### École Internationale Polytechnique de Genève (EIPG) e-Learning platform (LMS and LCMS)

The Platform supports 26 languages, it's compliant with the SCORM 1.2 and 2004 standards, and supports different didactic models, including: Blended, Self-Directed, Collaborative and even Social Learning through Chat, Wiki, Forums and other different functions.

#### BVMDAE(Bachelors Validation/Masters & Doctorates Access Exam)

You are in possession of Master's or Doctorate's Degrees from other institutions you still have to pass the BVMDAE.

Whether you studied at Oxford, Harvard, Cambridge, Yale, The Open University, UNISA, SAIM, IBS, ICS, Intec, via Alison, Coursera or Fun MOOC, you will have to pass the BVMDAE(Bachelor's Validation/Master's & Doctorate's Access Exam). The BVMDAE will become the future Educational level Evaluation tool. The BVMDAE has for goal to keep the highest standard possible. Take note that Oxford, Harvard, Cambridge and Yale students have difficulty to pass the BVMDAE.

EIPG Online Vocational education(and what is ...) can take place at the post-secondary, further education and higher education level; and can interact with the apprenticeship system. At the post-secondary level, vocational education is often provided by highly specialized trade, Technical schools, community colleges, colleges of further education UK, Universities, Institutes of technology / Polytechnic Institutes etc.

Until recently, almost all vocational education took place in the classroom, or on the job site, with students learning trade skills and trade theory from accredited professors or established professionals. However, online vocational education has grown in popularity, and made it easier than ever for students to learn various trade skills and soft skills from established professionals in the industry.

What about Recognition ... The BVMDAE(Bachelors Validation/Masters & Doctorates Access Exam) serve as an automated recognition exam. Most Public University students fails this exam so do not take this Examination for Granted.

### Faculty of Management

#### Doctor Of Science Business Management & Administration or D.Sc. BMA or DBMA

will be of great interest to students of Management, Administration, Managerial science and Entrepreneur in general, as well as any student wanting to expand their knowledge and understanding of Management & Administration.

Management, Administration, Managerial science and Entrepreneurship.

### Faculty of Medecine

#### Doctor Of Science Natural Medicine or D.Sc. nMed.

will be of great interest to students of anatomy, physiology, biomedical science and medicine in general, as well as any student wanting to expand their knowledge and understanding of Natural Medicine.

We developed a Research Protocol, with respecting to animal and human life, so with our vision and determination to be transparent in our research, this is what we do during our testing phase to elaborate a protocol in Nutrition and Medication-based on plants and minerals.

#### Doctor Of Science Nursing, Nutrition & Naturopathy or D.Sc. nNat.

will be of great interest to students of anatomy, physiology, biomedical science and nursing in general, as well as any student wanting to expand their knowledge and understanding of Nutrition & Naturopathy.

From taking our vital signs to conducting patient education and case management, nurses are the jack-of-all-trades in the medical field. Celebrated May 6-12, National Nurses Week is a time...

#### Doctor Of Science Psychiatrist & Naturopathy or D.Sc. pNat.

will be of great interest to students of Natural Medicine, Psychiatrist in general, as well as any student wanting to expand their knowledge and understanding of Psychiatrist.

General psychiatrists are medically qualified doctors who contribute to the management and treatment of adults with mental health problems.

#### Doctor Of Science Psychology & Naturopathy or D.Sc. psNat.

will be of great interest to students of psychology, naturopathy, biomedical science and medicine in general, as well as any student wanting to expand their knowledge and understanding of Psychology.

Forensic Psychology - The Judicial Error

Although crime dramas focus more and more on the forensic analysis of material evidence (DNA for example), the human element of the story, especially the evidence provided by the victims / witnesses , remains a determining element.

### Faculty of Science

#### Doctor Of Science Astronomy & Space Science or D.Sc. aSS.

will be of great interest to students of Aero and Space Engineering in general, as well as any student wanting to expand their knowledge and understanding of Astronomy & Space Science.

