The following formulas are much used in the course of this work. They are to be found in most treatises on Trigonometry, and may be demonstrated by the method given in the appendix to this volume.
R
a
d
i
u
s
=
{\displaystyle Radius=}
1.
[1] Int.
sin
.
2
z
=
−
1
2
⋅
{
cos
.2
z
−
1
}
{\displaystyle \sin .^{2}z=-{\frac {1}{2}}\cdot \lbrace \cos .2z-1\rbrace }
[2] Int. „
sin
.
3
z
=
−
1
2
2
⋅
{
sin
.3
z
−
3
⋅
sin
.
z
}
{\displaystyle \sin .^{3}z=-{\frac {1}{2^{2}}}\cdot \lbrace \sin .3z-3\cdot \sin .z\rbrace }
[3] Int. „
sin
.
4
z
=
1
2
3
⋅
{
cos
.4
z
−
4
⋅
cos
.2
z
+
1
2
⋅
4
⋅
3
1
⋅
2
}
{\displaystyle \sin .^{4}z={\frac {1}{2^{3}}}\cdot \lbrace \cos .4z-4\cdot \cos .2z+{\frac {1}{2}}\cdot {\frac {4\cdot 3}{1\cdot 2}}\rbrace }
[4] Int. „
sin
.
5
z
=
1
2
4
⋅
{
sin
.5
z
−
5
⋅
sin
.3
z
+
5
⋅
4
1
⋅
2
⋅
sin
.
z
}
{\displaystyle \sin .^{5}z={\frac {1}{2^{4}}}\cdot \lbrace \sin .5z-5\cdot \sin .3z+{\frac {5\cdot 4}{1\cdot 2}}\cdot \sin .z\rbrace }
[5] Int. „
sin
.
6
z
=
−
1
2
5
⋅
{
cos
.6
z
−
6
⋅
cos
.4
z
+
6
⋅
5
1
⋅
2
⋅
cos
.2
z
−
1
2
⋅
6
⋅
5
⋅
4
1
⋅
2
⋅
3
}
{\displaystyle \sin .^{6}z=-{\frac {1}{2^{5}}}\cdot \lbrace \cos .6z-6\cdot \cos .4z+{\frac {6\cdot 5}{1\cdot 2}}\cdot \cos .2z-{\frac {1}{2}}\cdot {\frac {6\cdot 5\cdot 4}{1\cdot 2\cdot 3}}\rbrace }
[6] Int. „
c
o
s
.
2
z
=
1
2
⋅
{
cos
.2
z
+
1
}
{\displaystyle cos.^{2}z={\frac {1}{2}}\cdot \lbrace \cos .2z+1\rbrace }
[7] Int. „
c
o
s
.
3
z
=
1
2
2
⋅
{
cos
.3
z
+
3
⋅
cos
.
z
}
{\displaystyle cos.^{3}z={\frac {1}{2^{2}}}\cdot \lbrace \cos .3z+3\cdot \cos .z\rbrace }
[8] Int. „
c
o
s
.
4
z
=
1
2
3
⋅
{
cos
4
z
+
4
⋅
cos
.2
z
+
1
2
⋅
4
⋅
3
1
⋅
2
}
{\displaystyle cos.^{4}z={\frac {1}{2^{3}}}\cdot \lbrace \cos 4z+4\cdot \cos .2z+{\frac {1}{2}}\cdot {\frac {4\cdot 3}{1\cdot 2}}\rbrace }
[9] Int. „
c
o
s
.
5
z
=
1
2
4
⋅
{
cos
.5
z
+
5
⋅
cos
.3
z
+
5
⋅
4
1
⋅
2
⋅
cos
.
z
}
{\displaystyle cos.^{5}z={\frac {1}{2^{4}}}\cdot \lbrace \cos .5z+5\cdot \cos .3z+{\frac {5\cdot 4}{1\cdot 2}}\cdot \cos .z\rbrace }
[10] Int. „
c
o
s
.
6
z
=
1
2
5
⋅
{
cos
.6
z
+
6
⋅
cos
.4
z
+
6
⋅
5
1
⋅
2
⋅
cos
.2
z
+
1
2
⋅
6
⋅
5
⋅
4
1
⋅
2
⋅
3
}
{\displaystyle cos.^{6}z={\frac {1}{2^{5}}}\cdot \lbrace \cos .6z+6\cdot \cos .4z+{\frac {6\cdot 5}{1\cdot 2}}\cdot \cos .2z+{\frac {1}{2}}\cdot {\frac {6\cdot 5\cdot 4}{1\cdot 2\cdot 3}}\rbrace }
H
y
p
.
l
o
g
.
c
=
{\displaystyle Hyp.log.c=}
1
[11] Int. „
s
i
n
.
z
=
c
z
⋅
−
1
−
c
−
z
⋅
−
1
2
⋅
−
1
{\displaystyle sin.z={\frac {c^{z\cdot {\sqrt {-}}1}-c^{-z\cdot {\sqrt {-}}1}}{2\cdot {\sqrt {-}}1}}}
[12] Int. „
c
o
s
.
z
=
c
z
⋅
−
1
+
c
−
z
⋅
−
1
2
{\displaystyle cos.z={\frac {c^{z\cdot {\sqrt {-}}1}+c^{-z\cdot {\sqrt {-}}1}}{2}}}
[13] Int. „
c
z
⋅
−
1
=
cos
.
z
+
−
1
⋅
z
+
sin
.
z
{\displaystyle c^{z\cdot {\sqrt {-}}1}=\cos .z+{\sqrt {-}}1\cdot z+\sin .z}
[14] Int. „
c
−
z
⋅
−
1
=
cos
.
z
−
−
1
⋅
z
+
sin
.
z
{\displaystyle c^{-z\cdot {\sqrt {-}}1}=\cos .z-{\sqrt {-}}1\cdot z+\sin .z}
