product of the tangents of the adjacent parts.

Rule 2: The sine of any middle part is equal to the product of the cosines of the opposite parts.

NAPIER’S RULE

Illustrate:

RIGHT SPHERICAL TRIANGLE

a

bco - A

co - cco - B

Where: co – A = complement of A co – B = complement of B co – c = complement of c

Napier’s Circle

Solution of oblique spherical triangle involves six cases, namely:

Case 1: Two sides and included angle are given. ( SAS )

Case 2: Two angles and the included side. (ASA)

Case 3: Two sides and an angle opposite one of them. (SSA )

Case 4: Two angles and a side opposite one of them. (AAS)

Case 5: Three sides are given. ( SSS)

Case 6: Three angles are given. ( AAA)

Application:Problem: Solve for the spherical triangle whose parts are a = 73°, b = 62°, and C = 90°.

Find the area of a spherical triangle of whose angles are 123°, 84°, and 73°. The radius of the sphere is 30 m.