Basic Maths( yhi complete phyiscs ka base h good luck for your journey from satrting )

BASIC MATHS

Basic Maths( yhi complete phyiscs ka base h good luck for your journey from satrting )

# Pythagoras theorem

$$ (P)^{2}+(B)^{2}=(H)^{2} $$

The following table:

SCT
PBP
HHB

  • Sin $37^{\circ}\rightarrow3/5$
  • Cos $37^{\circ}\rightarrow4/5$
  • tan $37^{\circ}\rightarrow3/4$
  • Sin $53^{\circ}\rightarrow4/5$
  • Cos $53^{\circ}\rightarrow3/5$
  • tan $53^{\circ}\rightarrow4/3$

# Conversion

Conversion from radian to degree ki Jagah 180°

  • (i) $\frac{\pi}{2}rad\rightarrow\frac{3\times180}{2}=270^{\circ}$
  • (ii) $\frac{4\pi}{3}rad\rightarrow\frac{4\times180}{3}=240^{\circ}$

# Conversion of degree into Radian !

  • (i) $60^{\circ}\rightarrow60\times\frac{\pi}{180}=\frac{\pi}{3}rad$
  • (ii) $90^{\circ}\rightarrow90\times\frac{\pi}{180}=\frac{\pi}{2}rad$

# Small angle approximation.

$$ \Rightarrow sin~\theta\approx\theta $$

$\frac{\pi}{180}$ Se Multiply kardo
(If $\theta$ is very small) toh radian
$\theta<5^{\circ}$ (Approx. Equal)
Equal toh $\theta=0^{\circ}$ Pe honge.
30° tak laga sakte

$\Rightarrow tan~\theta$ (If $\theta$ is very small) $\theta<10^{\circ}$

$\Rightarrow$ If $\theta$ is very small $(\theta<10^{\circ})$
$$ sin~\theta\approx tan~\theta\approx\theta $$

Trignometry Identities

  • • $sin^{2}\theta+cos^{2}\theta=1$
  • • $Sin(A+B)=Sin~A~cos~B+cos~A~sin~B$
  • • $Sin(A-B)=Sin~A~cos~B-cos~A~Sin~B$
  • • $cos(A+B)=cos~A~cos~B-sin~A~sin~B$
  • • $cos(A-B)=cos~A~cos~B+sin~A~sin~B$
  • • $tan(A+B)=\frac{tan~A+tan~B}{1-tan~A~tan~B}$
  • • $tan(A-B)=\frac{tan~A-tan~B}{1+tan~A~tan~B}$
  • • $1+tan^{2}\theta=sec^{2}\theta$
  • • $1+cot^{2}\theta=cosec^{2}\theta$
  • • $Sin~2~\theta=2~sin~\theta~cos~\theta$
  • • $cos~2~\theta=cos^{2}\theta-sin^{2}\theta = 1-2~sin^{2}\theta = 2~cos^{2}\theta-1$

FOUR QUADRANT

0° / 360° 90° 180° 270° 1st All (+ve) 2nd Sin (+ve) Cosec 3rd tan (+ve) Cot 4th Cos (+ve) Sec

# If $\theta$ is very small
$sin~\theta=\theta=tan~\theta$
$cos~\theta=\sqrt{1-\theta^{2}}$ (If $\theta$ is very small $cos~\theta \approx 1$)

  1. Sabse Pehle angle ko 180±, 360± me tod do.
  2. Quad Pata kar k +/- Lagao.
  3. (180±, 360±) bhul Jaao.

Agar humne $90^{\circ}$, $270^{\circ}$ me toda:
$sin \leftrightarrow cos$
$tan \leftrightarrow cot$
$Cosec \leftrightarrow Sec$

Example

  • 1. $sin~150^{\circ}=sin(180-30)=+\frac{1}{2}$
  • 2. $cos~210^{\circ}=cos(180+30)=-\frac{\sqrt{3}}{2}$
  • 3. $sin~330^{\circ}=sin(360-30)=-\frac{1}{2}$

