# 0.5 Mathematical formulas

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If $a{x}^{2}+bx+c=0,$ then $x=\frac{\text{−}b±\sqrt{{b}^{2}-4ac}}{2a}$

Geometry
Triangle of base $b$ and height $h$ Area $=\frac{1}{2}bh$
Circle of radius $r$ Circumference $=2\pi r$ Area $=\pi {r}^{2}$
Sphere of radius $r$ Surface area $=4\pi {r}^{2}$ Volume $=\frac{4}{3}\pi {r}^{3}$
Cylinder of radius $r$ and height $h$ Area of curved surface $=2\pi rh$ Volume $=\pi {r}^{2}h$

Trigonometry

Trigonometric Identities

1. $\text{sin}\phantom{\rule{0.2em}{0ex}}\theta =1\text{/}\text{csc}\phantom{\rule{0.2em}{0ex}}\theta$
2. $\text{cos}\phantom{\rule{0.2em}{0ex}}\theta =1\text{/}\text{sec}\phantom{\rule{0.2em}{0ex}}\theta$
3. $\text{tan}\phantom{\rule{0.2em}{0ex}}\theta =1\text{/}\text{cot}\phantom{\rule{0.2em}{0ex}}\theta$
4. $\text{sin}\left({90}^{0}-\theta \right)=\text{cos}\phantom{\rule{0.2em}{0ex}}\theta$
5. $\text{cos}\left({90}^{0}-\theta \right)=\text{sin}\phantom{\rule{0.2em}{0ex}}\theta$
6. $\text{tan}\left({90}^{0}-\theta \right)=\text{cot}\phantom{\rule{0.2em}{0ex}}\theta$
7. ${\text{sin}}^{2}\phantom{\rule{0.2em}{0ex}}\theta +{\text{cos}}^{2}\phantom{\rule{0.2em}{0ex}}\theta =1$
8. ${\text{sec}}^{2}\phantom{\rule{0.2em}{0ex}}\theta -{\text{tan}}^{2}\phantom{\rule{0.2em}{0ex}}\theta =1$
9. $\text{tan}\phantom{\rule{0.2em}{0ex}}\theta =\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \text{/}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta$
10. $\text{sin}\left(\alpha ±\beta \right)=\text{sin}\phantom{\rule{0.2em}{0ex}}\alpha \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\beta ±\text{cos}\phantom{\rule{0.2em}{0ex}}\alpha \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\beta$
11. $\text{cos}\left(\alpha ±\beta \right)=\text{cos}\phantom{\rule{0.2em}{0ex}}\alpha \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\beta \mp \text{sin}\phantom{\rule{0.2em}{0ex}}\alpha \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\beta$
12. $\text{tan}\left(\alpha ±\beta \right)=\frac{\text{tan}\phantom{\rule{0.2em}{0ex}}\alpha ±\text{tan}\phantom{\rule{0.2em}{0ex}}\beta }{1\mp \text{tan}\phantom{\rule{0.2em}{0ex}}\alpha \phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}\beta }$
13. $\text{sin}\phantom{\rule{0.2em}{0ex}}2\theta =2\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta$
14. $\text{cos}\phantom{\rule{0.2em}{0ex}}2\theta ={\text{cos}}^{2}\phantom{\rule{0.2em}{0ex}}\theta -{\text{sin}}^{2}\phantom{\rule{0.2em}{0ex}}\theta =2\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{2}\phantom{\rule{0.2em}{0ex}}\theta -1=1-2\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{2}\phantom{\rule{0.2em}{0ex}}\theta$
15. $\text{sin}\phantom{\rule{0.2em}{0ex}}\alpha +\text{sin}\phantom{\rule{0.2em}{0ex}}\beta =2\phantom{\rule{0.2em}{0ex}}\text{sin}\frac{1}{2}\left(\alpha +\beta \right)\text{cos}\frac{1}{2}\left(\alpha -\beta \right)$
16. $\text{cos}\phantom{\rule{0.2em}{0ex}}\alpha +\text{cos}\phantom{\rule{0.2em}{0ex}}\beta =2\phantom{\rule{0.2em}{0ex}}\text{cos}\frac{1}{2}\left(\alpha +\beta \right)\text{cos}\frac{1}{2}\left(\alpha -\beta \right)$

Triangles

1. Law of sines: $\frac{a}{\text{sin}\phantom{\rule{0.2em}{0ex}}\alpha }=\frac{b}{\text{sin}\phantom{\rule{0.2em}{0ex}}\beta }=\frac{c}{\text{sin}\phantom{\rule{0.2em}{0ex}}\gamma }$
2. Law of cosines: ${c}^{2}={a}^{2}+{b}^{2}-2ab\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\gamma$
3. Pythagorean theorem: ${a}^{2}+{b}^{2}={c}^{2}$

