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