Approximations

Irrational

Ratios and Angles

Basic Formulae

$\begin{aligned} \sin^2\theta + \cos^2\theta & = 1 \\ \tan^2\theta - \sec^2\theta & = 1 \\ \csc^2\theta - \cot^2\theta & = 1 \end{aligned}$

$\begin{aligned} \sin x & = \cos \left(90^{\circ }-x\right) & = {\dfrac {1}{\csc x}} \\ \cos x & = \sin \left(90^{\circ }-x\right) & = {\dfrac {1}{\sec x}} \\ \tan x & = \cot \left(90^{\circ }-x\right) & = {\dfrac {\sin x}{\cos x}} = {\dfrac {1}{\cot x}} \end{aligned}$

A system of rectangular coordinate axes divides a plane into four quadrants. An angle $\theta$ lies in one and only one of these quadrants.
The change and sign of the trigonometric ratios in the various quadrants are shown in Fig-1 below.

Sum and Difference

Two Angle

$\begin{aligned} \sin(A\pm B) & = \sin A\cos B\pm \cos A\sin B \\[3mu] \cos(A\pm B) & = \cos A\cos B\mp \sin A\sin B \end{aligned}$

$\begin{aligned} \tan\left(A\pm B\right) & = \dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B} \\[10mu] \cot\left(A\pm B\right) & = \dfrac{\cot A\cot B\mp 1}{\cot B\pm \cot A} \end{aligned}$

$\begin{aligned} \sin\left(A+B\right)\sin\left(A-B\right) && = \sin^2A-\sin^2B && = \cos^2B-\cos^2A \\ \cos\left(A+B\right)\cos\left(A-B\right) && = \cos^2A-\sin^2B && = \cos^2B-\sin^2A \end{aligned}$

Three Angle

$\begin{aligned} \sin\left(A+B+C\right) & = \sin A\cos B\cos C+\cos A\sin B\cos C+\cos A\cos B\sin C-\sin A\sin B\sin C \\[3mu] & = \cos A\cos B\cos C\left(\tan A+\tan B+\tan C-\tan A\tan B\tan C\right) \end{aligned}$

$\begin{aligned} \cos\left(A+B+C\right) & = \cos A\cos B\cos C-\sin A\sin B\cos C-\sin A\cos B\sin C-\cos A\sin B\sin C \\[3mu] & = \cos A\cos B\cos C(1-\tan A\tan B-\tan B\tan C-\tan C\tan A) \end{aligned}$

$\begin{aligned} \tan\left(A+B+C\right) & = \dfrac{\tan A+\tan B+\tan C-\tan A\tan B\tan C}{1-\tan A\tan B-\tan B\tan C-\tan C\tan A} \\[12mu] \cot\left(A+B+C\right) & = \dfrac{\cot A+\cot B+\cot C-\cot A\cot B\cot C}{1-\cot A\cot B-\cot B\cot C-\cot C\cot A} \end{aligned}$

General

$\begin{aligned} S_{0} & = 1 \\ S_{1} & = \sum\limits_{i} x_{i} && = \sum\limits_{i} \tan \theta_{i} \\ S_{2} & = \sum\limits_{i < j} x_{i} x_{j} && = \sum\limits_{i < j} \tan \theta_{i} \tan \theta_{j} \\ S_{3} & = \sum\limits_{i < j < k} x_{i} x_{j} x_{k} && = \sum\limits_{i < j < k} \tan \theta_{i} \tan \theta_{j} \tan \theta_{k} \\ & \quad \vdots && \quad \vdots \end{aligned}$

Then,

$\begin{aligned} \tan\left(\theta_1+\theta_2+\cdots+\theta_n\right) & = \dfrac{S_1-S_3+S_5-S_7+\cdots}{1-S_2+S_4-S_6+\cdots} \\[16mu] \sec\left(\theta_1+\theta_2+\cdots+\theta_n\right) & = {\dfrac {\sec \theta_1 \sec \theta_2 \cdots \sec \theta_n}{S_{0}-S_{2}+S_{4}-\cdots }} \\[16mu] \csc\left(\theta_1+\theta_2+\cdots+\theta_n\right) & = {\dfrac {\sec \theta_1 \sec \theta_2 \cdots \sec \theta_n}{S_{1}-S_{3}+S_{5}-\cdots }} \end{aligned}$

