Algebra
4. ax2 + bx + c =
!
!
√
√
−b + b2 − 4ac
−b − b2 − 4ac
a x−
x−
,
2a
2a
1. (a + b)2 = a2 + 2ab + b2
2. (a − b)2 = a2 − 2ab + b2
if b2 − 4ac ≥ 0
3. a2 − b2 = (a − b)(a + b)
Trigonometry
Hyperbolic functions
Definitions
Definitions
sin x
1. tan x =
cos x
2. cot x =
cos x
sin x
e x − e −x
1. sinh x =
1
sin x
3. csch x =
1
sinh x
4. sech x =
1
cosh x
5. tanh x =
sinh x
cosh x
6. coth x =
cosh x
sinh x
4. csc x =
Identities
2
1. cos2 x + sin2 x = 1
4. tan2 x + 1 = sec2 x
Identities
2. sin 2x = 2 sin x cos x
5. cot2 x + 1 = csc2 x
1. cosh2 x − sinh2 x = 1
3. cos 2x =
= cos2 x − sin2 x
= 2 cos2 x − 1
= 1 − 2 sin2 x
6. tan 2x =
2 tan x
1 − tan2 x
2. 1 − tanh2 x = sech2 x
7. cot 2x =
cot2 x − 1
2 cot x
3. coth2 x − 1 = csch2 x
cos x
7. (arcsin x)0 = √
2. (cos x)0 = − sin x
3. (tan x)0 =
0
sec2 x
2
1
1 − x2
8. (arccos x)0 = √
4. (cot x) = − csc x
5. (sec x)0 =
2. cosh x =
2
4. sinh 2x = 2 sinh x cosh x
5. cosh 2x =
= cosh2 x + sinh2 x
= 2 cosh2 x − 1
= 2 sinh2 x + 1
Derivatives
Derivatives
1. (sin x)0 =
e x + e −x
1
3. sec x =
cos x
−1
1 − x2
tan x sec x
6. (csc x)0 = − cot x csc x
9. (arctan x)0 =
1
1 + x2
Integrals
1.
R
cos x dx = sin x
2.
R
sin x dx = − cos x
3.
R
sec2 x dx = tan x
4.
R
csc2 x dx = − cot x
5.
R
cot x dx = ln | sin x|
6.
R
tan x dx =
= − ln | cos x|
R
7. sec x dx =
= ln | tan x + sec x|
R
8. csc x dx =
= − ln | cot x + csc x|
1. (sinh x)0 = cosh x
2. (cosh x)0 = sinh x
7. (arcsinh x)0 =
1
=√
2
x +1
3. (tanh x)0 = sech2 x
4. (coth x)0 = − csch2 x
5. (sech x)0 =
= − tanh x sech x
6. (csch x)0 =
= − coth x csch x
8. (arccosh x)0 =
1
=√
2
x −1
9. (arctanh x)0 =
1
= 2
x −1
Symmetry & Periodicity
1. sin(−x) = − sin(x)
5. cos(−x) = cos(x)
9. tan(−x) = − tan(x)
2. sin(x + π/2) = cos(x)
6. cos(x + π/2) = − sin(x)
10. tan(x + π/2) = − cot(x)
3. sin(x − π/2) = − cos(x)
7. cos(x − π/2) = sin(x)
11. tan(x − π/2) = − cot(x)
4. sin(x ± π) = − sin(x)
8. cos(x ± π) = − cos(x)
12. tan(x ± π) = tan(x)
x
−π
−π/2
0
π/6
π/4
π/2
√1
2
π/3
√
3
2
1
2π/3
√
3
2
1
2
5π/6
π
3π/2
2π
1
2
0
−1
0
−1
0
1
sin(x)
0
−1
0
cos(x)
−1
0
1
3
2
√1
2
1
2
0
− 12
tan(x)
0
–
0
√1
3
1
√
3
–
√
− 3
− √13
0
–
–
cot(x)
–
0
–
1
− √13
√
− 3
–
0
1
√
√
3
1
√1
3
√
−
3
2
Powers, Exponents & Logarithms
1. xa+b = xa · xb
xa
2. xa−b = b
x
1
3. x−a = a
x
4. xa·b = (xa )b
√
a
5. x /b = b xa
6. e x : (−∞, ∞) → (0, ∞)
9. ln(x) : (0, ∞) → (−∞, ∞)
10. ln(a · b) = ln(a) + ln(b)
7. e
ln(x)
=x
11. ln
8. ln(e x ) = x
a
b
= ln(a) − ln(b)
12. ln(ab ) = b ln(a)
Derivatives & Integrals
1. (xn )0 = nxn−1
0
3. (e ax ) = ae ax
5. (ln(x))0 =
Z
6.
Z
2.
xn+1
x dx =
, n 6= −1
n+1
n
Z
4.
Other Integrals
Z
1
1
x
1.
dx = arctan
2
2
x +a
a
|a|
Z
p
1
√
2.
dx = ln |x + x2 ± a2 |
x2 ± a2
e
ax
dx =
e ax
a
1
x
dx
= ln(x)
x
Z
7.
ln(x) dx = x ln(x) − x
Inverse trigonometric functions
√
arcsin x = arccos 1 − x2
y = arcsin(x)
sin(y)
x
y = arccos(x)
p
1 − x2
cos(y)
p
1 − x2
x
tan(y)
cot(y)
x
√
1 − x2
√
1 − x2
x
√
√
y = arctan(x)
x
−1
1
√
1 + x2
√
x2
y = arccot(x)
1
1 + x2
x
√
2
x −1
√
1 − x2
x
x
1
x
x
1 − x2
1
x
x
Inverse hyperbolic functions
y = arcsinh(x)
sinh(y)
cosh(y)
tanh(y)
coth(y)
x
p
y = arccosh(x)
p
x2 − 1
x2 + 1
x
√
x2 + 1
√
x2 + 1
x
y = arctanh(x)
x
√
1 − x2
1
√
1 − x2
y = arccoth(x)
sign x
√
x2 − 1
|x|
√
x2 − 1
x2 − 1
x
x
1
x
x
−1
1
x
x
x
√
√
x2