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2021-07-29 18:03:22 COMPETISHUN BY MOHIT TYAGI PRAVEEN (DROPPER BATCH) COMPLETE MATERIAL FOR JEE 2021


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2021-07-29 17:08:22 Notes on Trigonometric Equations and Identities


A function f(x) is said to be periodic if there exists some T > 0 such that f(x+T) = f(x) for all x in the domain of f(x).

In case, the T in the definition of period of f(x) is the smallest positive real number then this ‘T’ is called the period of f(x).

Periods of various trigonometric functions are listed below:

1) sin x has period 2π

2) cos x has period 2π

3) tan x has period π

4) sin(ax+b), cos (ax+b), sec(ax+b), cosec (ax+b) all are of period 2π/a

5) tan (ax+b) and cot (ax+b) have π/a as their period

6) |sin (ax+b)|, |cos (ax+b)|, |sec(ax+b)|, |cosec (ax+b)| all are of period π/a

7) |tan (ax+b)| and |cot (ax+b)| have π/2a as their period

Sum and Difference Formulae of Trigonometric Ratios

1) sin(a + ß) = sin(a)cos(ß) + cos(a)sin(ß)

2) sin(a – ß) = sin(a)cos(ß) – cos(a)sin(ß)

3) cos(a + ß) = cos(a)cos(ß) – sin(a)sin(ß)

4) cos(a – ß) = cos(a)cos(ß) + sin(a)sin(ß)

5) tan(a + ß) = [tan(a) + tan (ß)]/ [1 - tan(a)tan (ß)]

6)tan(a - ß) = [tan(a) - tan (ß)]/ [1 + tan (a) tan (ß)]

7) tan (π/4 + θ) = (1 + tan θ)/(1 - tan θ)

8) tan (π/4 - θ) = (1 - tan θ)/(1 + tan θ)

9) cot (a + ß) = [cot(a) . cot (ß) - 1]/ [cot (a) +cot (ß)]

10) cot (a - ß) = [cot(a) . cot (ß) + 1]/ [cot (ß) - cot (a)]

Double or Triple -Angle Identities

1) sin 2x = 2sin x cos x

2) cos2x = cos2x – sin2x = 1 – 2sin2x = 2cos2x – 1

3) tan 2x = 2 tan x / (1-tan 2x)

4) sin 3x = 3 sin x – 4 sin3x

5) cos3x = 4 cos3x – 3 cosx

6) tan 3x = (3 tan x - tan3x) / (1- 3tan 2x)

For angles A, B and C, we have

1) sin (A + B +C) = sinAcosBcosC + cosAsinBcosC + cosAcosBsinC - sinAsinBsinC

2) cos (A + B +C) = cosAcosBcosC- cosAsinBsinC - sinAcosBsinC - sinAsinBcosC

3) tan (A + B +C) = [tan A + tan B + tan C –tan A tan B tan C]/ [1- tan Atan B - tan B tan C –tan A tan C

4) cot (A + B +C) = [cot A cot B cot C – cotA - cot B - cot C]/ [cot A cot B + cot Bcot C + cot A cotC–1]


List of some other trigonometric formulas:

1) 2sinAcosB = sin(A + B) + sin (A - B)

2) 2cosAsinB = sin(A + B) - sin (A - B)

3) 2cosAcosB = cos(A + B) + cos(A - B)

4) 2sinAsinB = cos(A - B) - cos (A + B)

5) sin A + sin B = 2 sin [(A+B)/2] cos [(A-B)/2]

6) sin A - sin B = 2 sin [(A-B)/2] cos [(A+B)/2]

7) cosA + cos B = 2 cos [(A+B)/2] cos [(A-B)/2]

8) cosA - cos B = 2 sin [(A+B)/2] sin [(B-A)/2]

9) tanA ± tanB = sin (A ± B)/ cos A cos B

10)cot A ± cot B = sin (B ± A)/ sin A sin B

Method of solving a trigonometric equation:

1) If possible, reduce the equation in terms of any one variable, preferably x. Then solve the equation as you used to in case of a single variable.