$\cos2^r\theta\cdot\cos2^{r+1}\theta\cdot\cos2^{r+2}\theta\cdots\cos2^n\theta= \dfrac{\sin2^{n+1}\theta}{2^{n-r+1}\sin2^r\theta}$

$\begin{aligned} \sin(\alpha)+\sin(\alpha+\beta)+\sin(\alpha+2\beta)+\cdots+\sin\left(\alpha+n\beta\right) & = \dfrac{\sin \left((n+1)\frac{\beta}{2}\right)}{\sin\left(\frac{\beta}{2}\right)}\sin\left(\alpha+n\frac{\beta}{2}\right) \\[16mu] \cos(\alpha)+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+\cdots+\cos\left(\alpha+n\beta\right) & = \dfrac{\sin\left((n+1)\frac{\beta}{2}\right)}{\sin\left(\frac{\beta}{2}\right)}\cos\left(\alpha+n\frac{\beta}{2}\right) \end{aligned}$

$1+2\cos \theta+2\cos(2\theta)+2\cos(3\theta)+\cdots +2\cos(n\theta)={\dfrac {\sin \left(\left(n+{\frac {1}{2}}\right)\theta\right)}{\sin \left({\frac {\theta}{2}}\right)}}$

Generally not true! only for common values of ‘$\theta$’. USE WITH CAUTION

$\dfrac{\sin{\theta_1} \pm \sin{\theta_2} + \sin{\theta_3} + \cdots + \sin{\theta_n}}{\cos{\theta_1} \pm \cos{\theta_2} + \cos{\theta_3} + \cdots + \cos{\theta_n}} = \tan{\dfrac{\theta_1 \pm \theta_2 + \theta_3 + \cdots + \theta_n}{n} }$

Product to Sum

$\begin{aligned} 2\sin A\cos B & = \sin\left(A+B\right)+\sin\left(A-B\right) \\[6mu] 2\cos A\sin B & = \sin\left(A+B\right)-\sin\left(A-B\right) \\[6mu] 2\cos A\cos B & = \cos\left(A+B\right)+\cos\left(A-B\right) \\[6mu] 2\sin A\sin B & = \cos\left(A-B\right)-\cos\left(A+B\right) \end{aligned}$

Sum to Product

$\begin{aligned} \sin C+\sin D & = 2\sin\frac{C+D}2\cos\frac{C-D}2 \\[8mu] \sin C-\sin D & = 2\cos\frac{C+D}2\sin\frac{C-D}2 \\[8mu] \cos C+\cos D & = 2\cos\frac{C+D}2\mathrm{cos}\frac{C-D}2 \\[8mu] \cos C-\cos D & = -2\sin\frac{C+D}2\sin\frac{C-D}2 \end{aligned}$

Multiple Angle

Two/Three Angle

$\begin{aligned} \sin2\theta& = 2\sin \theta\cos \theta \\[3mu] & = \dfrac{2\tan \theta}{1+\tan^2\theta} \end{aligned}$

$\begin{aligned} \cos2\theta& = \cos^2\theta-\sin^2\theta \\[3mu] & = 2\cos^2\theta-1 \\[3mu] & = 1-2\sin^2\theta \\[3mu] & = \dfrac{1-\tan^2\theta}{1+\tan^2\theta} \end{aligned}$

$\tan2\theta= \dfrac{2\tan\theta}{1-\tan^2\theta}$

$\begin{aligned} \sin3\theta & = 3\sin \theta-4\sin^3\theta \\[3mu] & = 4\sin\left(60^\circ-\theta\right)\sin (\theta)\sin\left(60^\circ+\theta\right) \end{aligned}$

$\begin{aligned} \cos3\theta & = 4\cos^3\theta-3\cos \theta \\[3mu] & = 4\cos\left(60^\circ-\theta\right)\cos (\theta)\cos\left(60^\circ+\theta\right) \end{aligned}$

$\begin{aligned} \tan3\theta & = \dfrac{3\tan \theta-\tan^3\theta}{1-3\tan^2\theta} \\[3mu] & = \tan\left(60^{\circ}-\theta\right)\tan (\theta)\tan\left(60^{\circ}+\theta\right) \end{aligned}$

