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CBSE NOTES CLASS 11 PHYSICS

Mathematics In Physics

Basic Concepts and Important Formulae

Mathematical Signs and Symbols

Some Useful Concepts

Componendo

Dividendo

Alternetendo

Invertendo

Componendo and Dividendo

Geometry

Pythagoras theorem

Algebraic Identities

Roots of a Quadratic Equation

Logarithmic Functions

Binomial Theorem

Exponential Expantion (or Taylor’s expansion)

Logarithmic expansion

Important Trigonometric Concepts

Measurement of Angle

Degree Measure

Radian Measure

Trigonometric Functions

Values of trigonometric ratios of some common angles

Signs of Trigonometric Functions

Trends of Trigonometric Functions

Graphical Representation of Trigonometric Functions

Important Trigonometric Formulae

Sine formulae

Cosine formulae

Tips for use of sine and cosine formulae

Differentials and Integrals

CBSE NOTES CLASS 11 PHYSICS

Mathematics In Physics

Basic Concepts and Important Formulae

Mathematical Signs and Symbols

= equals

≅ equals approximately

~ is the order of magnitude of

≠ is not equal to

≡ is identical to, is defined as

> is greater than (>> is much greater than)

< is less than (<< is much less than)

≥ is greater than or equal to (or, is no less than)

≤ is less than or equal to (or, is no more than)

± plus or minus

∝ is proportional to

Σ the sum of

x̅ or < x > or xav - the average value of x

Some Useful Concepts

Componendo

If ab=cd, then a+bb=c+dd

Dividendo

If ab=cd , then a-bb=c-dd

Componendo and Dividendo

If ab=cd, then a+ba-b=c+dc-d.

Invertendo

If ab=cd, then  ba=dc

Alternetendo

If ab=cd, then  ac=bd

Geometry

For a circle of radius r:

For a sphere of radius r:

For right circular cylinder of radius r and height h:

For a triangle of base b and altitude h.

Pythagoras theorem

For a right triangle with hypotenuse = c and perpendicular sides a and b,

c2 = a2 + b2

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Algebraic Identities

(x + y)2 = x2 + 2xy + y2x2 + y2 = (x + y)2 - 2xy

(x – y)2 = x22xy + y2 x2 + y2 = (x - y)2 + 2xy

(x + y)2 = (x - y)2 + 4xy ⇒ (x - y)2 = (x + y)2 - 2xy

x2 – y2 = (x + y) (x – y)

(x + a) (x + b) = x2 + (a + b)x + ab

(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx

(x + y)3 = x3 + y3 + 3xy(x + y)

= x3 + 3x2y + 3xy2 + y3

(x – y)3 = x3 – y33xy(x – y)

= x3 – 3x2y + 3xy2y3

x3 + y3 + z33xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)

x3 + y3 = (x + y) (x2 + y2 - xy)

x3 - y3 = (x - y) (x2 + y2 + xy)

If x + y + z = 0 then x3 + y3 + z3 = 3xyz

Roots of a Quadratic Equation

Roots of equation ax2 + bx + c = 0 are given by

x=-b±b2-4ac2a

Case I: Two distinct real roots exist, if,

Descriminent D = b2-4ac>0

Case II: Two identical real roots exist, if,

Descriminent D = b2-4ac=0

Case III: No real roots exist, if,

Descriminent D = b2-4ac<0

Logarithmic Function

If a is a positive real number, other than unity, then,

fx=logbx, x>0

is defined as logarithmic function. b is called the base.

For 0 < b < 1, the exponential function is a strictly decreasing function and is negative

For b > 1, the exponential function is a strictly increasing function and is positive.

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Laws of logarithms

If

x = b y,

then

logb (x) = y

For example,

100 = 102 ⇒ log10 (100) = 2

Logarithm as inverse function of exponential function

The logarithmic function,

y = logb(x)

is the inverse function of the exponential function,

x = by= blogb x

Also x = logb (bx)

Natural logarithm (ln)

ln (x) = loge (x)

Inverse logarithm (antilog) calculation

The inverse logarithm (or anti logarithm) is calculated by raising the base b to the logarithm y

x = log-1(y) = b y

Logarithm rules

logbx  y= logbx+logby

logbxy= logbx- logby

logb xy= y  logbx

logbc= 1logcb

logbx=logcxlogc(b)

Binomial Theorem

For any real numbers n and x

1 ±  xn=1 ± nx1! ± nn-1x22!±

For x << 1,we ignore higher powers of x. Thus, we can write,

(1 + x)n = (1 + nx) and

(1 – x)n = (1 – nx)

Exponential Expantion (or Taylor’s expansion)

ex=1+x1!+x22!+x33!+,  -<x<

Logarithmic expansion

ln(1+x)=x- x22+x33+,  | x |<1

Important Trigonometric Concepts

Measurement of Angle

Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final position of the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex.

