NAPIER’S
RULE
INTRODUCTION
John Napier (1550–1617)
discovered a way to reduce 10
equations in spherical trig
down to 2 equations and to
make them easier to remember.
Draw a right triangle on a
sphere and label the
sides a, b, and c where c is
the hypotenuse. Let A be the
angle opposite side a, B the
angle opposite side b, and C
the right angle opposite the
hypotenuse c.
2
There are 10 equations relating the sides
and angles of the triangle:
sin a = sin A sin c = tan b cot B
sin b = sin B sin c = tan a cot A
cos A = cos a sin B = tan b cot c
cos B = cos b sin A = tan a cot c
cos c = cot A cot B = cos a cos b
Here’s how Napier reduced these equations
to a more memorable form. Arrange the parts
of the triangle in a circle as below.
3
Then Napier has two rules:
1. The sine of a middle part is equal to the
product of the cosines of the two opposite parts.
Sin-Co-Op
2. The sine of a middle part is equal to the
product of the tangents of the two adjacent parts.
Sin-Tan-Ad
4
1. Solve for the spherical triangle whose
parts are
a= 73°, b= 62°, and C= 90°.
5
2. Solve for the spherical triangle whose
parts are
B= 130°, a= 114°, and C= 90°.
6
3. Solve for the spherical triangle whose
parts are
a= 70°, b=52°, and c= 90°.
A
B
ac
bc
Cc
7
THE END