Multi(>3) Angle

$\begin{aligned} \sin4\theta & = 4\sin \theta\cos^3 \theta - 4\cos \theta\sin^3 \theta \\ \cos4\theta & = 8\cos^4\theta-8\cos^2\theta+1 \\ \tan4\theta & = \dfrac{4\tan\theta-4\tan^3\theta}{1-6\tan^2\theta+\tan^4\theta} \\ \sin5\theta & = 16\sin^5\theta-20\sin^3\theta+5\sin \theta \\ \cos5\theta & = 16\cos^5\theta-20\cos^3\theta+5\cos \theta \end{aligned}$

Half Angle

$\begin{aligned} \sin {\dfrac {\theta }{2}} & = \operatorname {sgn} \left(\sin {\dfrac {\theta }{2}}\right){\sqrt {\dfrac {1-\cos \theta }{2}}} \\ \cos {\dfrac {\theta }{2}} & = \operatorname {sgn} \left(\cos {\dfrac {\theta }{2}}\right){\sqrt {\dfrac {1+\cos \theta }{2}}} \end{aligned}$

$\begin{aligned} \tan {\dfrac {\theta }{2}} & ={\frac {1-\cos \theta }{\sin \theta }}\\[8mu] & ={\frac {\sin \theta }{1+\cos \theta }}\\[8mu] & =\csc \theta -\cot \theta \\[1mu] & ={\frac {\tan \theta }{1+\sec {\theta }}}\\[1mu] & =\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}} \end{aligned}$

Special Cases

$\tan {\dfrac {\eta \pm \theta }{2}}={\dfrac {\sin \eta \pm \sin \theta }{\cos \eta +\cos \theta }}$

$\tan \left({\dfrac {\theta }{2}}+{\dfrac {\pi }{4}}\right)=\sec \theta +\tan \theta$
${\sqrt {\dfrac {1-\sin \theta }{1+\sin \theta }}}=\left|{\dfrac {1-\tan {\dfrac {\theta }{2}}}{1+\tan {\dfrac {\theta }{2}}}}\right|$

Power Reduction

Special Cases

If $A+B+C=180^{\circ}=\pi$ , then

$\begin{aligned} \sin2A+\sin2B+\sin2C & = 4\sin A\sin B\sin C \\ \sin2A+\sin2B-\sin2C & = 4\cos A\cos B\sin C \\ \cos2A+\cos2B+\cos2C & = -1-4\cos A\cos B\cos C \\ \cos2A+\cos2B-\cos2C & = 1-4\sin A\sin B\cos C \\[20mu] \sin A+\sin B+\sin C & =4\cos\frac A2\cos\frac B2\cos\frac C2 \\ \sin A+\sin B-\sin C & =4\sin\frac A2\sin\frac B2\cos\frac C2 \\ \cos A+\cos B+\cos C & =1+4\sin\frac A2\sin\frac B2\sin\frac C2 \\ \cos A+\cos B-\cos C & =-1+4\cos\frac A2\cos\frac B2\sin\frac C2 \\[20mu] \sin^2A+\sin^2B-\sin^2C & = 2\sin A\sin B\cos C \\ \cos^2A+\cos^2B-\cos^2C & = 1-2\sin A\sin B\cos C \\ \cos^2A+\cos^2B+\cos^2C & = 1-2\cos A\cos B\cos C \\ \sin^2A+\sin^2B+\sin^2C & = 2+2\cos A\cos B\cos C \\[20mu] \tan A+\tan B+\tan C & = \tan A\tan B\tan C \\ \cot B\cot C+\cot C\cot A+\cot A\cos B & = 1 \\ \tan\frac B2\tan\frac C2+\tan\frac C2\tan\frac A2+\tan\frac A2.\tan\frac B2 & = 1 \\ \cot\frac A2+\cot\frac B2+\cot\frac C2 & = \cot\frac A2\cot\frac B2\cot\frac C2 \end{aligned}$

