0 30∨π6

45∨π4

60∨π3

90∨π2

cosθ 1 √32

1√2

12 0

sin θ 0 12

1√2

√32

1

If θ is a multiple of π2 the coordinate of the points on a unit circle involve 0 and ±1.

If θ is a multiple of π4 (but not a multiple of π2), the coordinates involve ± 1

√2 . If θ is a multiple of π6 (but not a multiple of π2), the

coordinate involve ± 12 and ± √3

2 .

y=A sin B ( x−C )+D

A = amplitudeB = PeriodC = Horizontal translation (by +C)D = Vertical translation

y=A sin B ( x+C )+D

A = amplitudeB = PeriodC = Horizontal translation (by -C)

D = Vertical translation

2πB = Period

A=max−¿min2

¿

D=max+min2

Area of an equilateral triangle –A=√3

4a2

Where ‘a’ is the side of the equilateral triangle.

Three identities in trigonometry – sin2 x+cos2 x=1 tan2 x+1=sec2 x 1+cot2 x=csc2 x

Where –- sec x= 1

cos x

- csc x= 1sin x

- cot x= 1tan x

Compound angle formulae –

sin(A + B) = sinAcosB + cosAsinB

sin(A - B) = sinAcosB - cosAsinB

cos(A + B) = cosAcosB - sinAsinBcos(A - B) = cosAcosB + sinAsinB

tan (A + B) =   tanA + tanB                  1 - tanAtanBtan (A - B) =    tanA - tanB                   1 + tanAtanB

Double angle formulae –

sin2A = 2sinAcosA

cos2A = cos2A - sin2A- cos2A = 1 - 2sin2A- cos2A = 2cos2A - 1

tan2A =      2tanA            1 - tan2A 

Derivative functions –

d ¿¿

d ¿¿

d ¿¿

d ¿¿

d ¿¿

d ¿¿

Differential Calculus –

f ' ( x ) : lim h→0Slope→ f ( x+h )−f ( x )h

f ' ( x ) : lim x→aSlope→ f (x )−f (a )x−a

Slopeof tangent= −1Slope of normal

i .e. slope of tangent to a point is the negative reciprocal of the slope of the normal to that point on the curve.

-(equation of the curve)+(equation of the tangent)=0 If f ( x )=¿¿; f ' ( x )=n (a×b ) ¿¿

Where (a×b) is the derivative of the the highest degree of polynomial.

If f(x)=xn , then f ’(x) = nxn-1

If f(x)= cu(x) (i.e. c times u(x)), then f ’(x) = c u’(x)

If f(x) = u(x)+v(x), then f ’(x) = u’(x)+v’(x)

Chain rule - dydx=dydu

× dudx

Product rule – if f(x)=u(x)v(x)Then f ' x=u' ( x ) v ( x )+v ' ( x )u ( x )

Quotient rule – if, Q(x)= v (x )u(x)

Q’(x)=u' ( x ) v ( x )−u (x ) v ' ( x)(v (x ))2

i.e. if y=u/vdydx

=u 'v−v 'uv2

Complex numbers – |z*|=|z| |z|2 = z z* |z1z2| = |z1||z2| |z1/z2|=|z1|/|z2| (provided z2≠0) |z1z2z3z4…zn|=|z1||z2||z3||z4|…|zn| |zn| = |z|n

If z = |z|cis θ then z* = ¿ z∨cis (−θ)

|z|cisθ=¿z∨eiθ

(Polar Form) (Euler Form)

cis A × cis B = cis (A+B)

cis(A)cis(B) = cis (A-B)

cis (θ+2kπ ¿ = cis (θ ¿

Arg(zw) = Arg(z) + Arg(w)

Arg(z/w) = Arg(z) – Arg(w)

Arg(zn) = n Arg(z)

(|z| cis θ)n = |z|ncis nθ

z = a± ib

Quadratic equation representing ‘z’ –x2−¿

where, sum = a+ ib+a−ib (2a)& product = (a+ ib)(a−ib) (a2+b2)