Asin $sin(360+\theta)=sin~\theta$
$cos(360+\theta)=cos~\theta$
$tan(360+\theta)=tan~\theta$

e.g. $sin~405^{\circ}=Sin(360+45) = +sin~45^{\circ} = \frac{1}{\sqrt{2}}$

$sin(-\theta)=-sin~\theta$

$cos(-\theta)=cos~\theta$

$tan(-\theta)=-tan~\theta$

$(sin~\theta)_{max}=+1$

$(sin~\theta)_{min}=-1$

$(cos~\theta)_{max}=+1$

$(cos~\theta)_{min}=-1$

$(sin^{2}\theta)_{max}=1$

$(sin^{2}\theta)_{min}=0$

$(cos^{2}\theta)_{max}=1$

$(cos^{2}\theta)_{min}=0$

$\rightarrow$ From $0^{\circ}$ to $90^{\circ}$

$\theta \uparrow = sin~\theta \uparrow$

$\theta \uparrow = tan~\theta \uparrow$

$\theta \uparrow = cos~\theta \downarrow$

Question

1. $y=2+3~sin~\theta$
$y_{max}=2+3\times1=5$
$y_{min}=2+3(-1)=-1$

3. $y=5+3tan~\theta$
$y_{max}=\infty$
$y_{min}=-\infty$

(i) $y=\frac{8}{5+3~sin~\theta}$
$y_{min}=\frac{8}{5+3}=1$
$y_{max}=\frac{8}{5-3}=4$

2. $y=3+4~cos\theta$
$y_{max}=3+4\times1=7$
$y_{min}=3-4=-1$

(ii) $y=\frac{10}{3+2~cos~\theta}$
$y_{max}=\frac{10}{3-2}=10$
$y_{min}=\frac{10}{3+2}=2$

# $y=a~sin~\theta+b~cos~\theta$

$y_{max}=\sqrt{a^{2}+b^{2}}$

$y_{min}=-\sqrt{a^{2}+b^{2}}$

Ques. $y=3~sin~\theta+4~cos~\theta$

$y_{max}=\sqrt{3^{2}+4^{2}}=5$
$y_{min}=-\sqrt{3^{2}+4^{2}}=-5$

Ques. Find value of $y_{max}$ and when $y=sin~3~\theta$

$y_{max}=1$
$3~\theta=90^{\circ} \implies \theta=30^{\circ}$

radian = 180 degree = 180x60 min = 180x60x60 sec

Ques CONVERT-

(i) 6 rad into min.
$\pi~rad=180~degree$
$\pi~rad=180\times60~min$
$1~rad=\frac{180\times60~min}{\pi}$
$6~rad=\frac{6\times10800~min}{\pi}$

(ii) 6 rad into second
$6~rad=\frac{6\times10800}{\pi}\times(60~sec)$

(iii) 6 degree into min
$1~degree=60~min$
$6~degree=6\times60~min=360~min$

(iv) 40 min into radian
$\pi~rad=180\times60~min$
$\pi~rad=10800~min$
$10800~min=\pi~rad$
$40~min=\frac{\pi}{10800}\times40~rad = \frac{\pi}{270}rad$

Ques. $\frac{sin~\theta+cos~\theta}{sin~\theta-cos~\theta}=\frac{9}{4}$. Find tan?

$\frac{sin~\theta}{cos~\theta}=\frac{9+4}{9-4}=\frac{13}{5}=tan~\theta$

# Distance between two point

2D: $A(x_{1}, y_{1})$ & $B(x_{2}, y_{2})$
$$ AB=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}} $$

3D: $A(x_{1}, y_{1}, z_{1})$ & $B(x_{2}, y_{2}, z_{2})$
$$ AB=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}} $$

Mid Point = $(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$

Distance of 'P' $(x,y,z)$ from origin:
$OA=\sqrt{x^{2}+y^{2}}$
$OA=\sqrt{x^{2}+y^{2}+z^{2}}$

Qus.

A(7,3), B(10,7)
$AB=\sqrt{(10-7)^{2}+(7-3)^{2}} = 5$

A(2,3,4), B(5,7,9)
$AB=\sqrt{(5-2)^{2}+(7-3)^{2}+(9-4)^{2}} = 5\sqrt{2}$

A(4,6), B(10,12). Mid Point C?
$C=(\frac{4+10}{2},\frac{6+12}{2}) = (7,9)$

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Basic Maths - Part 2

# Slope of Straight Line (Continued)

Equation of straight line

$$ y=mx+c $$

$m=slope$

$At~x=0$, $y=0+c \implies c = y$ intercept / $x=0$ Par 'y' ki value.

Slope of line joining A $(x_1, y_1)$ to B $(x_2, y_2)$:

$$ Slope=tan~\theta=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} $$

Ques. $y=\sqrt{3}x+10$

$m=\sqrt{3}=slope=tan~\theta \therefore \theta=60^{\circ}$

At $x=0$, $y=10 \implies c=10$

(0,10) 60°

Ques. $y=\frac{x}{\sqrt{3}}+5$

$m=\frac{1}{\sqrt{3}}=tan~\theta \therefore \theta=30^{\circ}$

At $x=0$, $y=5$ or $c=5$

(0,5) 30°

Qus. $y=x-10$

$m=1=tan~\theta=45^{\circ}$

(0,-10) 45°

Comparing Slopes

$\theta_{3} > \theta_{2} > \theta_{1}$

$tan~\theta_{3} > tan~\theta_{2} > tan~\theta_{1}$

$(Slope)_{3} > (Slope)_{2} > (Slope)_{1}$

3 2 1

More Equation Examples

# $3x+4y+5=0$
$4y=-3x-5 \implies y=-\frac{3}{4}x-\frac{5}{4}$
$Slope=-\frac{3}{4}$, $C=\frac{-5}{4}$

Ques. Find equation passing through (2,3) having slope +10.
$y=mx+c \implies 3=10(2)+c \implies C=-17$
$y=10x-17$