Series Expansion

$\begin{aligned} \sin x & = x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}+\cdots && = \sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{2n+1}}{(2n+1)!} \\ \cos x & = 1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!}+\cdots && = \sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{2n}}{(2n)!} \\ \tan x & = x+\dfrac{x^{3}}{3}+\dfrac{2x^{5}}{15}+\dfrac{17x^{7}}{315}+\dfrac{62x^{9}}{2835}\cdots \end{aligned}$

Complex Exponential Function

$e^{ix}=\cos x+i\sin x=\operatorname {cis}(\theta)$

$\begin{aligned} \cos x & = {\dfrac {e^{ix}+e^{-ix}}{2}} \\ \sin x & = {\dfrac {e^{ix}-e^{-ix}}{2i}} \end{aligned}$

References

• Wikipedia: Trigonometric Functions
• Wikipedia: List of Trigonometric Identities

Special Values

Common & Sub-Angle Values

References

• Wikipedia: Exact Trigonometric Values

Indefinite Integration

Basic Formulas

1. Power Rule

$\quad (n\not ={-1})$

$\displaystyle\int (f(x))^n f'(x) \, dx = \dfrac{(f(x))^{n+1}}{n+1} + C$

$\displaystyle\int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C$

$\displaystyle\int |x|^n \, dx = \dfrac{x |x|^{n}}{n+1} + C$

2. Logarithmic Integration

$\displaystyle\int \dfrac{f'(x)}{f(x)} \, dx = \ln |f(x)| + C$

$\displaystyle\int \dfrac{1}{x} \, dx = \ln |x| + C$

3. Trigonometric Functions

$\begin{aligned} \displaystyle\int \sin x \, dx &&& = -\cos x + C \\ \displaystyle\int \cos x \, dx &&& = \sin x + C \\ \displaystyle\int \tan x \, dx &&& = -\ln |\cos x| + C && = \ln |\sec x| + C \\ \displaystyle\int \cot x \, dx &&& = \ln |\sin x| + C && = -\ln |\csc x| + C \\ \displaystyle\int \sec x \, dx &&& = \ln |\sec x + \tan x| + C && = \ln \left| \tan \left(\frac{\pi}{4} + \frac{x}{2}\right) \right| + C \\ \displaystyle\int \csc x \, dx &&& = \ln |\csc x - \cot x| + C && = \ln \left| \tan \frac{x}{2} \right| + C \end{aligned}$

4. Exponential Function

$\displaystyle\int e^x(f(x)+f'(x)) \, dx = e^x f(x) + C$

$\displaystyle\int a^x \, dx = \dfrac{a^x}{\ln a} + C$

$\displaystyle\int e^x \, dx = e^x + C$

5. Special

$\begin{aligned} \displaystyle\int \dfrac{dx}{a^2 + x^2} & = \dfrac{1}{a} \tan^{-1} \dfrac{x}{a} + C \\[8mu] \displaystyle\int \dfrac{dx}{a^2 - x^2} & = \dfrac{1}{2a} \ln \left| \dfrac{a+x}{a-x} \right| + C \\[25mu] \displaystyle\int \dfrac{dx}{\sqrt{a^2 - x^2}} & = \sin^{-1} \dfrac{x}{a} + C \\[25mu] \displaystyle\int \dfrac{dx}{\sqrt{x^2 + a^2}} & = \ln \left|x + \sqrt{x^2 + a^2}\right| + C \\[8mu] \displaystyle\int \dfrac{dx}{\sqrt{x^2 - a^2}} & = \ln \left| x + \sqrt{x^2 - a^2} \right| + C \\[25mu] \displaystyle\int \dfrac{dx}{x\sqrt{x^2 - a^2}} & = \dfrac{1}{a} \sec^{-1} \dfrac{x}{a} + C \\[25mu] \displaystyle\int \sqrt{a^2 - x^2} \, dx & = \dfrac{x}{2} \sqrt{a^2 - x^2} + \dfrac{a^2}{2} \sin^{-1} \dfrac{x}{a} + C \\[8mu] \displaystyle\int \sqrt{x^2 + a^2} \, dx & = \dfrac{x}{2} \sqrt{x^2 + a^2} + \dfrac{a^2}{2} \ln \left|x + \sqrt{x^2 + a^2}\right| + C \\[8mu] \displaystyle\int \sqrt{x^2 - a^2} \, dx & = \dfrac{x}{2} \sqrt{x^2 - a^2} - \dfrac{a^2}{2} \ln \left|x + \sqrt{x^2 - a^2}\right| + C \\[8mu] \end{aligned}$

Constants

Universal / Physical Constants

Derived / Composite Constants

Empirical / Local Constants

References

• Wikipedia: List of physical constants

Note: This page includes several unconventional approximations, often tailored for exams where calculators aren’t permitted.

