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## #176 2022-11-23 01:30:18

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

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

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #177 2022-11-23 12:51:04

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

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

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #178 2022-11-23 13:13:32

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

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

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #179 2022-11-23 15:10:41

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Pythagorean Trigonometric Ratios Identities

These can be derived: Pythagorean trigonometric ratios identities:

{sin}^2\theta + {cos}^2\theta = 1

gives

1 + {tan}^2\theta = {sec}^2\theta

gives

.

1 + {cot}^2\theta = {cosec}^2\theta

gives

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #180 2022-11-23 15:40:22

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Sum, Difference, Product Trigonometric Ratios Identities

The sum, difference, and product trigonometric ratios identities include the formulas of sin(A+B), sin(A-B), cos(A+B), cos(A-B), etc.

sin (A + B) = sin A cos B + cos A sin B

gives

sin (A - B) = sin A cos B - cos A sin B

gives

cos (A + B) = cos A cos B - sin A sin B

gives

cos (A - B) = cos A cos B + sin A sin B

gives

tan (A + B) = (tan A + tan B)/ (1 - tan A tan B)

gives

tan (A - B) = (tan A - tan B)/ (1 + tan A tan B)

gives

cot (A + B) = (cot A cot B - 1)/(cot B - cot A)

gives

cot (A - B) = (cot A cot B + 1)/(cot B - cot A)

gives

2 sin A⋅cos B = sin(A + B) + sin(A - B)

gives

2 cos A⋅cos B = cos(A + B) + cos(A - B)

gives

2 sin A⋅sin B = cos(A - B) - cos(A + B)

gives

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #181 2022-11-23 19:02:16

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Note:

So, the new set of formulas for trigonometric ratios is:

sin \theta = 1/cosec\theta

is

cos \theta = 1/sec\theta

is

tan \theta = 1/cot \theta

is

cosec \theta = 1/sin \theta

is

sec \theta = 1/cos \theta

is

cot \theta = 1/tan \theta

is

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #182 2022-11-23 20:03:39

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Double Angle Identities

Double angle identities for reference:

sin 2\theta = 2 sin\theta cos\theta

gives

cos 2\theta = {cos}^2\theta - {sin}^2\theta

gives

tan 2\theta = (2 tan\theta)/(1 - {tan}^2\theta)

gives

sec 2\theta = \dfrac{{sec}^{2}\theta}{(2-{sec}^2\theta)}

gives

cosec 2\theta = \dfrac{(sec \theta \cdot cosec \theta)}{2}

gives

cot 2\theta = \dfrac{(cot \theta - tan \theta)}{2}

gives

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #183 2022-11-24 00:30:17

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Triple Angle Trigonometric Ratios Identities

sin 3\theta = 3sin \theta - 4{sin}^3\theta

gives

cos 3\theta = 4{cos}^3\theta - 3cos\theta

gives

tan 3\theta = (3tan\theta - {tan}^3\theta)/(1 - 3{tan}^2\theta)

gives

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #184 2022-11-24 18:47:21

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Distance between any Two Points in a Cartesian Plane

PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

gives

or

PQ = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}

gives

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #185 2022-11-24 23:00:32

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Coordinate Geometry

OP = \sqrt{x^2 + y^2}

is

.

\left(\dfrac{m_1x_1 + m_2x_2}{m_1 + m_2}, \dfrac{m_1y_1+ m_2y_2}{m_1 + m_2}\right)

gives

and

\left(\dfrac{m_1x_1 - m_2x_2}{m_1 - m_2}, \dfrac{m_1y_1- m_2y_2}{m_1 - m_2}\right)

gives

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #186 2022-11-25 00:33:42

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Coordinates of the mid-point

\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right).

gives

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #187 2022-11-25 01:20:16

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Slope

One way of finding Slope

We can find the slope of the line using different methods. The first method to find the value of the slope is by using the equation is given as,

m = \dfrac{(y_2 - y_1)}{(x_2 - x_1)}

gives

where m is the slope of the line.

y = mx + b

gives

.

tan\theta = |\dfrac{m_1 - m_2}{1 + m_1m_2}|

gives

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #188 2022-11-25 02:11:09

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Slope

tan\theta = \dfrac{y_2 - y_1}{x_2 - x_1}

gives

m = tan\theta

gives

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #189 2022-11-25 02:57:53

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Equation of Straight Line : Two Point Form

\dfrac{y - y_1}{y_2 - y_1} = \dfrac{x - x_1}{x_2 - x_1}

gives

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #190 2022-11-25 20:52:41

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

The centroid of a Triangle

The centroid of a triangle is the point of intersection of medians of a triangle. (Median is a line joining the vertex of a triangle to the mid-point of the opposite side.). The centroid of a triangle having its vertices

A(x_1,y_1), B(x_2, y_2), C(x_3, y_3)

gives

is obtained from the following formula:

(x, y) = \left(\dfrac{x_1 + x_2 + x_3}{3}, \dfrac{y_1 + y_2 + y_3}{3}\right)

gives

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #191 2022-11-25 21:57:56

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Distance Formulas - I

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

is

.