(z*)*= z (zn)* = (z*)n

for n = 1,2 & 3

(z1+z2+z3+z4…+zn)* = (z1)*+(z2)*+(z3)*+(z4)*…+(zn)*

P(x )ax+b

=Q ( x )+ R (x)ax+b

where – Q(x) – quotient R(x) – remainder ax+b – divisor

When P(x) is divided by a factor x-k until a constant remainder ‘R’ is obtained – P(k) = R

P(x) = Q(x)(x-k)+RP(k)= Q(k)(k-k)+R as k-k=0P(k)= R

If ‘k’ is a zero of P(x), then (x-k) is a factor of P(x)

Counting and binomial expansion – t r+ 1=( Cr

n ) xn−rar

where, t r+ 1 is the (r+1)th term of the polynomial – (x+a)n

nCr=n!

r! (n−r )!

Quadratic functions and equations –

Discriminant (∆) = b2-4ac(ax2+bx+c)if ∆>0 – two distinct real rootsif ∆<0 – no real rootsif ∆=0 – single real repeated rootif ∆≥0 – two real roots

If ∆>0, the graph cuts x-axis twice ∆<0, does not touch the x-axis

∆=0, the graph touches the x-axis once

If ax2+bx+c has two roots – α∧β

Then – α+β=−b

a

andα× β= c

a

f(x) = ax2+bx+c, if a¿0 and ∆<0the graph is a negative parabola and if a¿0 and ∆<0 the graph is a positive parabola.

Logarithm –

(no negative numbers in logarithmic equations) log aa

n=n

log cA +logcB=logc ¿¿¿

log cA−logcB=logc (AB ¿)¿

n log cA=logc (An)

ln e x=x∧e ln x=x

(examples – ln1 = lne0 = 0lne = ln e1 = 1)

ax=e x ln a

log b A=logc Alogcb

Exponentials –

Laws of Indices –1. am×an = am+n

2. am

an=am−n , where a ≠ 0

3. ( ab)n

=an

bn, where b≠0

4. (am)n = amn

5. (ab)n = anbn

6. a0 = 1, where a ≠ 07. a−n= 1

an , where a ≠ 08. a

1n=n√a

9. amn=

n√am

10. If ax = ay, then x = y

Matrices –

We can add or subtract two matrices only if they are of the same order i.e. both are 2× 2, 3 × 3, etc.

¿

Matrix – A × BOrder of matrix A - n×mOrder of matrix B - u×vOrder of matrix AB = n×vProvided m=u, if m≠u – AB cannot be multiplied.

Matrix A = a bc d

Then |A| (determinant of matrix A) = ad – bc

If |A| = 0, the matrix is singular and has no inverse.

Matrix A = a bc d , then

Matrix A-1 = 1

¿ A∨¿×( d −b−c a )¿

Equations – ax+ by = c dx+ ey = fcan be written as –

(for 2 equations)Last row – all zero’s – infinite solutions Two zero – no solutionsOne zero or no zero – unique solutions

(for 3 equations)All zero’s – infinite3 zero’s (last value) – no solution

one zero, two zero or no zero – unique solutions

Matrices – A & B(AB)-1 = B-1A-1

Sequences and series –

Arithmetic sequence – un = u1+ (n-1)d

Sum of first ‘n’ terms of an arithmetic sequence –

n2(2u1+(n−1)d )

where, u1 – the first term d – the common difference n – the no. of terms

Arithmetic sequence – d = un+1-un

Geometric sequence – un= u1 × rn-1

Sum of ‘n’ terms in a geometric progression – u1(r

n−1)r−1 or u1(1−rn)

1−r

Sum of infinite Geometric progression –u1

1−r

Compound Interest –un+1=u1+r

n

where, u1 is the principal and r is the rate and ‘n’ is the time.

Superannuation –Amount at the end ¿ ai { (1+ i¿¿¿n−1 }

Amount at the beginning = ai (1+i){(1+i ¿¿¿n−1 }

Principal at beginning =ai(1+i){1− (1+i ¿¿¿−n }

Principal at end = ai (1+i){1− (1+i ¿¿¿−n }