To Find equation if two points are given: (2,3) and (4,5)
$Slope = \frac{5-3}{4-2} = 1 = m$
$y=x+c \implies 3=2+c \implies c=1$
$\therefore y=x+1$

# Graphs & Curves

$y=x^{2}$

$y^{2}=x$

$y=-x^{2}$

$y^{2}=-x$

# $xy=c$ (e.g. $xy=4$) $\implies$ Rectangular Hyperbola

# Geometric Progression

General Form: $a, ar, ar^{2}, ar^{3}$

First term = $a$, $r$ = common ratio

$$ S_{\infty}=\frac{a}{1-r} $$ (For $|r| < 1$)

Ques. Find $1+\frac{1}{4}+\frac{1}{16}+\frac{1}{32}+....\infty$

$a=1$, $r=\frac{1}{4}$ $\implies S_{\infty}=\frac{1}{1-\frac{1}{4}}=\frac{4}{3}$

# Thermodynamics Graphs ($PV=nRT$)

V vs T Graph (At constant Pressure)

$V = (\frac{nR}{P})T \implies y=mx$

T V $\theta$

$tan~\theta = \frac{nR}{P}$

P vs V Graph (Isotherms)

V P $T_{1} > T_{2}$

# Photoelectric Effect Graphs

$E=\Phi+(KE)_{max} \implies hv=\Phi+eV_{0}$

$V_{0} = (\frac{h}{e})v - \frac{\Phi}{e}$

Comparing with $y=mx-c \implies Slope = \frac{h}{e}$

# Quadratic Equation

$$ ax^{2}+bx+c=0 $$

$b^{2}>4ac$ (two real roots)

$$ x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} $$

Ques. $x^{2}-6x+5=0$

$a=1, b=-6, c=5$

$x_{1}=\frac{-(-6)+\sqrt{36-4(1)(5)}}{2(1)} = \frac{6+4}{2} = 5$

$x_{2}=\frac{-(-6)-\sqrt{36-20}}{2} = \frac{6-4}{2} = 1$

# DIFFERENTIATION (Start)

$$ y=x^{n} \implies \frac{dy}{dx}=nx^{n-1} $$

  • $sin~x\rightarrow cos~x$
  • $cos~x\rightarrow -sin~x$
  • $tan~x\rightarrow sec^{2}x$
  • $e^{x}\rightarrow e^{x}$
  • Constant $\rightarrow 0$
  • $ln~x\rightarrow\frac{1}{x}$
  • $cot~x\rightarrow -cosec^{2}x$
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Basic Maths - Part 3

# Differentiation (Examples)

Tangent at a Point

Ques. $y=x^{2}$. Find $\frac{dy}{dx}$ at $x=\frac{1}{2}$

$\frac{dy}{dx}=2x \implies 2(\frac{1}{2})=1$

$\frac{dy}{dx}=1=slope=tan~\theta \implies \theta=45^{\circ}$

$\theta$ $x=\frac{1}{2}$ $\frac{1}{4}$

PRODUCT RULE

$y=u\cdot v \implies \frac{dy}{dx}=u\cdot\frac{dv}{dx}+v\cdot\frac{du}{dx}$

Q. $y=x^{2}\sin x \implies \frac{dy}{dx}=x^{2}\cdot\cos x + \sin x\cdot(2x)$

QUOTIENT RULE

$y=\frac{u}{v} \implies \frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}$

Q. $y=\frac{x^{2}}{\sin x} \implies \frac{dy}{dx}=\frac{(\sin x)(2x)-x^{2}\cos x}{(\sin x)^{2}}$

# Double Differentiation

1. $y=x^{3}+\sin x \implies y'=3x^{2}+\cos x \implies y''=6x-\sin x$

2. $y=x^{5}+e^{x}+\sin x \implies y'=5x^{4}+e^{x}+\cos x \implies y''=20x^{3}+e^{x}-\sin x$

# Chain Rule & Geometrical Meaning

1. $y=\sin(x^{3}) \implies \frac{dy}{dx}=\cos(x^{3})\times 3x^{2}$

2. $y=ln(\sin(x^{2}+2)) \implies y'=\frac{1}{\sin(x^{2}+2)}\times\cos(x^{2}+2)\times(2x)$

3. $y=A\sin(\omega t+\phi) \implies \frac{dy}{dt}=A\omega\cos(\omega t+\phi)$

kisi bhi curve k kisi bhi point Par dy/dx ka matlab hai us point Par tangent ka slope.

