Answer to Question #146201 in Trigonometry for Raffy

Question

Establish two formulas using Napier’s rules I and II if Angle B is taken as the middle part.

Answer 1

It is necessary to establish two formulas using Napier’s rules I and II if Angle B is taken as the middle part.

To begin, you must understand that Napier formula used in spherical trigonometry. All formulas can be derived from the cosine rule in spherical trigonometry. Cosine rule is a fundamental identity of spherical trigonometry.

$cosa=cos b * cos c + sinb* sinc*cosA$

Consider Napier's first rule when the angle is 90 degrees or $Pi/2$ . Draw a spherical triangle with a right angle at the vertex C. (Left in the picture)

To find the formula for finding the midline, you can use the "Napier pentagon". (On the right in the picture)

For any selection of three adjacent portions, one (middle portion) will be adjacent to two portions and opposite to the other two portions. Thus, we find the angle of the arc b: based on the figure and Napier's rule:

sine of the middle part = the product of the tangent adjacent parts.

sine of the middle part = the product of the cosines of the opposite parts.

$sin(b) = tan(\Pi/2-A)tan(a)=cot(B)tan(b)$

$sin(b)=cos(\Pi/2-c)cos(\Pi/2-B)=sin(c)sin(B)$

Well we find the middle part B. Napier’s rules I

$cos(B)=sin(A)cos(b)$

$sin(B)= sin(b)/sin(c)$

Let us now consider the second rule Napier, about quadrant spherical triangles.

A quadrant spherical triangle is defined as a spherical triangle in which one of the sides makes an angle of $\Pi/2$ radians at the center of the sphere: on a unit sphere, a side is $\Pi/2$ long.

And Middle part B for Napier’s rules II:

$sin(B) = sin(b)sin(C)$

$cos(B) = cos(b)/sin(a)$