Conversions

Metric Prefixes

Conversions

Length

$\begin{aligned} 1 \:m &= 39.37( \approx 243/8) \:in &&= 3.28( \approx 105/32) \:ft &&= 1.094( \approx 11/10) \:yd \\ 1 \:in &= 2.54 \:cm &&= 1/12 \:ft \\ 1 \:ft &= 12 \:in &&= 0.3048 \:m \\ 1 \:km &= 0.6214 \:mi &&= 3281 \:ft \\ 1 \:mi &= 5280 \:ft &&= 1.609 \:km \\ 1 \:\text{light-year} &= 9.461 \times 10^{12} \:km \end{aligned}$

Temperature

$\begin{aligned} \text{Kelvin, } & K &&= {}^\circ C + 273.15 \\ \text{Celsius, } & {}^\circ C &&= K - 273.15 &&= \dfrac{5}{9}({}^\circ F - 32) \\ \text{Fahrenheit, } & {}^\circ F &&= \dfrac{9}{5}{}^\circ C + 32 \\ \text{Rankine, } & {}^\circ R &&= {}^\circ F + 459.67&&= \dfrac{5}{9}K \end{aligned}$

Speed

$\begin{aligned} km/h &= \dfrac{5}{18} \:m/s , & m/s &= \dfrac{18}{5} \:km/h \\ mi/h &= 0.447 \:m/s , &ft/s &= 0.305 \:m/s \\ \end{aligned}$

Mass

$\begin{aligned} 1 \:kg &= 2.204 \:lb &&= 35.274 \:oz \\ 1 \:lb &= 0.4536 \:kg &&= 16 \:oz \\ 1 \:oz &= 0.0283 \:kg \\ 1 \:amu &= 1.66 \times 10^{-27} \:kg \end{aligned}$

Force

$\begin{aligned} 1 \:N &= 10^5 \:dyn &&= 0.2248 \:lbf \\ 1 \:dyn &= 10^{-5} \:N \\ 1 \:lbf &= 4.448 \:N \end{aligned}$

Area

$\begin{aligned} 1 \:m^2 &= 10.764 \:ft^2 &&= 1550 \:in^2 \\ 1 \:in^2 &= 6.45 \:cm^2 \\ 1 \:acre &= 4047 \:m^2 &&= 43560 \:ft^2 \\ 1 \:hectare &= 10^4 \:m^2 \\ 1 \:mi^2 &= 2.59 \:km^2 &&= 640 \:acres \end{aligned}$

Volume

$\begin{aligned} 1 \:m^3 &= 10^3 \:L &&= 35.315 \:ft^3 &&= 264.2 \:gal \\ 1 \:cm^3 &= 1 \:mL &&= 0.061 \:in^3 \\ 1 \:L &= 10^3 \:cm^3 &&= 0.264 \:gal \\ 1 \:ft^3 &= 7.48 \:gal &&= 28.317 \:L \\ 1 \:gal &= 3.785 \:L &&= 231 \:in^3 \end{aligned}$

Pressure

$\begin{aligned} 1 \:kPa &= 10^3 \:N/m^2 &&= 10^{-2} \:bar &&= 9.87 \times 10^{-3} \:atm \\ 1 \:atm &= 101.325 \:kPa &&= 1.013 \:bar &&= 760 \:\text{mmHg (Torr)} \\ 1 \:bar &= 10^2 \:kPa &&= 14.5 \:psi \\ 1 \:psi &= 6.895 \:kPa \\ 1 \:\text{Torr} &= 0.133 \:Pa && (\vec{g} = 9.80665 \:m/s^2) \end{aligned}$

Work/Heat

$\begin{aligned} 1 \:J &= 624.15 \times 10^{10} \:MeV &&= 10^7 \:erg \\ 1 \:eV &= 1.602 \times 10^{-19} \:J \\ 1 \:cal &= 4.184 \:J \\ 1 \:Btu &= 1055 \:J \\ 1 \:\text{kWh} &= 3.6 \times 10^6 \:J &&= 3412 \:Btu \end{aligned}$

