LaTeX - A Crash Course

Bob · 2022-11-02 21:20:11

hi Temporary username

Welcome to the forum.

I think the poster didn't intend the LaTex to show here; but rather on the Dr Who site given.

Years ago MIF-Forum used a different server but had to change. Some LaTex that worked on the orignal now fails because the commands aren't implemented now. Cannot do a lot about it I'm afraid

Bob

Jai Ganesh · 2022-11-04 02:46:52

A quadratic equation solution is

or

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

x = (-b \pm \sqrt(b^2 - 4ac))/(2a).

Jai Ganesh · 2022-11-04 17:11:56

Square, cube, and nth power; square root, cube root and nth root

a^2

gives

a^3

gives

a^n

gives

\sqrt{n}

gives

\sqrt[3]{n}

gives

\sqrt[a]{n}

gives

For example,

\sqrt[6]{64}

gives

which is 2.

Jai Ganesh · 2022-11-05 01:22:53

Some Algebraic Expansions

(a + b)^2 = a^2 + 2ab + b^2

gives

(a - b)^2 = a^2 - 2ab + b^2

gives

(a + b)^3 = a^3 + 3a^2b + 3ab^3 + b^3

gives

(a - b)^3 = a^3 - 3a^3 + 3ab^2 - b^3

gives

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

gives

a^3 - b^3  = (a - b)(a^2 + ab + b^2

gives

.

Jai Ganesh · 2022-11-05 21:40:10

Factorial, Permutations, and Combinations

n! = n \times (n - 1) \times (n - 2) \times .... 3 \times 2 \times 1 = n!

gives

nP_r = \dfrac{n!}{(n - r)!}

gives

nC_r = \dfrac{n!}{(n - r)!r!}

gives

.

Jai Ganesh · 2022-11-06 02:39:18

Summation

\Sigma \ n = \dfrac{n(n+ 1)}{2}

gives

\Sigma \ n^2 = \dfrac{n(n + 1)(2n + 1)}{6}

gives

\Sigma \ n^3 = \left[\dfrac{n(n +1)}{2}\right]^2

gives

Jai Ganesh · 2022-11-06 15:49:22

Arithmetic Progression

nth term of a Arithmetic Progression is

a_n = a + (n - 1)d

given by

where a is the first term, n is the number of terms, d is the common difference, and

is the nth term.

Sum of n terms of an Arithmetic Progression :

S_n = n/2[2a + (n - 1)d]

is

.

Jai Ganesh · 2022-11-06 20:53:59

Geometric Progression

nth term is

a_n = ar^{n - 1}

gives

Sum of the terms

\Sigma = a\left(\dfrac{1 - r^n}{1 - r}\right)

gives

or

\Sigma = a\left(\dfrac{r^n - 1}{r - 1}\right)

gives

.

Jai Ganesh · 2022-11-11 20:03:42

Circle, Hemisphere, and Sphere

Area of a Circle:

\pi{r^2}

written as

is the Area of a circle.

2\pi{r}

written as

is the Circumference of a circle.

\dfrac{2}{3}

written as

is the Volume of a Hemisphere.

\dfrac{4}{3}\pi{r^3}

written as

is the Volume of a Sphere.

{3}\pi{r^2}

written as

is the Surface area of a Hemisphere.

4\pi{r^2}

written as

is the Surface area of a sphere.

Jai Ganesh · 2022-11-12 21:23:30

Two Dimensions

Rectangle :

lb

is

Square:

a^2

is

Triangle:

\dfrac{1}{2}bh

is

where b is base and h is height.

Hero's formula for Area of a Triangle:

\sqrt{s(s - a)(s - b)(s - c)}

is

where a, b, and c are side lengths and s is semi-perimeter (half of perimeter).

Jai Ganesh · 2022-11-13 15:41:32

Parallelogram, Trapezium, Kite, and Quadrilateral

Area of a Parallelogram:

Area = Base \times Height

is

where b is the base and h is the height.

Perimeter:

P = 2(a + b)

is

where a and b are the two sides of a Parallelogram.

Area of Rhombus:

Area = \dfrac{d_1 \times d_2}{2}

is

where d1 and d2 are length of diagonals.

Perimeter:

4a

is

.

Kite:

Area = \dfrac{pq}{2}

is

where p and q are the diagonals.

Perimeter :

Perimeter: 2 x (sum of lengths of the sides)

is

Trapezium:

Area = \dfrac{a + b}{2}h

is

where a, b are sides and h is the height.

