Factoring
Arithmetic Series
2
• ak = a + (k − 1)d
2
• a − b = (a − b)(a + b)
• a2 + b2 is prime
2
• Sn =
2
• a + 2ab + b = (a + b)
2
n
X
[a + (k − 1)d] =
k=1
• a2 − 2ab + b2 = (a − b)2
• Sn =
• a3 + b3 = (a + b) a2 − ab + b2
• a3 − b3 = (a − b) a2 + ab + b2
n
X
n
[2a + (n − 1)d]
2
[a + (k − 1)d] = n
k=1
a + an
2
Geometric Series
Analytic Geometry
y2 − y1
• slope: m =
x2 − x1
• an = arn−1
• Sn =
n−1
X
k=0
• equation of a line: y − y1 = m (x − x1 )
q
2
2
• distance: d = (x2 − x1 ) + (y2 − y1 )
• S=
∞
X
• ax+y = ax ay
k=0
y
• (ax ) = axy
adjacent
hypotenuse
• tan A =
opposite
adjacent
• a0 = 1 if a 6= 0
1
if a 6= 0
ax
ax
if a 6= 0
ay
Logarithm Rules
• logb x = y ⇐⇒ x = b
if r 6= 1
a
if |r| < 1
1−r
• cos A =
• (ab)x = ax bx
• ax−y =
ark =
Trigonometry
Exponent Rules
• a−x =
1 − rn
ar = a
1−r
k
opposite
hypotenuse
0◦
30◦
45◦
60◦
90◦
0
π
6
π
4
√
2
2
√
2
2
π
3
√
3
2
π
2
sin
0
cos
1
tan
0
y
• blogb x = x
• sin A =
1
2
√
3
2
√
3
3
1
1
1
2
√
0
3
undefined
• logb bx = x
• logb 1 = 0
Pythagorean Identities
• logb b = 1
• cos2 A + sin2 A = 1
• logb xy = logb x + logb y
• 1 + tan2 A = sec2 A
x
= logb x − logb y
y
• 1 + cot2 A = csc2 A
• logb
• logb xy = y logb x
loga x
• logb x =
loga b
Ratio Identities
• tan A =
sin A
cos A
• cot A =
cos A
sin A
Reciprocal Identities
• sec A =
1
cos A
• csc A =
Sum-to-Product Identities
1
sin A
• cot A =
1
tan A
• sin A + sin B = 2 sin
A+B
2
cos
A−B
2
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
A+B
A−B
sin
2
2
A−B
A+B
cos
• cos A + cos B = 2 cos
2
2
A+B
A−B
• cos A − cos B = −2 sin
sin
2
2
• sin A − sin B = 2 cos
Product-to-Sum Identities
• sin A cos B =
1
2
[sin(A + B) + sin(A − B)]
• cos 2A = 2 cos2 A − 1
• cos A cos B =
1
2
[cos(A + B) + cos(A − B)]
• cos 2A = 1 − 2 sin2 A
• sin A sin B =
1
2
[cos(A − B) − cos(A + B)]
• cos 2A = cos2 A − sin2 A
• sin 2A = 2 cos A sin A
• tan 2A =
2 tan A
1 − tan2 A
Half Angle Identities
r
A
1 + cos A
• cos = ±
2
2
r
A
1 − cos A
• sin = ±
2
2
• tan
A
1 − cos A
sin A
=
=
2
sin A
1 + cos A
Sums of Sines and Cosines
√
• A cos x + B sin x = A2 + B 2 sin(x + φ) where
B
A
and sin φ = √
cos φ = √
2
2
2
A +B
A + B2
√
• A cos x + B sin x = A2 + B 2 cos(x − φ) where
A
B
cos φ = √
and sin φ = √
2
2
2
A +B
A + B2
Laws of Sines and Cosines
• c2 = a2 + b2 − 2ab cos C
•
b
c
a
=
=
sin A
sin B
sin C
Triple Angle Identities
• cos 3A = 4 cos3 A − 3 cos A
Area of a Triangle
• sin 3A = 3 sin A − 4 sin3 A
For a triangle with sides a, b, c and angles 6 A, 6 B,
and 6 C,
p
• Area = s(s − a)(s − b)(s − c) where
a+b+c
s=
2
Power Reduction Identities
• cos2 A =
1 + cos 2A
2
1 − cos 2A
• sin A =
2
• Area =
1
ab sin C
2
1 − cos 2A
• tan2 A =
1 + cos 2A
• Area =
c2 sin A sin B
2 sin C
2
• cos3 A =
3 cos A + cos 3A
4
3 sin A − sin 3A
• sin A =
4
Circular Section
• Arc length: s = rθ
3
• Area: A =
1 2
r θ
2