EIPG Orbiter II - micro-satellite

Our micro-satellite(Illustrations) with Gyroscope, Geiger Counter, Thermostat and other Scientific Instrument for Pressure / Vacuum Measurement, Electromagnetism, Gas Composition etc. Fully Designed and Developed by Prof. Horstmann and his team of researchers. Cost per unit launched into space 12 000.00€

#### Doctor Of Science Chemistry or D.Sc. Chem.

will be of great interest to students of Chemistry, Analytical Chemistry, Biochemistry, Chemical Engineering, Food Science & Nutrition, Petroleum Chemistry, Pharmaceutical Chemistry and Pharmacovigilance.

Analytical Chemistry. Analytical chemistry involves the quantitative and qualitative measurement of atoms, molecules, particles, and cells from the chemical standpoint. As such, all Chemists, indeed most scientists use Analytical Chemistry in some form.

#### Doctor Of Science Ecology & Environmental Science or D.Sc. eEnv.

will be of great interest to students of ecology, environmental, sustainable development and conservation in general, as well as any student wanting to expand their knowledge and understanding of Ecology & Environmental Science.

the branch of science concerned with the physical, chemical, and biological conditions of the environment and their effect on organisms.

#### Doctor Of Science General Science - biology, chemistry, physics & mathematics or D.Sc. Gen

will be of great interest to students of biology, physics & chemistry and science in general, as well as any student wanting to expand their knowledge and understanding of this vast field of study.

The DSc General Science degree(biology, chemistry, physics & mathematics)
is to explore your interests in the vast field of science. This degree offers a solid foundation in all core areas from biological sciences, chemistry and biochemistry, computer science, earth and environmental sciences, mathematics, physics, and economics. There is room in the degree to explore other areas of interest as well.

#### Doctor Of Science Mathematics or D.Sc. Math.

will be of great interest to students of Chemistry, Engineering, Physic as well as any student wanting to expand their knowledge and understanding of Mathematics.

Vogel’s approximation method for solving transportation problems.

The Vogel Approximation Method is an improved version of the Minimum Cell Cost Method and the Northwest Corner Method that in general produces better initial basic feasible solution, which are understood as basic feasible solutions that report a smaller value in the objective (minimization) function of a balanced Transportation Problem (sum of the supply = sum of the demand).

#### Doctor Of Science Physics or D.Sc. Phys.

will be of great interest to students of Physics, as well as any student wanting to expand their knowledge and understanding of Physics.

Standard model of physics
CERN - all of our current understanding of the universe condensed into a single infographic

### Faculty of Engeneering

#### Doctor Of Science Agricultural Engineering or D.Sc. agriEng.

will be of great interest to students of agricultural, as well as any student wanting to expand their knowledge and understanding of Agricultural Engineering.

What is the importance of c language in agricultural engineering?
As evident from the above picture there are many applications of Software and Information technology across the main verticals of agriculture engineering. As an agricultural engineer , one is supposed to be familiar with these technologies and hence knowledge of programming is necessary.

#### Doctor Of Science Aeronautics & Astronautics Engineering or D.Sc. aEng.

will be of great interest to students of Aero and Space Engineering in general, as well as any student wanting to expand their knowledge and understanding of Aeronautics & Astronautics.

1600 years before any flying machine could ever fly and before Newton announced the third mechanical law, Hero of Alexandria invented the Aeolipile. Nothing serious(just a toy) but the modern jet engine use it, same physics principle!
In Greek, the word Aeolipile means Wind Ball. How does this stuff work? A closed vessel filled with water is heating. After a few minutes, steam is produced and passes through two vertical outlets into the ball. Steam began to escape from the directional nozzle and the ball can rotate.

Liquid rocket motors

We locked our main engine choice about a year ago. This occurred after a series of static tests with medium and large liquid propellant rocket engines using LOX (Liquid Oxygen) and ethanol / water as a propellant.

#### Doctor Of Science Biomedical Engineer or D.Sc. bioTEng.

will be of great interest to students of Biotechnology, as well as any student wanting to expand their knowledge and understanding of Biotechnology.

Synthetic Stingray May Lead To A Better Artificial Heart Using gold, silicone and heart cells from a rat, scientists have made a tiny artificial stingray. The engineering involved in propelling it could help make a heart that's more than a mechanical pump.

#### Doctor Of Science Chemical Engineer or D.Sc. chemEng.

will be of great interest to students of Chemical Engineering, as well as any student wanting to expand their knowledge and understanding of Chemical Engineering.

