a3 − b3 = (a − b) (a2 + ab + b2). Analytic Geometry. • slope: m = y2 − y1 x2 − x1. •
equation of a line: y − y1 = m (x − x1). • distance: d = √. (x2 − x1). 2. + (y2 − y1).
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