2) Try to derive the linear/algebraic simultaneous equations from the given trigonometric equations and solve them as algebraic simultaneous equations.

3) At times, you might be required to make certain substitutions. It would be beneficial when the system has only two trigonometric functions.

Some results which are useful for solving trigonometric equations:
1) sin θ = sina and cosθ = cosa ⇒ θ = 2nπ + a

2) sin θ = 0 ⇒ θ = nπ

3) cosθ = 0 ⇒ θ = (2n + 1)π/2

4) tan θ = 0 ⇒ θ = nπ

5) sinθ = sina⇒ θ = nπ + (-1)na where a ∈ [–π/2, π/2]

6) cosθ= cos a ⇒ θ = 2nπ ± a, where a ∈[0,π]

7) tanθ = tana⇒ θ = nπ+ a, where a ∈[–π/2, π/2]

8) sinθ = 1 ⇒ θ= (4n + 1)π/2

9) sin θ = -1 ⇒ θ = (4n - 1) π /2

10) sin θ = -1 ⇒ θ = (2n +1) π /2

11) |sinθ| = 1⇒ θ =2nπ

12) cosθ = 1 ⇒ θ =(2n + 1)

13) |cosθ| = 1⇒ θ =nπ
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2021-07-29 17:04:40 Class 12 Board I Removed Questions and Exercises of
Physics Chemistry Mathematics 2021-22

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2021-07-29 10:27:35 GUYS END OF WAIT AT LAST OUR OFFICIAL MATERIALS BOT LAUNCHED WITH ALL NEW FACILITIES,MATERIALS AND ESPECIALLY REFER AND WIN SYSTEM

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2021-07-29 10:01:26 Properties of Cathode Rays
(i) Cathode rays are not electromagnetic rays.

(ii) Cathode rays are deflected by electric field and magnetic field.

(iii) Cathode rays produce heat in metals when they fall on them.

(iv) Cathode rays can pass through thin aluminium or gold foils without puncturing them.

(v) Cathode rays can produce physical and chemical change.

(vi) Cathode rays travel in straight line with high velocity
momentum and energy and cast shadow of objects placed in
their path.

(vii) On striking the target of high atomic weight and high melting point, they produce X-rays.

(viii) Cathode rays produce fluorescence and phosphorescence in
certain substance and hence affect photographic plate.

(ix) Specific charge of an electron was determined by JJ Thomson using perpendicular magnetic and electric field applied on a beam of electrons, at the same place.

(x) Specific charge of electron e/m = E^2/V^2B^2 where, E = electric field, B = magnetic field and V = potential difference applied across ends of tube.

(xi) The value of specific charge of an electron is1.7589 ×10^11 C/kg. Open / Comment

2021-07-29 08:59:41 Open / Comment

2021-07-29 08:34:18 https://drive.google.com/file/d/1IIOKJn0ofANCbkB0YBLEypXpirv7bVGB/view?usp=sharing

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class 12 physics mind maps

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2021-07-29 08:22:12 Important Points For NEET

Evidences from comparative morphology and anatomy :-

Examples of homologous organs:

(i) Forelimbs of mammals
(ii) Thorn of Bougainvillea and tendril of Cucurbita both are modification of axillary bud.
(iii)Vertebrate hearts or brains
(iv)Mouth parts of insects :
Cockroach Honey bee Mosquito
(Biting & chewing) (Chewing & lapping) (Piercing & Sucking)
(v) Testes in male and ovaries in female
(vi) Potato and Ginger - both are modified shoot
(vii) Radish and Carrot - both are modified roots

Examples of Analogous organs :

(i) Wings of butterfly and birds
(ii) Eye of the octopus and of mammals
(iii) Flippers of Penguins and Dolphins
(iv) Sweet potato (root modification) and potato (stem modification)
(v) Sting of bee and scorpion
(vi) Chloragogen cells of earthworm and liver of vertebrates

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2021-07-29 08:05:34 https://drive.google.com/file/d/12d9yZ6tel2kT_DPWhe3c3HmqhuGd1KFC/view?usp=sharing
PHYSICS MINDMAPS

CLASS 11

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