If the direction of rotation is anticlockwise, the angle is said to be positive and if the direction of rotation is clockwise, then the angle is negative

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Degree: If a rotation from the initial side to terminal side isth 1360th of a revolution, the angle is said to have a measure of one degree (1°).

One degree is divided into 60 minutes, and a minute is divided into 60 seconds. That is, one sixtieth of a degree is called a minute, written as 1′, and one sixtieth of a minute is called a second, written as 1″.

Radian: Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of 1 radian.

One complete revolution of the initial side subtends an angle of 2π radian.

In a circle of radius r, an arc of length l will subtend an angle θ radian at the centre, given by,

θ =lr rad, or l = r θ

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2π radians = 360° 

Or π radians = 180°

1 radian = 180°π= 57° 16' approximately

1° =π180= 0.01746 radians approximately

Radian measure = π180× Degree measure

Degree measure = 180π× Radian measure

1° = 60′ and 1′= 60′′

Some Important Angle Measures in Degree and Radian

deg

0

30

45

60

90

180

270

360

rad

0

π6

π4

π3

π2

π

3π2

2 π

Trigonometric Functions

Trigonometric ratios for an angle are the ratio of sides of a right angled triangle.

sinx=ph

cosx=bh

cosec x =hp=1sinx, x ≠ nπ

tanx=pb=sinxcosx            x 2n + 1π2

secx=hb=1cosx, x ≠ 2n + 1π2

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cotx=bp=cosxsinx, x ≠ nπ

sinx= 0  x = nπ

cosx= 0  x =2n + 1π2, where n is any integer

sin2 x + cos2 x = 1

1 + tan2 x = sec2 x

1 + cot2 x = cosec2 x

Values of trigonometric ratios of some common angles

0

π6

π4

π3

π2

π

3π2

2π

sin

0

12

12

32

1

0

-1

0

cos

1

32

12

12

0

-1

0

1

tan

0

13

1

3

nd

0

nd

0

cot

nd

3

1

13

0

nd

0

nd

sec

1

23

2

2

nd

-1

nd

1

cosec

nd

2

2

23

1

nd

-1

nd

Signs of Trigonometric Functions

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Trends of Trigonometric Functions

f

Q I

Q II

Q III

Q IV

sin

increases from

0 to 1

decreases from

1 to 0

decreases from

0 to -1

increases from

-1 to 0

cos

decreases from

1 to 0

decreases from

0 to -1

increases from

-1 to 0

increases from

0 to 1

tan

increases from

0 to ∞

increases from

-∞ to 0

increases from

0 to ∞

increases from

-∞ to 0

cot

decreases from

∞ to 0

decreases from

0 to -∞

decreases from

∞ to 0

decreases from

0 to -∞

sec

increases from

1 to ∞

increases from -∞ to -1

decreases from -1 to -∞

decreases from

∞ to 1

cosec

decreases from

∞ to 1

increases from

1 to ∞

increases from

-∞ to -1

decreases from

-1 to-∞

Graphical Representation of Trigonometric Functions

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Important Trigonometric Formulae