Chemical engineering is a branch of engineering that applies physical sciences (physics and chemistry), life sciences (microbiology and biochemistry), together with applied mathematics and economics to produce, transform, transport, and properly use chemicals, materials and energy. A chemical engineer designs large-scale processes that convert chemicals, raw materials, living cells, microorganisms and energy into useful forms and products.
Chemical engineers are involved in many aspects of plant design and operation, including safety and hazard assessments, process design and analysis, control engineering, chemical reaction engineering, construction specification and operating instructions.

#### Doctor Of Science Civil Engineer or D.Sc. cEng.

will be of great interest to students of civil engineering, construction, civil engineering in general, as well as any student wanting to expand their knowledge and understanding of Construction(civil) Engineering.

Civil Engeneering Courses Masters of Engineering Concepts of Engineering Practice: It is a graduate-level category that covers the role of engineering in fashionable society. Lecture notes cowl ethics, construction and project management. Students will transfer case studies and find out about the project analysis method.

#### Doctor Of Science Design Engineer or D.Sc. dEng.

will be of great interest to students of Design Engineering, as well as any student wanting to expand their knowledge and understanding of Design Engineering.

Design Engineer in Action.
3D Printing for the Design and Manufacturing Engineers are a must in today's fast developing society.

#### Doctor Of Science Electrical & Computer Engineer or D.Sc. eEng.

will be of great interest to students of electrical control systems, programmers, computer science in general, as well as any student wanting to expand their knowledge and understanding of Electrical, electronic and computer Systems.

Embedded software designs such as those for avionics and automotive systems have become highly complex to develop, test and certify. As a result, the traditional document...

#### Doctor Of Science Electronics & Communication Engineer or D.Sc. ecEng.

will be of great interest to students of Electronics & Communication, as well as any student wanting to expand their knowledge and understanding of Electronics & Communication Engineer.

Semiconductor pn junction diagram
The junction here is drawn with photon injection condition like in a solar cell, so we use the quasi-fermi levels.

#### Doctor Of Science Geology & Geophysics Engineering or D.Sc. gEng.

will be of great interest to students of Climate, Ocean Engineering, as well as any student wanting to expand their knowledge and understanding of Geology & Geophysics Engineering.

The Geological Society
Various pathways in Geology & Geophysics. You have a vast zone of Activities to choose from ...

#### Doctor Of Science Information Systems Engineer or D.Sc. isEng.

will be of great interest to students of Information systems, programers, computer science in general, as well as any student wanting to expand their knowledge and understanding of Information Systems.

EIPG Mos All in One PC (21") (HMT2160AGOS)
The EIPG Mos All in One PC (21") (HMT2160AGOS) Product Description Model HMT2160AGOS; Display Size 21.6 inch; Resolution 1360*768@60Hz or 1920*1080 Display Ratio 16: 9; Mother Board Intel G41 ICH7; CPU Intel Atom@ Dual Core HT E5500 2.8Hz; Memory DDR2-800, 2GB and above; Hard Disk S-ATAII 250GB/500GB/8MB/7200rpm; Graphic Card Intel Graphics Media Accelerator 3150; Network Card 1000M(Built-in wireless card); Audio Card (Realtek HD audio output) audio system; Input/Output USB2.0*4 / AUDIO / EARPHONE / MIC /RJ45 LAN / SD, MMC, MS / VGA / HDMI; Operating System: Mos64 atom version 1.0

#### Doctor Of Science Control & Instrumentation Engineer or D.Sc. ciEng.

will be of great interest to students of Control & Instrumentation Engineering, as well as any student wanting to expand their knowledge and understanding of Control & Instrumentation Engineering.

Control and Instrumentation Illustrations
As Control and instrumentation engineers (C&I engineers) you are responsible for designing, developing, installing, managing and maintaining equipment which is used to monitor and control engineering systems, machinery and processes.

#### Doctor Of Science Mechanical Engineer or D.Sc. mEng.

will be of great interest to students of Structural engineering, construction, Automotive engineering in general, as well as any student wanting to expand their knowledge and understanding of Construction(Mechanical) Engineering.