Power

$\begin{aligned} 1 \:W &= 1 \:J/s &&= 0.7376 \:ft \cdot lbf/s \\ 1 \:hp &= 745.7 \:W \end{aligned}$

Angle

$\begin{aligned} 1^\circ \text{ (degree)} &= \dfrac{\pi}{180} \:\text{rad} &&= 0.01745 \:\text{rad} \\ 1^\circ &= 60'\text{ (minutes)} \\ 1' &= 60''\text{ (seconds)} \\ 1 \:\text{rad} &= \dfrac{180^\circ}{\pi} \: &&= 57.30^\circ \\ 1 \:\text{revolution} &= 360 \:{}^\circ &&= 2 \pi \:\text{rad} \\ 1 \:\text{rev/min (rpm)} &= 0.1047\:\text{rad/s} \end{aligned}$

References

• Wikipedia: Metric Prefixes
• Openstax Org: University Physics Volume 1

Units

Derived Units

Units Named After People

References

• Physics Info: System International
• Physics Info: System Gaussian

Induction

Electromagnetic Induction

Faraday’s Law

• Whenever the flux of magnetic field through the area bounded by a closed conducting loop changes, and emf in produced in the loop.

• EMF induced $(\mathcal{E})$:
  $\mathcal{E} = -\dfrac{d\Phi}{dt}$

• Flux ($\Phi$):
  $\Phi = \int{\vec{B}\cdot \vec{dS}} = BA \cos \theta$

Lenz’s Law

• The direction of induced current is such theat it opposes the change that has induced it.

Motional EMF

• EMF in a conductor moving with velocity $v$ in magnetic field $B$:
  $\mathcal{E} = vBl$

Induced Electric Field

• Induced electric field $E$ around a loop: $\oint E \, dl = -\dfrac{d\Phi}{dt}$

Eddy Current

• Electromagnetic damping.
• Circular currents induced in conductors due to changing magnetic flux.
• $i \propto \left|\dfrac{d\Phi}{dt}\right|$.

Self-Induction

• $\Phi = Li$

• Induced EMF $(\mathcal{E})$ in coil due to its own current $I$:
  $\mathcal{E} = -L \dfrac{di}{dt}$

Inductors

Self-Inductance of a Long Solenoid

• $L = \mu_0\:n^2Al$

• $n$: Turns per unit length,
  $A$: Cross-sectional area,
  $l$: Length of solenoid.

Growth and Decay of Current in an LR Circuit

1. Growth:
   $i = i_0 \biggr(1 - e^{-t/\tau} \biggr)$
2. Decay:
   $i = i_0\: e^{-t/\tau}$
3. Time constant ($\tau$):
   $\tau = \dfrac{L}{R}$

At $t = \tau$,
Growth: $i = i_0 (1-\dfrac{1}{e}) = 0.63 i_0$
Decay: $i = i_0 \dfrac{1}{e} = 0.37 i_0$

Energy Stored in an Inductor

• Energy $(U)$: $U = \dfrac{1}{2} L i^2$

Energy Density in a Magnetic Field

• $B = \mu_0\:ni$

• $u = \dfrac{B^2}{2 \mu_0}$

Mutual Induction

• Mutual Inductance $(M)$:
  $M = \dfrac{\mu_0 N_1 N_2 A}{l}$

• Induced EMF $\mathcal{E}_2$ in $\text{coil}_2$ due to change in current $i_1$ in $\text{coil}_1$:
  $\mathcal{E}_2 = -M \dfrac{di_1}{dt}$

IUPAC Naming

IUPAC Naming System

Naming organic compounds using the IUPAC system follows four key steps:

1. Identify the main functional group and assign it the highest priority.
2. Find the longest carbon chain that includes this group, numbering it to give the lowest possible number to the priority group.
3. List and number any substituents (branches), arranging them alphabetically in the name.
4. Specify stereochemistry if applicable (E/Z, R/S, cis/trans, etc.).

