Basic Formulae

Question 1

Geometric sum

Answer 1

Sn = a + ar + ar^2 +...+ ar^(n-1)
Sn = a(1-r^n)/(1-r)

Question 2

Geometric sum to infinity

Answer 2

S = a/(1-r)

when |r| > 1

Question 3

Binomial Expansion (1+x)^n

Answer 3

(1+x)^n = 1 + nx + n(n-1)/2! x^2 + n(n-1)(n-2)/3! x^3 +…

Question 4

Binomial Expansion (a+b)^n

Answer 4

(a+b)^n = a^n + nC1 a^(n-1) b + nC2 a^(n-2) b^2 +…+ b^n

Question 5

Maclaurin Series

Answer 5

f(x) = f(0) + xf’(0) + x^2 f’‘(0)/2! + x^3 f’’‘(0)/3! +…

Question 6

Taylor Series f(a+x)

Answer 6

f(a+x) = f(a) + xf’(a) + x^2 f’‘(a)/2! + x^3 f’’‘(a)/3! +…

Question 7

Taylor Series f(x)

Answer 7

f(x) = f(a) + (x-a)f’(a) + (x-a)^2 f’‘(a)/2! + (x-a)^3 f’’‘(a)/3! +…

Question 8

Power Series e^x

Answer 8

e^x = 1 + x + x^2 /2! + x^3 /3! +…+ x^n /n! +…

Question 9

Power Series ln(1+x)

Answer 9

ln(1+x) = x - x^2 /2 + x^3 /3 -…+ (-1)^(n+1) x^n /n +…

Question 10

Power Series sin(x)

Answer 10

sin(x) = x - x^3 /3! + x^5 /5! -…

Question 11

Power Series cos(x)

Answer 11

cos(x) = 1 - x^2 /2! + x^4 / 4!

Question 12

De Moivre’s Theorem

Answer 12

(cos(θ) + isin(θ))^n = cos(nθ) + isin(nθ)

Question 13

Trig in a circle

Answer 13

cos^2 (A) + sin^2 (A) = 1
sec^2(A) = tan^2 (A) + 1
cosec^2 (A) = cot^2 (A) + 1

Question 14

Double angle formulae

Answer 14

sin(2A) = 2sin(A)cos(A)
cos(2A) = cos^2 (A) - sin^2 (A)
= 2 cos^2 (A) - 1
= 1 - 2sin^2 (A)

Question 15

sin(A + B)

sin(A - B)

Answer 15

sin(A)cos(B) + cos(A)sin(B)

sin(A)cos(B) - cos(A)sin(B)

Question 16

cos(A + B)

cos(A - B)

Answer 16

cos(A)cos(B) - sin(A)sin(B)

cos(A)cos(B) + sin(A)sin(B)

Question 17

tan(A + B)

tan(A - B)

Answer 17

(tan(A) + tan(B))/(1 - tan(A)tan(B))

tan(A) - tan(B))/(1 + tan(A)tan(B)

Question 18

cos(A)cos(B)

Answer 18

[cos(A + B) + cos(A - B)]/2

Question 19

sin(A)sin(B)

Answer 19

[cos(A - B) - cos(A + B)]/2

Question 20

sin(A)cos(B)

Answer 20

[sin(A + B) + sin(A - B)]/2

Question 21

sin(A) + sin(B)

Answer 21

2sin((A + B)/2)cos((A - B)/2)

Question 22

sin(A) - sin(B)

Answer 22

2sin((A - B)/2)cos((A + B)/2)

Question 23

cos(A) + cos(B)

Answer 23

2cos((A + B)/2)cos((A - B)/2)

Question 24

cos(A) - cos(B)

Answer 24

-2sin((A + B)/2)sin((A - B)/2)

Question 25

Sine Rule

Answer 25

a/sin(A) = b/sin(B) = c/sin(C)
sin(A)/a = sin(B)/b = sin(C)/c

Question 26

Cosine Rule

Answer 26

a^2 = b^2 + c^2 - 2bccos(A)
cos(A) = (b^2 + c^2 - a^2)/2bc

Question 27

Product Rule

Answer 27

d(uv)/dx = vu’ + uv’

Question 28

Quotient Rule

Answer 28

d(u/v)/dx = (vu’ - uv’)/v^2

Question 29

Differential of x^n

Answer 29

nx^(n-1)

Question 30

Differential of e^(kx)

Answer 30

ke^(kx)

Question 31

Differential of ln(x)

Answer 31

1/x

Question 32

Differential of e^(f(x))

Answer 32

f’(x)e^(f(x))

Question 33

Differential of ln(f(x))

Answer 33

f’(x)/f(x)

Question 34

Differential of sin(x)

Answer 34

cos(x)

Question 35

Differential of cos(x)

Answer 35

-sin(x)

Question 36

Differential of tan(x)

Answer 36

sec^2 (x)

Question 37

Differential of sec(x)

Answer 37

sec(x)tan(x)

Question 38

Differential of cot(x)

Answer 38

-cosec^2 (x)

Question 39

Differential of cosec(x)

Answer 39

-cosec(x)cot(x)

Question 40

Integral of sec^2(x)

Answer 40

tan(x)

Question 41

Integral of sec(x)tan(x)

Answer 41

sec(x)

Question 42

Integral of -cosec^2 (x)

Answer 42

cot(x)

Question 43

Integral of -cosec(x)cot(x)

Answer 43

cosec(x)

Question 44

Quadratic Formula roots

Answer 44

x = (-b +/- sqrt(b^2 - 4ac))/2a

Question 45

Logarithm Rules

Answer 45

log(xy) = log(x) + log(y)
log(x/y) = log(x) - log(y)
log(x^k) = klog(x)

Question 46

Surface area of a sphere

Answer 46

4πr^2

Question 47

Volume of a sphere

Answer 47

4/3 πr^3