If the beta is greater than 90 degrees, the pressure increases with increasing flow. Such blades are called forward curved blades. If beta is less than 90 degrees. Since the second term in LHS of the head vs. flow equation is positive in this case, the pressure head decreases with increasing flow. These types of impellers are called backward curved blades. Thus backward curved blades that achieved self-stabilizing characteristics in energy consumption is the most preferred in the industry.

#### Doctor Of Science Materials Engineering or D.Sc. mesEng.

will be of great interest to students of Materials Engineering, as well as any student wanting to expand their knowledge and understanding of Materials Engineering.

Resources Presentation
Material resources include wood, glass (which comes from sand), metals, edible plants, and plastics (which are made from natural chemicals). Renewable material resources, like glass, can be re-created easily.

#### Doctor Of Science Metallurgical Engineer or D.Sc. mtesEng.

will be of great interest to students of Metallurgical Engineer, as well as any student wanting to expand their knowledge and understanding of Metallurgical Engineer.

Metallurgical Engineer in Action.
... Nonrenewable resources Recyclable resources The degree of availability of ... Almost all home scrap, as well as a sizeable part of prompt industrial scrap, are ... etc.

#### Doctor Of Science Mining Engineer or D.Sc. minEng.

will be of great interest to students of Mining Engineer, as well as any student wanting to expand their knowledge and understanding of Mining Engineer.

Mining Engineer in Action.
Various mining resources are available exp. Gold, Silver, Diamonds, Platinum, Uranium etc.

#### Doctor Of Science Ocean & Atmospheric Engineering or D.Sc. oaEng.

will be of great interest to students of Ocean Engineering, as well as any student wanting to expand their knowledge and understanding of Ocean Engineering.

Non-Convectional Energy Resources Presentation
Various Ocean Energy resources are available exp. Ocean Thermal Energy Conversion as illustrated above.

#### Doctor Of Science Bio Petrochemicals & Petroleum Engineer or D.Sc. pEng.

will be of great interest to students of biofuel, alternative, ecology & environmental science in general, as well as any student wanting to expand their knowledge and understanding of Bio Petrochemical Industries.

HIBPCI - Growing Algae for Biodiesel Use So, we've talked about the chemical process that takes algae and turns it into biodiesel fuel.

#### Doctor Of Science Production & Industrial Engineer or D.Sc. piEng.

will be of great interest to students of Industries, engineering, construction, production in general, as well as any student wanting to expand their knowledge and understanding of Production & Industrial Engineering.

EIPG Candela I - Electric Car project Hands on Production & Industrial engineering experience by our students, as part of their practical formation.

#### Doctor Of Science Textile Engineer or D.Sc. tEng.

will be of great interest to students of Textile Engineer, as well as any student wanting to expand their knowledge and understanding of Textile Engineer.

Textile Engineering
Various chemical, mechanical and physical processes are involved throughout the textile manufacturing process. Textile equipment, chemicals, fibres, polymers and other synthetic materials are involved in it. Textile engineering deals with these processes, synthetic materials, machinery and overall production. These days, computers play a prominent role in textile industry. Computers are used to design and manufacture (process control) textile products!

#### Doctor Of Science Vehicle Engineer or D.Sc. vEng.

will be of great interest to students of Automotive engineering, construction, Automotive engineering in general, as well as any student wanting to expand their knowledge and understanding of Construction(Automotive) Engineering.

EIPG Presents Candela I - Electric Car
Customer choice upholstery/color vehicles will be build by our Students, Doctor Of Science Vehicle Engineer or D.Sc. vEng., as part of their practical experience during formation.

### Faculty of Arts

#### Doctor Of Science Architecture and Planning or D.Sc. pArch.

will be of great interest to students of architect, Architectural planning, Architecture and Planning in general.

Architectural planning: The architect usually begins to work when the site and the type and cost of a building have been determined. The site involves the varying behaviour of the natural environment that must be adjusted to the unvarying physical needs of human beings; the type is the generalized form ...

### Faculty of Theology

#### Doctor Of Science Theology or D.Sc. Th.

will be of great interest to students of biblical studies in general, as well as any student wanting to expand their knowledge and understanding of Theology.

your gateway to an exciting, fulfilling and ... future! Learn to apply biblical principles in a secular world, and help families and individuals in your community find Christ-centered solutions to difficult life issues. All study materials are provided free-of-charge.

