Trigonometry
When one of the angles, say C, of a spherical triangle is equal to π/2 the various identities given above are considerably simplified. There are ten identities relating three elements chosen from the set a, b, c, A, B.
Napier provided an elegant mnemonic aid for the ten independent equations: the mnemonic is called Napier's circle or Napier's pentagon (when the circle in the above figure, right, is replaced by a pentagon).
First, write the six parts of the triangle (three vertex angles, three arc angles for the sides) in the order they occur around any circuit of the triangle: for the triangle shown above left, going clockwise starting with a gives aCbAcB. Next replace the parts that are not adjacent to C (that is A, c, B) by their complements and then delete the angle C from the list. The remaining parts can then be drawn as five ordered, equal slices of a pentagram, or circle, as shown in the above figure (right). For any choice of three contiguous parts, one (the middle part) will be adjacent to two parts and opposite the other two parts. The ten Napier's Rules are given by
sine of the middle part = the product of the tangents of the adjacent parts
sine of the middle part = the product of the cosines of the opposite parts
The key for remembering which trigonometric function goes with which part is to look at the first vowel of the kind of part: middle parts take the sine, adjacent parts take the tangent, and opposite parts take the cosine. For an example, starting with the sector containing
a
{\displaystyle a}
we have:
sin
a
=
tan
(
π
/
2
−
B
)
tan
b
=
cos
(
π
/
2
−
c
)
cos
(
π
/
2
−
A
)
=
cot
B
tan
b
=
sin
c
sin
A
.
{\displaystyle \sin a=\tan(\pi /2-B)\,\tan b=\cos(\pi /2-c)\,\cos(\pi /2-A)=\cot B\,\tan b=\sin c\,\sin A.}
The full set of rules for the right spherical triangle is (Todhunter, Art.62)
(R1)
cos
c
=
cos
a
cos
b
,
(R6)
tan
b
=
cos
A
tan
c
,
(R2)
sin
a
=
sin
A
sin
c
,
(R7)
tan
a
=
cos
B
tan
c
,
(R3)
sin
b
=
sin
B
sin
c
,
(R8)
cos
A
=
sin
B
cos
a
,
(R4)
tan
a
=
tan
A
sin
b
,
(R9)
cos
B
=
sin
A
cos
b
,
(R5)
tan
b
=
tan
B
sin
a
,
(R10)
cos
c
=
cot
A
cot
B
.
{\displaystyle {\begin{alignedat}{4}&{\text{(R1)}}&\qquad \cos c&=\cos a\,\cos b,&\qquad \qquad &{\text{(R6)}}&\qquad \tan b&=\cos A\,\tan c,\\&{\text{(R2)}}&\sin a&=\sin A\,\sin c,&&{\text{(R7)}}&\tan a&=\cos B\,\tan c,\\&{\text{(R3)}}&\sin b&=\sin B\,\sin c,&&{\text{(R8)}}&\cos A&=\sin B\,\cos a,\\&{\text{(R4)}}&\tan a&=\tan A\,\sin b,&&{\text{(R9)}}&\cos B&=\sin A\,\cos b,\\&{\text{(R5)}}&\tan b&=\tan B\,\sin a,&&{\text{(R10)}}&\cos c&=\cot A\,\cot B.\end{alignedat}}}
A quadrantal spherical triangle is defined to be a spherical triangle in which one of the sides subtends an angle of π/2 radians at the centre of the sphere: on the unit sphere the side has length π/2. In the case that the side c has length π/2 on the unit sphere the equations governing the remaining sides and angles may be obtained by applying the rules for the right spherical triangle of the previous section to the polar triangle A'B'C' with sides a',b',c' such that A' = π − a, a' = π − A etc. The results are:
(Q1)
cos
C
=
−
cos
A
cos
B
,
(Q6)
tan
B
=
−
cos
a
tan
C
,
(Q2)
sin
A
=
sin
a
sin
C
,
(Q7)
tan
A
=
−
cos
b
tan
C
,
(Q3)
sin
B
=
sin
b
sin
C
,
(Q8)
cos
a
=
sin
b
cos
A
,
(Q4)
tan
A
=
tan
a
sin
B
,
(Q9)
cos
b
=
sin
a
cos
B
,
(Q5)
tan
B
=
tan
b
sin
A
,
(Q10)
cos
C
=
−
cot
a
cot
b
.
{\displaystyle {\begin{alignedat}{4}&{\text{(Q1)}}&\qquad \cos C&=-\cos A\,\cos B,&\qquad \qquad &{\text{(Q6)}}&\qquad \tan B&=-\cos a\,\tan C,\\&{\text{(Q2)}}&\sin A&=\sin a\,\sin C,&&{\text{(Q7)}}&\tan A&=-\cos b\,\tan C,\\&{\text{(Q3)}}&\sin B&=\sin b\,\sin C,&&{\text{(Q8)}}&\cos a&=\sin b\,\cos A,\\&{\text{(Q4)}}&\tan A&=\tan a\,\sin B,&&{\text{(Q9)}}&\cos b&=\sin a\,\cos B,\\&{\text{(Q5)}}&\tan B&=\tan b\,\sin A,&&{\text{(Q10)}}&\cos C&=-\cot a\,\cot b.\end{alignedat}}}
Logarithm
Given a positive real number b such that b ≠ 1, the logarithm of a positive real number x with respect to base b is the exponent by which b must be raised to yield x. In other words, the logarithm of x to base b is the unique real number y such that by = x.
The logarithm is denoted "logb x" (pronounced as "the logarithm of x to base b", "the base-b logarithm of x", or most commonly "the log, base b, of x").
An equivalent and more succinct definition is that the function
f
:
x
→
l
o
g
b
x
{\displaystyle f\colon x\to log_{b}x}
is the inverse function to the function
f
:
x
→
b
x
.
{\displaystyle f\colon x\to b^{x}.}