The final IUPAC name follows this structure:

# - Stereochemistry - # - Prefix F.G(s). - # - Substituent(s) - Parent Chain - Multiple Bond - Suffix F.G.

Examples of IUPAC naming:

• (E)-4-Hydroxy-3-methylpent-2-en-1-one
• (R)-3,4-Dihydroxy-5-methylcyclohex-2-en-1-one
• (Z)-4-Bromo-3-methylhex-2-en-1-ol
• (S)-2,5-Diamino-3-hydroxy-4-methylhexanoic acid
• (E)-5-Ethyl-4-hydroxy-3,6-dimethylhept-2-en-1-one
• (R,E)-3,5-Dimethoxy-4-methylhex-2-enal
• (S)-4-Ethyl-2-hydroxy-3,5-dimethylhexanoic acid
• (E)-2-Bromo-5-ethyl-4-methylhex-2-enamide
• (Z)-3,6-Dihydroxy-4,8-dimethylundec-2-enoic acid

This method applies to all organic compounds.

Prefix Table (Carbon Chain)

Suffix Table (Bond Types)

Functional Groups

The below table is in decreasing order of priority.

Additional Naming Rules & Notes

1. Multiple Functional Groups:

   • The highest priority group gets the suffix.

   • All other functional groups are named as prefixes.

   • Example 1:

     • Compound: $\ce{CH3CH(OH)CH2COOH}$
     • IUPAC Name: 3-Hydroxybutanoic acid
     • Priority Order: Carboxyl ($\ce{-COOH}$) > Hydroxyl ($\ce{-OH}$)

   • Example 2:

     • Compound: $\ce{CH3CH(Br)CH(OH)CH3}$
     • IUPAC Name: 2-Bromo-butan-3-ol
     • Priority Order: Hydroxyl ($\ce{-OH}$) > Halo ($\ce{-Br}$)

2. Numbering of Chains:

   • The longest continuous chain is chosen.
   • The functional group gets the lowest possible number.

3. Multiple Bonds (Double & Triple):

   • If both double $\ce{C=C}$ and triple $\ce{C#C}$ bonds are present, the double bond is given priority in numbering.

4. Cyclic Compounds:

   • Prefix: Cyclo-
   • Example: Cyclohexanol → $\ce{C6H11OH}$.

Exceptions & Special Cases

1. Common Names Used

Some compounds have retained their common names as IUPAC-approved names.

2. Modified Functional Groups Naming

Some groups have unique suffixes or prefixes in IUPAC.

3. Aromatic System Naming Exceptions

Some aromatic compounds have special historical names.

4. Special Aliphatic Naming Exceptions

Some chains do not follow usual alkane naming.

6. Special Amino Acid Naming

Amino acids often do not follow usual amine and acid naming.

7. Special Oxoacid Naming

Some oxoacids have irregular naming.

Key Takeaways

• Some common names are still IUPAC-approved (e.g. acetone, toluene, glycerol).
• Carboxylic acids and esters retain some historical names.
• Aromatic compounds often have non-systematic names.
• Amino acids and oxoacids follow special conventions.

References

• Wikipedia: IUPAC Nomenclature

Common Names

Organic Chemistry: Common Names

Common names are often used colloquially and in everyday life but lack a systematic approach like IUPAC nomenclature. These seemingly arbitrary names can hinder your understanding and ability to solve problems, as exam questions may use common names that you might not be familiar with. To overcome this, it’s essential to familiarize yourself with common names and their corresponding structures.

Resources

PDFs

1. Common Names Google Drive Collection:
   • PDFs I collected that included common names and their corresponding structures. (Sorry, but I didn’t consider noting the sources of some of these PDFs at that time.)

Online Courses

1. Khan Academy Organic Chemistry Course:
   • Khan Academy is the best. (My favourite)

Websites

1. IUPAC Nomenclature of Organic Compounds:
   • Full text explaination of IUPAC Nomenclature of Organic Compounds.

Memorization Techniques

To effectively memorize common names, try these strategies:

1. Spaced Repetition:
   • Revise common names at increasing intervals to reinforce your memory.
2. Associations:
   • Link common names to familiar objects or concepts to enhance recall.
3. Flashcards: (Totally Optional)
   • Create flashcards with common names on one side and structures on the other. Try out Anki if you wanna make them digitally.