### Private School - Saint Amrane Of Jerusalem

From Kindergarten to 12 Grade - Homeschool for Isolated individuals

#### Diploma of Science High School or Dip.Sc. Hs.

High school graduation is a rite of passage in modern society. Whether you want to impress your friends, get the job of your dreams or qualify for an upcoming promotion, we can help. Saint Amrane Church of Jerusalem has partnered with EIPG Online Academy to complete your primary education with a Christian degree. Avoid the stigma attached to GEDs with a diploma that says "I finished High School." All study materials are provided free-of-charge.

Saint Amrane Of Jerusalem High School

EIPG for "Saint Amrane Of Jerusalem" - High school students raising hands, in classroom with professor

### General Information ...

Magnesium aluminate, Synthetic spinel, accidentally produced in the mid-18th century, has been more recently described in scientific publications in 2000 and 2004. By 2015, transparent spinel was made into sheets and other forms by sintering. Synthetic spinel, which looks like glass but has a significantly higher resistance to pressure, can also have applications for military and commercial purposes.

At EIPG our Students produce different Turbo Machines Transparent like Glass, Light as Feathers and Hard as Tanks from Spinel(Magnesium aluminate) formula MgAl2O4.
We are at the point of realizing a 3D Printer for Simply Printing Pumps and PD Pumps(Positive Displacement Pumps) in Spinel at 650°C.

#### Periodic Table of Chemical Elements

Periodic Table of Chemical Elements for EIPG students

## MathML Integral Formula

$\begin{array}{r}f\left(a\right)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f\left(z\right)}{z-a}dz\end{array}$

## or Math API Tex base ... Integral Formula

\begin{align} f(a) = \frac{1}{2\pi i} \oint_{\gamma}\frac{f(z)}{z-a}dz \end{align}

## Dynamic Equations - step by step

Expand the following: \begin{align} (x 1)^2 &\cssId{Step1}{= (x 1)(x 1)}\\ &\cssId{Step2}{= x(x 1) 1(x 1)}\\ &\cssId{Step3}{= (x^2 x) (x 1)}\\ &\cssId{Step4}{= x^2 (x x) 1}\\ &\cssId{Step5}{= x^2 2x 1}\\ \end{align}

\begin{align} When $a \ne 0$, there are two solutions to $ax^2 bx c = 0$ and they are \end{align}

\begin{align} x = {-b \pm \sqrt{b^2-4ac} \over 2a}. \end{align}

## Double angle formula for Cosines

\begin{align} \cos(θ φ)=\cos(θ)\cos(φ)−\sin(θ)\sin(φ) \end{align}

## Gauss' Divergence Theorem

\begin{align} \int_D ({\nabla\cdot} F)dV=\int_{\partial D} F\cdot ndS \end{align}

## Curl of a Vector Field

\begin{align} \vec{\nabla} \times \vec{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \mathbf{i} \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \mathbf{j} \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \mathbf{k} \end{align}

## Standard Deviation

\begin{align} \sigma = \sqrt{ \frac{1}{N} \sum_{i=1}^N (x_i -\mu)^2} \end{align}

## Definition of Christoffel Symbols

\begin{align} (\nabla_X Y)^k = X^i (\nabla_i Y)^k = X^i \left( \frac{\partial Y^k}{\partial x^i} \Gamma_{im}^k Y^m \right) \end{align}

## The Lorenz Equations

\begin{align} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z xy \end{align}

## Cauchy's Integral Formula

\begin{align} f(a) = \frac{1}{2\pi i} \oint\frac{f(z)}{z-a}dz \end{align}

## The Cauchy-Schwarz Inequality

$\left( \sum_{k=1}^n a_k b_k \right)^{\!\!2} \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$

## A Cross Product Formula

$\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\ \end{vmatrix}$

## The probability of getting $k$ heads when flipping $n$ coins is:

$P(E) = {n \choose k} p^k (1-p)^{ n-k}$

## An Identity of Ramanujan

$\frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} = 1 \frac{e^{-2\pi}} {1 \frac{e^{-4\pi}} {1 \frac{e^{-6\pi}} {1 \frac{e^{-8\pi}} {1 \ldots} } } }$

