Topic 21

This time I chose to write all the basic mathematics formulae which everyone studies in their higher secondary school. I chose to write this because at some point or the other many of us need to brush the dust out from our brains to remember them. So, a quick online reference to them will certainly be helpful to many. And also another important thing in this is, 'to write mathematics symbols in HTML'. You can view the source of this page to see how this can be done. Alternatively you can obtain full details about writing mathematical symbols in HTML from this comprehensive HTML 4.0 Unicode Character Entity reference.

Algebraic Identities
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a − b)³ = a³ − 3a²b + 3ab² − b³
(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
(a − b)⁴ = a⁴ − 4a³b + 6a²b² − 4ab³ + b⁴
(a² − b²) = (a − b)(a + b)
(a³ − b³) = (a − b)(a² + 2ab + b²)
(a³ + b³) = (a + b)(a² − 2ab + b²)

Trigonometrical Identities
(1) sin²θ + cos²θ = 1
     sec²θ = 1 + tan²θ
     cosec²θ = 1 + cot²θ
(2) sin (A + B) = sin A cos B + cos A sin B
     sin (A − B) = sin A cos B − cos A sin B
     cos (A + B) = cos A cos B − sin A sin B
     cos (A − B) = cos A cos B + sin A sin B

tan (A + B) =

tan A + tan B 1 − tan A tan B

tan (A − B) =

tan A − tan B 1 + tan A tan B

(3) sin 2A = 2sin A cos A
     cos 2A = cos²A − sin²A
                 = 1 − 2sin²A
                 = 2cos²A − 1

tan 2A =

2 tan A 1 − tan²A

(4) sin C + sin D = 2 sin

C + D 2

cos

C − D 2

sin C − sin D = 2 cos

C + D 2

sin

C − D 2

cos C + cos D = 2 cos

C + D 2

cos

C − D 2

−cos C + cos D = 2 sin

C + D 2

sin

C − D 2

(5) 2 sin A cos B = sin (A + B) + sin (A − B)
     2 cos A sin B = sin (A + B) − sin(A − B)
     2 cos A cos B = cos (A + B) + cos(A − B)
     2 sin A sin B = cos (A − B) − cos(A + B)
(6) Negative angles:
     sin (−θ) = −sin θ
     cos (−θ) = cos θ
     tan (−θ) = −tan θ
(8) Angles having the same trig. ratios:
     Same sine:      θ and (180° − θ)
     Same cosine:  θ and (360° − θ), i.e. (−θ)
     Same tangent: θ and (180° + θ)
(8) a sin θ + b cos θ = A sin (θ + α)
     a sin θ − b cos θ = A sin (θ − α)
     a cos θ + b sin θ = A cos (θ − α)
     a cos θ − b sin θ = A cos (θ + α)
where

{	A = √(a² + b²) α = tan−1 b/a (0° < α < 90°)

Standard Curves
(1) Straight line:

Slope, m =

dy dx

=	y2 − y1 x2 − x1

Angle between two lines, tanθ =

m2 − m1 1 + m1m2

     For parallel lines, m₂ = m₁
     For perpendicular lines, m₁m₂ = −1
     Equation of straight line (slope = m)
     (i) Intercept c on real y-axis: y = mx + c
     (ii) Passing through (x₁, y₁): y − y₁ = m(x − x₁)

(iii) Joining (x1, y1) and (x2, y2):

y − y1 y2 − y1

=	x − x1 x2 − x1

(2) Circle:
     Centre at origin, radius r: x² + y² = r²
     Centre (h, k), radius r: (x − h)² + (y − k)² = r²
     General equation: x² + y² + 2gx + 2fy + c = 0 with centre (−g, −f): radius = √(g² = f² − c)
     Parametric equations: x = r cosθ, y = r sinθ
(3) Parabola:
     Vertex at origin, focus (a, 0): y² = 4ax
     Parametric equations: x = at², y = 2at
(4) Ellipse:

Centre at origin, foci (±√[a² + b²], 0):

x² a²

+

y² b²

= 1 where a = semi major axis, b = semi minor axis

     Parametric equations: x = a cosθ, y = b sinθ
(5) Hyperbola:

Centre at origin, foci (±√[a² + b²], 0):

x² a²

−

y² b²

= 1

     Parametric equations: x = a secθ, y = b tanθ
     Rectagular hyperbola:

Centre at origin, vertex ±

(

a √2

,

a √2

)

: xy =

a² 2

= c² i.e., xy = c² where c =

a √2

Parametric equations: x = ct, y =

c t

Laws of Mathematics
(1) Associative laws - for addition and multiplication
     a + (b + c) = (a + b) + c
     a(bc) = (ab)c
(2) Commutative laws - for addition and multiplication
     a + b = b + a
     ab = ba
(3) Distributive laws - for multiplication and division
     a(b + c) = ab + ac

b + c a

=

b a

+

c a

(provided a ≠ 0)

V Rama Aravind,
2006-02-08.
Posted On: 2006-02-15.