A 60°

$\frac{dy}{dx}$ at A = slope of tangent at A = $\tan 60^{\circ} = \sqrt{3}$

Binomial Expansion

If $x \ll 1$

$$ (1+x)^{n} \approx 1+nx $$

1. $(1.001)^{3} = (1+.001)^{3} = 1+3\times(.001) = 1.003$

Exponential Decay Graph

$y=e^{-x}$ (for $x \ge 0$)

# Maxima & Minima

Q. $y=x^{3}-3x^{2}+6x$, Find $y_{min}$ & $y_{max}$

$\frac{dy}{dx}=3x^{2}-6x=0 \implies 3x(x-2)=0 \implies x=0, 2$

$\frac{d^{2}y}{dx^{2}}=6x-6$

At $x=0 \rightarrow 6(0)-6=-6$ (Negative, Maxima). $y_{max}=0-0+0=6$

At $x=2 \rightarrow 6(2)-6=6$ (Positive, Minima). $y_{min}=2^{3}-3(2)^{2}+6(2)=2$

# Integration

  • $\int x^{n}dx=\frac{x^{n+1}}{n+1}+c$
  • $\int\frac{1}{x}dx=ln~x+c$
  • $\int e^{x}dx=e^{x}+c$
  • $\int\cos x~dx=\sin x+c$
  • $\int\sin x~dx=-\cos x+c$
  • $\int\sec^{2}x~dx=\tan x+c$

Definite Integral

1. $\int_{2}^{4}xdx=[\frac{x^{2}}{2}]_{2}^{4}=[\frac{16}{2}-\frac{4}{2}]=8-2=6$

2. $\int_{0}^{\pi}\sin\theta~d\theta=[-\cos\theta]_{0}^{\pi}=-(\cos\pi-\cos 0^{\circ})=-(-1-1)=2$

Substitution Form

$\int\sin(ax+b)dx=\frac{-\cos(ax+b)}{a}+c$

$\int e^{ax+b}dx=\frac{e^{ax+b}}{a}+c$

Trigonometry Table

$\theta$ 0° 30° 45° 60° 90° 120° 135° 150° 180°
Sin 0 1/2 $1/\sqrt{2}$ $\sqrt{3}/2$ 1 $\sqrt{3}/2$ $1/\sqrt{2}$ 1/2 0
Cos 1 $\sqrt{3}/2$ $1/\sqrt{2}$ 1/2 0 -1/2 $-1/\sqrt{2}$ $-\sqrt{3}/2$ -1
Tan 0 $1/\sqrt{3}$ 1 $\sqrt{3}$ $\infty$ $-\sqrt{3}$ -1 $-1/\sqrt{3}$ 0

Sinx Graph (Integration Area)

+1 -1 $\pi$ $2\pi$

$\int_{0}^{\pi/2}\sin x~dx=1$ | $\int_{0}^{\pi}\sin x~dx=2$

Cosx Graph (Integration Area)

+1 -1 $\pi/2$ $3\pi/2$

$\int_{0}^{\pi}\cos x~dx=1-1=0$

Integration of $\sin^{2}x$ & $\cos^{2}x$

Formula used: $\cos 2x=1-2\sin^{2}x \implies \sin^{2}x=\frac{1}{2}(1-\cos 2x)$

$\int\sin^{2}xdx=\int\frac{1}{2}(1-\cos 2x)dx = \frac{x}{2}-\frac{\sin 2x}{4}+c$

$\int\cos^{2}xdx=\int\frac{1}{2}(1+\cos 2x)dx = \frac{x}{2}+\frac{\sin 2x}{4}+c$

# LOGARITHM

$log_{b}a=c \implies a=b^{c}$

Natural log $(ln)$: $ln~x=y \implies log_{e}x=y \implies x=e^{y}$

Example: $log_{2}8=3 \implies 2^{3}=8$

$log_{a}a=1$ | $ln~e=1$ | $log_{10}10=1$

1. $log~m+log~n=log(m\cdot n)$

2. $log(\frac{m}{n})=log~m-log~n$

3. $log_{b}a^{n}=n~log_{b}a$

Q. $log~8-log~4+log~2 = log(\frac{8}{4}\times 2)=log~4$

Q. $log_{5}125=log_{5}5^{3}=3~log_{5}5=3$

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