Quadrilateral:

Area: 1/2 x diagonal x (sum of perpendicular heights)

is

Perimeter: a + b + c + d

is

.

Perimeter: sum of lengths sides of the quadrilateral.

is

Jai Ganesh · 2022-11-16 18:11:04

Cylinder and Cone

Right Circular Cylinder

Volume:

\pi{r^2}h

gives

.

Surface Area:

2\pr{r}h

gives

where r is radius, h is height.

Right Circular Cone

Volume:

\dfrac{1}{3}\pi{r^2}h

gives

.

Area:

\pi{r}(r + l)

gives

where l is slant height.

Slant height:

\sqrt(r^2 + h^2)

gives

.

Jai Ganesh · 2022-11-17 17:14:16

Hollow Sphere, Hollow Right Cylinder, Hollow Right Circular Cone

Volume

Sphere = \dfrac{4}{3}\pi(R^3 - r^3)

gives

where R and r are external and internal radii.

Right Circular Cylinder = \pi({R^2 - r^2})h

gives

.

Right Circular Cone = \dfrac{1}{3}\pi(R^2 - r^2)h

gives

Jai Ganesh · 2022-11-17 21:38:31

Exponents

a^m \times a^2 = a^{m + n}

gives

\dfrac{a^m}{a^n} = a^{m - n}

gives

(a^m)^{n}) = a^{mn}

gives

a^0 = 1, a \neq 0

gives

.

Jai Ganesh · 2022-11-18 17:43:55

Rational Numbers

\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}

gives

\dfrac{a}{b} - \dfrac{c}{d} = \dfrac{ad - bc}{bd}

gibes

\frac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}

gives

\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{ad}{bc}

gives

.

Jai Ganesh · 2022-11-19 02:24:12

Quadratic Equation

Standard form : ax^2 + bx + c = 0

gives

.

Forming a Quadratic Equation:

We will learn the formation of the quadratic equation whose roots are given.

To form a quadratic equation, let

\alpha and \beta

is

and be the two roots.

Let us assume that the required equation be

ax^2  + bx + c = 0, a \neq 0

is

According to the problem, roots of this equation are

\alpha and \beta

gives

and .

Therefore,

\alpha + \beta = -\dfrac{b}{a} and \alpha\beta = \dfrac{c}{a}

gives

and

Now,

ax^2 + bx + c = 0

gives

x^2 + \dfrac{b}{a}x + \dfrac{c}{a}a = 0 (Since, a \neq 0)

gives

.

Jai Ganesh · 2022-11-20 17:22:23

Surface Area

Sphere:

Curved surface area (CSA) :

The Curved surface area of hollow sphere is the area of the paper that can completely cover the surface of the hollow sphere. It is equal to the CSA of inner sphere subtracted from the CSA of outer sphere.

CSA of hollow sphere, = CSA of outer sphere - CSA of inner sphere

= 4\pi(R^2) - 4\pi(r^2)

is

= 4 \pi(R^2-r^2)

is

.

Total surface area of hollow sphere :

The total surface area of a hollow sphere is equal to the CSA of hollow sphere as a hollow sphere has only one surface that constitutes it.

Thus CSA=TSA for a hollow sphere

Jai Ganesh · 2022-11-20 18:46:27

Surface Area

Cylinder:

Total:

Area = 2\pi{r^2} + 2\pi{r}h

gives

where r is radius, h is the height.

Curved Surface Area:

Area = 2\pi{r}h

gives

.

Jai Ganesh · 2022-11-20 19:17:58

Cone

Surface Area

The surface area of a cone is equal to the curved surface area plus the area of the base:

\pi{r^2} + \pi{l}r

gives

, where r denotes the radius of the base of the cone, and L denotes the slant height of the cone. The curved surface area is also called the lateral area.

l = \sqrt{r^2 + h^2}

gives

where r is radius and h height.

Curved Surface Area:

Area = \pi{r}l

gives

l

Jai Ganesh · 2022-11-21 17:26:59

Set Formulas

If A, B, and C are three sets, then the number of elements

n(A \cup B) = n(A) + n(B) - n(A \cap B)

gives

.

If

A \cap B = \phi, then n(A \cup B) = n(A) + n(B)

gives

n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)

gives

.