Series expansions

1. Binomial theorem: ${\left(a+b\right)}^{n}={a}^{n}+n{a}^{n-1}b+\frac{n\left(n-1\right){a}^{n-2}{b}^{2}}{2\text{!}}+\frac{n\left(n-1\right)\left(n-2\right){a}^{n-3}{b}^{3}}{3\text{!}}+\text{···}$
2. ${\left(1±x\right)}^{n}=1±\frac{nx}{1\text{!}}+\frac{n\left(n-1\right){x}^{2}}{2\text{!}}±\text{···}\left({x}^{2}<1\right)$
3. ${\left(1±x\right)}^{\text{−}n}=1\mp \frac{nx}{1\text{!}}+\frac{n\left(n+1\right){x}^{2}}{2\text{!}}\mp \text{···}\left({x}^{2}<1\right)$
4. $\text{sin}\phantom{\rule{0.2em}{0ex}}x=x-\frac{{x}^{3}}{3\text{!}}+\frac{{x}^{5}}{5\text{!}}-\text{···}$
5. $\text{cos}\phantom{\rule{0.2em}{0ex}}x=1-\frac{{x}^{2}}{2\text{!}}+\frac{{x}^{4}}{4\text{!}}-\text{···}$
6. $\text{tan}\phantom{\rule{0.2em}{0ex}}x=x+\frac{{x}^{3}}{3}+\frac{2{x}^{5}}{15}+\text{···}$
7. ${e}^{x}=1+x+\frac{{x}^{2}}{2\text{!}}+\text{···}$
8. $\text{ln}\left(1+x\right)=x-\frac{1}{2}{x}^{2}+\frac{1}{3}{x}^{3}-\text{···}\left(|x|<1\right)$

Derivatives

1. $\frac{d}{dx}\left[af\left(x\right)\right]=a\frac{d}{dx}f\left(x\right)$
2. $\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}f\left(x\right)+\frac{d}{dx}g\left(x\right)$
3. $\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}g\left(x\right)+g\left(x\right)\frac{d}{dx}f\left(x\right)$
4. $\frac{d}{dx}f\left(u\right)=\left[\frac{d}{du}f\left(u\right)\right]\frac{du}{dx}$
5. $\frac{d}{dx}{x}^{m}=m{x}^{m-1}$
6. $\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}x=\text{cos}\phantom{\rule{0.2em}{0ex}}x$
7. $\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}x=\text{−}\text{sin}\phantom{\rule{0.2em}{0ex}}x$
8. $\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}x={\text{sec}}^{2}\phantom{\rule{0.2em}{0ex}}x$
9. $\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{cot}\phantom{\rule{0.2em}{0ex}}x=\text{−}{\text{csc}}^{2}\phantom{\rule{0.2em}{0ex}}x$
10. $\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{sec}\phantom{\rule{0.2em}{0ex}}x=\text{tan}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sec}\phantom{\rule{0.2em}{0ex}}x$
11. $\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{csc}\phantom{\rule{0.2em}{0ex}}x=\text{−}\text{cot}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{csc}\phantom{\rule{0.2em}{0ex}}x$
12. $\frac{d}{dx}{e}^{x}={e}^{x}$
13. $\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}x=\frac{1}{x}$
14. $\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{-1}\phantom{\rule{0.2em}{0ex}}x=\frac{1}{\sqrt{1-{x}^{2}}}$
15. $\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{-1}x=-\frac{1}{\sqrt{1-{x}^{2}}}$
16. $\frac{d}{dx}\phantom{\rule{0.2em}{0ex}}{\text{tan}}^{-1}x=-\frac{1}{1+{x}^{2}}$

