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Factoring
- a^2 − b^2 = ( a − b )( a + b )
- a^2 + b^2 is prime
- a 2 + 2 ab + b 2 = ( a + b ) 2
- a 2 − 2 ab + b 2 = ( a − b ) 2
- a 3 + b 3 = ( a + b )
a 2 − ab + b 2
a 2
Analytic Geometry
y 2 − y 1
x 2 − x 1
- equation of a line: y − y 1 = m ( x − x 1 )
- distance: d =
( x 2 − x 1 )
2
2
Exponent Rules
- a x + y = a x a y
- ( ab ) x^ = axbx
- ( ax )
y = axy
- a^0 = 1 if a 6 = 0
- a − x^ =
ax^
if a 6 = 0
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
∑^ n
k =
[ a + ( k − 1) d ] =
n
2
[2 a + ( n − 1) d ]
∑^ n
k =
[ a + ( k − 1) d ] = n
a + an
2
Geometric Series
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
adjacent
hypotenuse
opposite
hypotenuse
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
sin A
cos A
cos A
sin A
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Reciprocal Identities
cos A
sin 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
A
1 + cos A
2
A
1 − cos A
2
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
1 + cos 2 A
2
1 − cos 2 A
2
1 − cos 2 A
1 + cos 2 A
3 cos A + cos 3 A
4
3 sin A − sin 3 A
4
Sum-to-Product Identities
A + B
cos
A − B
A + B
sin
A − B
A + B
cos
A − B
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( A − B ) − cos( A + B )]
Sums of Sines and Cosines
A^2 + B^2 sin( x + φ ) where
cos φ =
B
A^2 + B^2
and sin φ =
A
A^2 + B^2
A^2 + B^2 cos( x − φ ) 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 ,
s ( s − a )( s − b )( s − c ) where
s =
a + b + c
2
ab sin C
c 2 sin A sin B
2 sin C
Circular Section
- Arc length: s = rθ
- Area: A =
r^2 θ
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