Jai Ganesh · 2022-11-22 01:08:49

Simple Interest

Simple Interest = Prt where P is Principal, r is rate of Interest, and t time (months, quarters, years etc.)

gives

,

Compound Interest

Compound Interest

gives

A = P\left(1 + \dfrac{r}{n}\right)^{nt}

gives

where:

* A is the final amount
* P is the original principal sum
* r is the nominal annual interest rate
* n is the compounding frequency
* t is the overall length of time the interest is applied (expressed using the same time units as r, usually years).

Jai Ganesh · 2022-11-22 20:38:51

Profit and Loss

Profit = Selling Price - Cost Price

gives

Loss = Cost Price - Selling Price

gives

Profit Percentage = \dfrac{Profit}{Cost \ Price} \ times \ 100\%

gives

Loss Percentage = \dfrac{Loss}{Cost \ Price} \times 100\%

gives

Jai Ganesh · 2022-11-23 01:30:18

Pythagoras Theorem

Pythagoras Theorem: In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two sides.

A right angled Triangle ABC, right angled at B.

gives

{AC}^2 = {AB}^2 + {BC}^2

gives

.

3^2 square units+ 4^2 square units = 5^2 square units

gives

Jai Ganesh · 2022-11-23 12:51:04

Trigonometric Ratios Formulas

Trigonometric ratios can be calculated by taking the ratio of any two sides of the right-angled triangle. We can evaluate the third side using the Pythagoras theorem, given the measure of the other two sides. We can use the abbreviated form of trigonometric ratios to compare the length of any two sides with the angle in the base. The angle θ is an acute angle (θ < 90º) and in general is measured with reference to the positive x-axis, in the anticlockwise direction. The basic trigonometric ratios formulas are given below,

sin\theta = Perpendicular / Hypotenuse

gives

cos\theta = Base / Hypotenuse

gives

tan\theta = Perpendicular / Base

gives

sec\theta = Hypotenuse / Base

gives

cosec\theta  = Hypotenuse / Perpendicular

gives

cot\theta  = Base / Perpendicular

gives

.

Jai Ganesh · 2022-11-23 13:13:32

Trigonometric Ratios of Complementary Angles Identities

The complementary angles are a pair of two angles such that their sum is equal to 90°. The complement of an angle θ is (90° - θ). The trigonometric ratios of complementary angles are:

sin (90°- \theta) = cos \theta

gives

cos (90°- \theta) = sin \theta

gives

cosec (90°- \theta) = sec \theta

gives

sec (90°- \theta) = cosec \theta

gives

tan (90°- \theta) = cot \theta

gives

cot (90°- \theta) = tan \theta

gives

.

Math Is Fun Forum

#151 2022-11-02 21:20:11

Re: LaTeX - A Crash Course

#152 2022-11-04 02:46:52

Re: LaTeX - A Crash Course

#153 2022-11-04 17:11:56

Re: LaTeX - A Crash Course

#154 2022-11-05 01:22:53

Re: LaTeX - A Crash Course

#155 2022-11-05 21:40:10

Re: LaTeX - A Crash Course

#156 2022-11-06 02:39:18

Re: LaTeX - A Crash Course

#157 2022-11-06 15:49:22

Re: LaTeX - A Crash Course

#158 2022-11-06 20:53:59

Re: LaTeX - A Crash Course

#159 2022-11-11 20:03:42

Re: LaTeX - A Crash Course

#160 2022-11-12 21:23:30

Re: LaTeX - A Crash Course

#161 2022-11-13 15:41:32

Re: LaTeX - A Crash Course

#162 2022-11-16 18:11:04

Re: LaTeX - A Crash Course

#163 2022-11-17 17:14:16

Re: LaTeX - A Crash Course

#164 2022-11-17 21:38:31

Re: LaTeX - A Crash Course

#165 2022-11-18 17:43:55

Re: LaTeX - A Crash Course

#166 2022-11-19 02:24:12

Re: LaTeX - A Crash Course

#167 2022-11-20 17:22:23

Re: LaTeX - A Crash Course

#168 2022-11-20 18:46:27

Re: LaTeX - A Crash Course

#169 2022-11-20 19:17:58

Re: LaTeX - A Crash Course

#170 2022-11-21 17:26:59

Re: LaTeX - A Crash Course

#171 2022-11-22 01:08:49

Re: LaTeX - A Crash Course

#172 2022-11-22 20:38:51

Re: LaTeX - A Crash Course

#173 2022-11-23 01:30:18

Re: LaTeX - A Crash Course

#174 2022-11-23 12:51:04

Re: LaTeX - A Crash Course

#175 2022-11-23 13:13:32

Re: LaTeX - A Crash Course

Board footer