RELATION OF ANGULAR FUNCTIONS IN TERMS OF ONE ANOTHER
Trigonometric Functions
Function
sin α
cos α
tan α
cot α
sec α
csc α
sin α
sin α
±
√
1 −cos
2
α
tan α
±
√
1 +tan
2
α
1
±
√
1 +cot
2
α
±
√
sec
2
α
−1
sec α
1
csc α
cos α
±
√
1 −sin
2
α
cos α
1
±
√
1 +tan
2
α
cot α
±
√
1 +cot
2
α
1
sec α
±
√
csc
2
α
−1
csc α
tan α
sin α
±
√
1−sin
2
α
±
√
1−cos
2
α
cos α
tan α
1
cot α
±
√
sec
2
α
−1
1
±
√
csc
2
α
−1
cot α
±
√
1−sin
2
α
sin α
cos α
±
√
1−cos
2
α
1
tan α
cot α
1
±
√
sec
2
α
−1
±
√
csc
2
α
−1
sec α
1
±
√
1−sin
2
α
1
cos α
±
√
1+tan
2
α
±
√
1+cot
2
α
cot α
sec α
csc α
±
√
csc
2
α
−1
csc α
1
sin α
1
±
√
1−cos
2
α
±
√
1+tan
2
α
tan α
±
√
1+cot
2
α
sec α
±
√
sec
2
α
−1
csc α
Note: The choice of sign depends upon the quadrant in which the angle terminates.
Hyperbolic Functions
Function
sinh x
cosh x
tanh x
sinh x =
sinh x
±
�
cosh
2
x − 1
tanh x
√
1−tanh
2
x
cosh x =
�
1 + sinh
2
x
cosh x
1
√
1−tanh
2
x
tanh x =
sinh x
√
1+sinh
2
x
±
√
cosh
2
x−1
cosh x
tanh x
cosech x =
1
sinh x
±
1
√
cosh
2
x−1
√
1−tanh
2
x
tanh x
sech x =
1
√
1+sinh
2
x
1
cosh x
�
1 − tanh
2
x
coth x =
√
1+sinh
2
x
sinh x
± cosh x
√
cosh
2
x−1
1
tanh x
Function
cosech x
sech x
coth x
sinh x =
1
cosech x
±
√
1−sech
2
x
sech x
±1
√
coth
2
x−1
cosh x =
±
√
cosech
2
x+1
cosech x
1
sech x
±
coth x
√
coth
2
x−1
tanh x =
1
√
cosech
2
x+1
±
�
1 + sech
2
x
1
coth x
cosech x =
cosech x
±
sech x
√
1−sech
2
x
±
√
coth
2
x−1
1
sech x =
±
cosech x
√
cosech
2
x+1
sech x
±
√
coth
2
x−1
coth x
coth x =
�
cosech
2
x + 1
±
1
√
1−sech
2
x
coth x
Whenever two signs are shown, choose + sign if x is positive, − sign if x is negative.
A-8
DERIVATIVES
In the following formulas u, v, w represent functions of x, while a, c, n represent fixed real numbers. All arguments in the
trigonometric functions are measured in radians, and all inverse trigonometric and hyperbolic functions represent principal values
*Let y = f (x) and
dy
dx
=
d[ f (x)]
dx
= f
�
(x) define, respectively, a function and its derivative for any value x in their common domain.
The differential for the function at such a value x is accordingly defined as
dy = d[ f (x)] =
dy
dx
dx =
d[ f (x)]
dx
dx = f
�
(x) dx
Each derivative formula has an associated differential formula. For example, formula 6 below has the differential formula
d(uvw) = uv dw + vw du + uw dv
1.
d
dx
(a) = 0
2.
d
dx
(x) = 1
3.
d
dx
(au) = a
du
dx
4.
d
dx
(u + v − w) =
du
dx +
dv
dx −
dw
dx
5.
d
dx
(uv) = u
dv
dx +
v
du
dx
6.
d
dx
(uvw) = uv
dw
dx +
vw
du
dx +
uw
dv
dx
7.
d
dx
� u
v
�
=
v
du
dx
− u
dv
dx
v
2
=
1
v
du
dx −
u
v
2
dv
dx
8.
d
dx
(u
n
) = nu
n−1
du
dx
9.
d
dx
�√
u
�
=
1
2√u
du
dx
10.
d
dx
� 1
u
�
= −
1
u
2
du
dx
11.
d
dx
� 1
u
n
�
= −
n
u
n+1
du
dx
12.
d
dx
� u
n
v
m
�
=
u
n−1
v
m+1
�
nv
du
dx −
mu
dv
dx
�
13.
d
dx
(u
n
v
m
) = u
n−1
v
m−1
�
nv
du
dx +
mu
dv
dx
�
14.
d
dx
[ f (u)] =
d
du
[ f (u)] ·
du
dx
15.
d
2
dx
2
[ f (u)] =
d f (u)
du ·
d
2
u
dx
2
+
d
2
f (u)
du
2
·
� du
dx
�
2
16.
d
n
dx
n
[uv] =
�n
0
�
v
d
n
u
dx
n
+
�n
1
� dv
dx
d
n−1
u
dx
n−1
+
�n
2
� d
2
v
dx
2
d
n−2
u
dx
n−2
+ · · · +
�n
k
� d
k
v
dx
k
d
n−k
u
dx
n−k
+ · · · +
�n
n
�
u
d
n
v
dx
n
where
�
n
r
�
=
n!
r!(n−r)!
is the binomial coefficient, n non-negative integer, and
�
n
0
�
= 1.
17.
du
dx =
1
dx
du
if
dx
du �=
0
18.
d
dx
(log
a
u) = (log
a
e)
1
u
du
dx
A-9