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Pre-Calculus Reference Sheet: Formulas and Identities, Cheat Sheet of Calculus

Formula sheet with factoring, arithmetic’s series, analytic geometry, geometric series, exponent and logarithms rules, trigonometry, power reduction identities and circular sections.

Typology: Cheat Sheet

2021/2022

Uploaded on 02/07/2022

jesus33
jesus33 🇺🇸

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Factoring
a2b2= (ab)(a+b)
a2+b2is prime
a2+ 2ab +b2= (a+b)2
a22ab +b2= (ab)2
a3+b3= (a+b)a2ab +b2
a3b3= (ab)a2+ab +b2
Analytic Geometry
slope: m=y2y1
x2x1
equation of a line: yy1=m(xx1)
distance: d=q(x2x1)2+ (y2y1)2
Exponent Rules
ax+y=axay
(ab)x=axbx
(ax)y=axy
a0= 1 if a6= 0
ax=1
axif a6= 0
axy=ax
ayif a6= 0
Logarithm Rules
logbx=y x=by
blogbx=x
logbbx=x
logb1=0
logbb= 1
logbxy = logbx+ logby
logb
x
y= logbxlogby
logbxy=ylogbx
logbx=logax
logab
Arithmetic Series
ak=a+ (k1)d
Sn=
n
X
k=1
[a+ (k1)d] = n
2[2a+ (n1)d]
Sn=
n
X
k=1
[a+ (k1)d] = na+an
2
Geometric Series
an=arn1
Sn=
n1
X
k=0
ark=a1rn
1rif r6= 1
S=
X
k=0
ark=a
1rif |r|<1
Trigonometry
cos A=adjacent
hypotenuse sin A=opposite
hypotenuse
tan A=opposite
adjacent
030456090
0π
6
π
4
π
3
π
2
sin 0 1
2
2
2
3
21
cos 1 3
2
2
2
1
20
tan 0 3
313undefined
Pythagorean Identities
cos2A+ sin2A= 1
1 + tan2A= sec2A
1 + cot2A= csc2A
Ratio Identities
tan A=sin A
cos Acot A=cos A
sin A
CSUEB Math Lab
Pre-Calculus Reference Sheet
pf2

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Factoring

  • a^2 − b^2 = ( ab )( a + b )
  • a^2 + b^2 is prime
  • a 2 + 2 ab + b 2 = ( a + b ) 2
  • a 2 − 2 ab + b 2 = ( ab ) 2
  • a 3 + b 3 = ( a + b )

a 2 − ab + b 2

  • a 3 − b 3 = ( ab )

a 2

  • ab + b 2

Analytic Geometry

  • slope: m =

y 2 − y 1

x 2 − x 1

  • equation of a line: yy 1 = m ( xx 1 )
  • distance: d =

( x 2 − x 1 )

2

  • ( y 2 − y 1 )

2

Exponent Rules

  • a x + y = a x a y
  • ( ab ) x^ = axbx
  • ( ax )

y = axy

  • a^0 = 1 if a 6 = 0
  • ax^ =

ax^

if a 6 = 0

  • axy^ =

a x

ay^

if a 6 = 0

Logarithm Rules

  • log b x = y ⇐⇒ x = b y
  • b log b x = x
  • log b b x = x
  • log b 1 = 0
  • log b b = 1
  • log b xy = log b x + log b y
  • log b

x

y

= log b x − log b y

  • log b x y = y log b x
  • log b x =

log a x

log a b

Arithmetic Series

  • ak = a + ( k − 1) d
  • Sn =

∑^ n

k =

[ a + ( k − 1) d ] =

n

2

[2 a + ( n − 1) d ]

  • Sn =

∑^ n

k =

[ a + ( k − 1) d ] = n

a + an

2

Geometric Series

  • an = arn −^1
  • Sn =

n ∑− 1

k =

ar k = a

[

1 − r n

1 − r

]

if r 6 = 1

• S =

∑^ ∞

k =

ar k =

a

1 − r

if | r | < 1

Trigonometry

  • cos A =

adjacent

hypotenuse

  • sin A =

opposite

hypotenuse

  • tan A =

opposite

adjacent

◦ 30 ◦ 45 ◦ 60 ◦ 90 ◦

π

6

π

4

π

3

π

2

sin 0

cos 1

tan 0

3 undefined

Pythagorean Identities

  • cos 2 A + sin 2 A = 1
  • 1 + tan 2 A = sec 2 A
  • 1 + cot 2 A = csc 2 A

Ratio Identities

  • tan A =

sin A

cos A

  • cot A =

cos A

sin A

CSUEB Math Lab

Pre-Calculus Reference Sheet

Reciprocal Identities

  • sec A =

cos A

  • csc A =

sin A

  • cot A =

tan A

Sum and Difference Identities

  • cos( A ± B ) = cos A cos B ∓ sin A sin B
  • sin( A ± B ) = sin A cos B ± cos A sin B
  • tan( A ± B ) =

tan A ± tan B

1 ∓ tan A tan B

Double Angle Identities

  • cos 2 A = cos 2 A − sin 2 A
  • cos 2 A = 2 cos 2 A − 1
  • cos 2 A = 1 − 2 sin 2 A
  • sin 2 A = 2 cos A sin A
  • tan 2 A =

2 tan A

1 − tan 2 A

Half Angle Identities

  • cos

A

1 + cos A

2

  • sin

A

1 − cos A

2

  • tan

A

1 − cos A

sin A

sin A

1 + cos A

Triple Angle Identities

  • cos 3 A = 4 cos 3 A − 3 cos A
  • sin 3 A = 3 sin A − 4 sin 3 A

Power Reduction Identities

  • cos^2 A =

1 + cos 2 A

2

  • sin^2 A =

1 − cos 2 A

2

  • tan 2 A =

1 − cos 2 A

1 + cos 2 A

  • cos 3 A =

3 cos A + cos 3 A

4

  • sin^3 A =

3 sin A − sin 3 A

4

Sum-to-Product Identities

  • sin A + sin B = 2 sin

A + B

cos

A − B

  • sin A − sin B = 2 cos

A + B

sin

A − B

  • cos A + cos B = 2 cos

A + B

cos

A − B

  • cos A − cos B = −2 sin

A + B

sin

A − B

Product-to-Sum Identities

  • sin A cos B = 1 2 [sin( A^ +^ B ) + sin( A^ −^ B )]
  • cos A cos B = 1 2 [cos( A^ +^ B ) + cos( A^ −^ B )]
  • sin A sin B = 1 2 [cos( AB ) − cos( A + B )]

Sums of Sines and Cosines

  • A cos x + B sin x =

A^2 + B^2 sin( x + φ ) where

cos φ =

B

A^2 + B^2

and sin φ =

A

A^2 + B^2

  • A cos x + B sin x =

A^2 + B^2 cos( xφ ) where

cos φ =

A

A^2 + B^2

and sin φ =

B

A^2 + B^2

Laws of Sines and Cosines

  • c 2 = a 2 + b 2 − 2 ab cos C

a

sin A

b

sin B

c

sin C

Area of a Triangle

For a triangle with sides a , b , c and angles 6 A , 6 B , and 6 C ,

  • Area =

s ( sa )( sb )( sc ) where

s =

a + b + c

2

  • Area =

ab sin C

  • Area =

c 2 sin A sin B

2 sin C

Circular Section

  • Arc length: s =
  • Area: A =

r^2 θ

CSUEB Math Lab