sin x= sinx

cosec x= -cosec x

cos x=cosx

sec -x=secx

tan-x= -tanx

cot -x= -cotx

cosπ2 x=sinx

secπ2- x= cosec x

sinπ2x=cosx

 cosec π2- x=secx

tanπ2x=cotx

cotπ2x=tanx

cosπ2+ x=-sinx

secπ2+ x= -cosec x

sinπ2+ x=cosx

cosec π2+ x =secx

tanπ2+ x=-cotx

cotπ2+ x= -tanx

cos π - x=-cosx

secπ - x= -secx

sinπ - x=sinx

cosec π - x= cosec x

tanπ - x= -tanx

cotπ - x= -cotx

cos π + x= -cosx

secπ + x= -secx

sinπ + x= -sinx

cosecπ + x=-cosec x

tanπ + x=tanx

cotπ +x=cotx

cos3π2+ x=sinx

sec3π2+ x= cosec x

sin3π2+ x= -cosx

 cosec 3π2+ x=-secx

tan3π2+ x= -cotx

cot3π2+x=-tanx

cos 2π  x=cosx

sec 2π - x=secx

sin2π  x= sinx

cosec 2π  x=-cosec x

tan (2π + x) = -tan x

cot (2π +x) =-cot x

cosx + y=cosxcosysinxsiny

cosx  y=cosxcosy+sinxsiny

sinx + y=sinxcosy+cosxsiny

sinx  y=sinxcosycosxsiny

If none of the angles x, y and (x + y) is an odd multiple of π2, then

tan x + y= tan x + tan y1  tan x tan y

tan x- y= tan x- tan y1+ tan x tan y

If none of the angles x, y and (x + y) is a multiple of π, then

cot  x + y=cot x cot y  1cot y+ cot x

cot  x- y=cot x cot y+ 1cot y- cot x

cos 2x = cos2x – sin2 x

= 2 cos2 x – 1

= 1 – 2 sin2 x

=1-tan2x1+tan2x, x  nπ+π2

sin 2x = 2 sin x cos x

=2 tan x1+tan2x, x  nπ+π2

tan 2x=2 tan x1-tan2x,  2x  nπ+π2

sin 3x = 3 sin x – 4 sin3 x

cos 3x = 4 cos3 x – 3 cos x

tan 3x= 3tan x- tan3 x1  3tan2 x

cos x + cos y = 2cosx + y2 cosx  y2

cos x  cos y = 2sinx + y2sinx  y2

sin x + sin y = 2sinx + y2cosx  y2

sin x  sin y = 2cosx + y2sinx  y2

2 cos x cos y = cos (x + y) + cos (xy)

–2 sin x sin y = cos (x + y) – cos (xy)

2 sin x cos y = sin (x + y) + sin (xy)

2 cos x sin y = sin (x + y) – sin (xy)

Trigonometric expansions

sin x=- x33!+x55!-

cos x=1- x22!+x44!-

tan x=+ x33+2x55+

Sine formulae

In any triangle, sides are proportional to the sines of the opposite angles. Let A, B and C be angles of a triangle and a, b and c be lengths of sides opposite to angles A, B and C respectively, then

sinAa=sinBb=sinCc

Cosine formulae

Let A, B and C be angles of a triangle and a, b and c be lengths of sides opposite to angles A, B and C respectively, then

a2 = b2 + c2 – 2bc cos A

b2 = c2 + a2 – 2ca cos B

c2 = a2 + b2 – 2ab cos C

That is,

cosA= b2+c2-a22bc

cosB= c2+a-b22ac

cosC= a2+b2-c22ab

Tips for use of sine and cosine formulae

isinAa=sinBb=sinCc=k

Or

iiasinA=bsinB=csinC=k

(iii) A + B + C = π

ivA+B+C2 = π2

(v) Use formulae for π - x and π2 x

Differentials and Integrals

Derivatives dfdx or f

Integrals (Anti derivatives)

d(xn)dx=n xn-1 

xn dx= xn+1n+1+ C

d(x)dx=1

dx= x+C

d(c)dx=0

d(kf(x))dx=kd(f(x))dx

kfx dx= kfx dx+C

d(sinx)dx=cos x

cosxdx=sinx+C

d(cosx)dx=- sin x

sinxdx=-cosx+C

d(tanx)dx=sec2 x

sec2 xdx=tanx+C

d(cotx)dx=- cosec2 x

cosec2 x dx=-cotx+C

d(secx)dx=sec xtanx

sec xtanx dx=secx+C

d(cosecx)dx=- cosec xcotx

cosec xcotx dx=-cosecx+C

d(ex)dx=ex

ex dx=ex+C

d(log|x|)dx=1x

1x dx=log|x|+C

Chain Rule

Chain rule is rule to differentiate composites of functions.

dydt=dydx×dxdt

dydx= dy/dtdx/dt

uv formulae

(u v)' = u' v + u v'

uv' = u'v- u v'v2