## A Rogers-Ramanujan Identity

$1 \frac{q^2}{(1-q)} \frac{q^6}{(1-q)(1-q^2)} \cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j 2})(1-q^{5j 3})}, \quad\quad \text{for |q|<1}.$

## Maxwell's Equations

\begin{align} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, \, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{align}

## In-line Mathematics

Finally, while display equations look good for a page of samples, the ability to mix math and text in a paragraph is also important. This expression $\sqrt{3x-1} (1 x)^2$ is an example of an inline equation. As you see, equations can be used this way as well, without unduly disturbing the spacing between lines.

## Trig Identity Formulas

Use these fundemental formulas of trigonometry to help solve problems by re-writing expressions in another equivalent form.

### Basic Identities

$\sin(x)=\frac{1}{\csc(x)}$

$\cos(x)=\frac{1}{\sec(x)}$

$\tan(x)=\frac{1}{\cot(x)}$

$\sec(x)=\frac{1}{\cos(x)}$

$\csc(x)=\frac{1}{\sin(x)}$

$\cot(x)=\frac{1}{\tan(x)}$

$\tan(x)=\frac{\sin(x)}{\cos(x)}$

$\sin(-x)=-\sin(x)$

$\cos(-x)=\cos(x)$

$\tan(-x)=-\tan(x)$

### Pythagorean Identities

$\sin^2(x) \cos^2(x)=1$

$1 \tan^2(x)=\sec^2(x)$

$1 \cot^2(x)=\csc^2(x)$

### Sum and Difference Formulas

$\sin(a b)=\sin(a)\cos(b) \cos(a)\sin(b)$

$\sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b)$

$\cos(a b)=\cos(a)\cos(b)-\sin(a)\sin(b)$

$\cos(a-b)=\cos(a)\cos(b) \sin(a)\sin(b)$

$\tan(a b)=\frac{\tan(a) \tan(b)}{1-\tan(a)\tan(b)}$

$\tan(a-b)=\frac{\tan(a)-\tan(b)}{1 \tan(a)\tan(b)}$

$\sin(x) \sin(y)=2\sin(\frac{x y}{2})\cos(\frac{x-y}{2})$

$\sin(x)-\sin(y)=2\cos(\frac{x y}{2})\sin(\frac{x-y}{2})$

$\cos(x) \cos(y)=2\cos(\frac{x y}{2})\cos(\frac{x-y}{2})$

$\cos(x)-\cos(y)=-2\sin(\frac{x y}{2})\sin(\frac{x-y}{2})$

### Double Angle Formulas

$\sin(2x)=2\sin(x)\cos(x)$

$\cos(2x)=\cos^2(x)-\sin^2(x)=1-2\sin^2(x) = 2\cos^2(x)-1$

### Half Angle Formulas

$\sin(\frac{x}{2})=\pm\sqrt{\frac{1-\cos(x)}{2}}$

$\cos(\frac{x}{2})=\pm\sqrt{\frac{1 \cos(x)}{2}}$

$\tan(\frac{x}{2})=\pm\sqrt{\frac{1-\cos(x)}{1 \cos(x)}}=\frac{1-\cos(x)}{\sin(x)}=\frac{\sin(x)}{1 \cos(x)}$

### Trigonometric Products

$\sin(x)\cos(y)=\frac{\sin(x y) \sin(x-y)}{2}$

$\cos(x)\cos(y)=\frac{\cos(x y) \cos(x-y)}{2}$

$\sin(x)\sin(y)=\frac{\cos(x-y)-\cos(x y)}{2}$

## In-line Chemical Equations

Finally, while display equations look good for a page of samples, the ability to mix Chemical Equations and text in a paragraph is also important. This expression $\ce{Zn^2 <=>[ 2OH-][ 2H ] \underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}} <=>[ 2OH-][ 2H ] \underset{\text{Hydroxozikat}}{\ce{[Zn(OH)4]^2-}}}$ is an example of an inline equation. As you see, equations can be used this way as well, without unduly disturbing the spacing between lines.