Integrals

1. $\int af\left(x\right)dx=a\int f\left(x\right)dx$
2. $\int \left[f\left(x\right)+g\left(x\right)\right]dx=\int f\left(x\right)dx+\int g\left(x\right)dx$
3. $\begin{array}{cc}\hfill \int {x}^{m}dx& =\frac{{x}^{m+1}}{m+1}\phantom{\rule{0.2em}{0ex}}\left(m\ne \text{−}1\right)\hfill \\ & =\text{ln}\phantom{\rule{0.2em}{0ex}}x\left(m=-1\right)\hfill \end{array}$
4. $\int \text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}dx=\text{−}\text{cos}\phantom{\rule{0.2em}{0ex}}x$
5. $\int \text{cos}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}dx=\text{sin}\phantom{\rule{0.2em}{0ex}}x$
6. $\int \text{tan}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}dx=\text{ln}|\text{sec}\phantom{\rule{0.2em}{0ex}}x|$
7. $\int {\text{sin}}^{2}\phantom{\rule{0.2em}{0ex}}ax\phantom{\rule{0.2em}{0ex}}dx=\frac{x}{2}-\frac{\text{sin}\phantom{\rule{0.2em}{0ex}}2ax}{4a}$
8. $\int {\text{cos}}^{2}\phantom{\rule{0.2em}{0ex}}ax\phantom{\rule{0.2em}{0ex}}dx=\frac{x}{2}+\frac{\text{sin}\phantom{\rule{0.2em}{0ex}}2ax}{4a}$
9. $\int \text{sin}\phantom{\rule{0.2em}{0ex}}ax\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}ax\phantom{\rule{0.2em}{0ex}}dx=-\frac{\text{cos}2ax}{4a}$
10. $\int {e}^{ax}\phantom{\rule{0.2em}{0ex}}dx=\frac{1}{a}{e}^{ax}$
11. $\int x{e}^{ax}dx=\frac{{e}^{ax}}{{a}^{2}}\left(ax-1\right)$
12. $\int \text{ln}\phantom{\rule{0.2em}{0ex}}ax\phantom{\rule{0.2em}{0ex}}dx=x\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}ax-x$
13. $\int \frac{dx}{{a}^{2}+{x}^{2}}=\frac{1}{a}\phantom{\rule{0.2em}{0ex}}{\text{tan}}^{-1}\frac{x}{a}$
14. $\int \frac{dx}{{a}^{2}-{x}^{2}}=\frac{1}{2a}\phantom{\rule{0.2em}{0ex}}\text{ln}|\frac{x+a}{x-a}|$
15. $\int \frac{dx}{\sqrt{{a}^{2}+{x}^{2}}}={\text{sinh}}^{-1}\frac{x}{a}$
16. $\int \frac{dx}{\sqrt{{a}^{2}-{x}^{2}}}={\text{sin}}^{-1}\frac{x}{a}$
17. $\int \sqrt{{a}^{2}+{x}^{2}}\phantom{\rule{0.2em}{0ex}}dx=\frac{x}{2}\sqrt{{a}^{2}+{x}^{2}}+\frac{{a}^{2}}{2}\phantom{\rule{0.2em}{0ex}}{\text{sinh}}^{-1}\frac{x}{a}$
18. $\int \sqrt{{a}^{2}-{x}^{2}}\phantom{\rule{0.2em}{0ex}}dx=\frac{x}{2}\sqrt{{a}^{2}-{x}^{2}}+\frac{{a}^{2}}{2}\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{-1}\frac{x}{a}$

عند قذف جسم إلى أعلى بسرعة إبتدائية فإنه سيصل إلى ارتفاع معين (أقصى ارتفاع) ثم يعود هابطاً نحو سطح الأرض .   إذا قُذِفَ جسم إلى أعلى ووجد أن سرعته 18 م / ث عندما قطع 1/4 المسافة التي تمثل أقصى ارتفاع سيصله فالمطلوب إيجاد السرعة التي قُذِف بها بالمتر / ث . إن هذه السرعة هي واحدة من الإجابات التالية
what is light
light is a kind of radiation That stimulates sight brightness a source of illumination.
kenneth
Electromagnet radiation creates space 7th, 8th, and 9th dimensions at the rate of c.
John
That is the reason that the speed of light is constant.
John
This creation of new space is "Dark Energy".
John
The first two sets of three dimensions, 1 through 6, are "Dark Matter".
John
As matter decays into luminous matter, a proton, a neutron, and an electron creat deuterium.
John
There are three sets of three protons, 9.
John
There are three sets of three neutrons, 9.
John
A free neutron decays into a proton, an electron, and a neutrino.
John
There are three sets of five neutrinoes, 15.
John
Neutrinoes are two dimensional.
John
A positron is composed of the first three dimensions.
John
An electron is composed of the second three dimensions.
John
What is photoelectric
light energy (photons) through semiconduction of N-P junction into electrical via excitation of silicon purified and cristalized into wafers with partially contaminated silicon to allow this N-P function to operate.
Michael
i.e. Solar pannel.
Michael
If you lie on a beach looking at the water with your head tipped slightly sideways, your polarized sunglasses do not work very well.Why not?
it has everything to do with the angle the UV sunlight strikes your sunglasses.
Jallal
this is known as optical physics. it describes how visible light, ultraviolet light and infrared light interact when they come into contact with physical matter. usually the photons or light upon interaction result in either reflection refraction diffraction or interference of the light.
Jallal
I hope I'm clear if I'm not please tell me to clarify further or rephrase
Jallal
what is bohrs model for hydrogen atom
hi
Tr
Hello
Youte
Hi
Nwangwu-ike
hi
Siddiquee
hi
Omar
helo
Mcjoi
what is the value of speed of light
1.79×10_¹⁹ km per hour
Swagatika
what r dwarf planet
what is energy
কাজের একক কী
Jasim
কাজের একক কী
Jasim
Energy is ability so capacity to do work.
kenneth
friction ka direction Kaise pata karte hai
friction is always in the opposite of the direction of moving object
Punia
A twin paradox in the special theory of relativity arises due to.....? a) asymmetric of time only b) symmetric of time only c) only time
b) symmetric of time only
Swagatika
fundamental note of a vibrating string
every matter made up of particles and particles are also subdivided which are themselves subdivided and so on ,and the basic and smallest smallest smallest division is energy which vibrates to become particles and thats why particles have wave nature
Alvin
what are matter waves? Give some examples
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
Manikant
a centeral part of theory of quantum mechanics example:just like a beam of light or a water wave
Swagatika
Mathematical expression of principle of relativity
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave
What is the importance of de-broglie's wavelength?
he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
Zahid
it is key of quantum
Manikant