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_2)^2}

is

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #192 2022-11-26 02:04:12

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Distance Formula - II

Distance From a Point To a Line in 2D

The distance formula to calculate the distance from a point to a line is the length of the perpendicular line segment that is drawn from the point to the line. Let us consider a line L in a two-dimensional plane with the equation ax + by + c =0 and consider a point

.

Then the distance (d) from P to L is,

d = \dfrac{|ax_1 + by_1 + c|}{\sqrt{x^2 + y^2}}

gives

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #193 2022-11-26 14:57:02

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Area of a Triangle

\left\{\dfrac[{1}{2}x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)]\right\}

gives

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #194 2022-11-26 22:01:15

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Collinearity of three points

x_1(y_2 - y_3) + y_2(y_3 - y_1) + x_3(y_1 - y_2) = 0

gives

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #195 2022-11-28 00:22:35

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

A = (1/2) ⋅ {(x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1) - (x_2y_1 + x_3y_2 + x_4y_3 + x_1y_4 )}

gives

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #196 2022-11-30 21:10:34

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Angle Between Straight Lines

tan \ \theta = \left|\dfrac{m_1 - m_2}{1 + m_1{m_2}}\right|

gives .

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #197 2022-12-01 21:55:19

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Angle between Straight Lines

tan \ \theta = \left|\dfrac{{a_2}{b_1} - {a_1}{b_2}}{{a_1}{a_2} + {b_1}{b_2}}

gives

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #198 2022-12-01 22:37:44

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Angle between two straight lines

\theta = {Tan}^{-1} \dfrac{m_1 - m_2}{1 + m_1 \cdot m_2}

gives

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #199 2022-12-02 19:41:00

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Circle

A circle is a curved plane figure. Every point on the circle is equidistant from a fixed point known as the center of the circle. It is a 2D shape and is measured in terms of radius. The word ‘Circle’ is derived from the Latin word 'circulus' meaning small ring.

What is Circle?

A circle is a two-dimensional figure formed by a set of points that are at a constant or at a fixed distance (radius) from a fixed point (center) on the plane. The fixed point is called the origin or center of the circle and the fixed distance of the points from the origin is called the radius.

Parts of a Circle

There are many parts or components of a circle that we should know to understand its properties. A circle has mainly the following parts:

* Circumference: It is also referred to as the perimeter of a circle and can be defined as the distance around the boundary of the circle.

* Radius of Circle: Radius is the distance from the center of a circle to any point on its boundary. A circle has many radii as it is the distance from the center and touches the boundary of the circle at various points.

* Diameter: A diameter is a straight line passing through the center that connects two points on the boundary of the circle. We should note that there can be multiple diameters in the circle, but they should:

** pass through the center.
** be straight lines.
** touch the boundary of the circle at two distinct points which lie opposite to each other.

* Chord of a Circle: A chord is any line segment touching the circle at two different points on its boundary. The longest chord in a circle is its diameter which passes through the center and divides it into two equal parts.

* Tangent: A tangent is a line that touches the circle at a unique point and lies outside the circle.

* Secant: A line that intersects two points on an arc/circumference of a circle is called the secant.

* Arc of a Circle: An arc of a circle is referred to as a curve, that is a part or portion of its circumference.

* Segment in a Circle: The area enclosed by the chord and the corresponding arc in a circle is called a segment. There are two types of segments - minor segment, and major segment.

* Sector of a Cirlce: The sector of a circle is defined as the area enclosed by two radii and the corresponding arc in a circle. There are two types of sectors - minor sector, and major sector.

Properties of Circle

Here is a list of properties of a circle:

* A circle is a closed 2D shape that is not a polygon. It has one curved face.
* Two circles can be called congruent if they have the same radius.
* Equal chords are always equidistant from the center of the circle.
* The perpendicular bisector of a chord passes through the center of the circle.
* When two circles intersect, the line connecting the intersecting points will be perpendicular to the line connecting their center points.
* Tangents drawn at the endpoints of the diameter are parallel to each other.

Let's see the list of important formulae pertaining to any circle.

Area of a Circle Formula: The area of a circle refers to the amount of space covered by the circle. It totally depends on the length of its radius :

Area = \pi{r^2} square units

gives .

.

Circumference of a Circle Formula: The circumference is the total length of the boundary of a circle.

Circumference = 2\pi{r} units

gives

Arc Length Formula: An arc is a section (part) of the circumference.

Length \ of \ an \ arc = \theta \times r. \ Here, \theta \ is \ in \radians.

gives

Area of a Sector Formula: If a sector makes an angle θ (measured in radians) at the center, then the area of the sector of a circle =

(\theta \times r^2) \div 2. \ Here, \theta \ is \ in \ radians.

gives

Length of Chord Formula: It can be calculated if the angle made by the chord at the center and the value of radius is known.

Length \ of \ chord = 2 r sin\dfrac{\theta}{2}. \ Here, \ \theta \ is \ in \ radians.

gives

Area of Segment Formula: The segment of a circle is the region formed by the chord and the corresponding arc covered by the segment.

The \ area \ of \ a \ segment = r^2(\theta − sin\theta) \div 2. Here, \theta \ is \ in \ radians.

gives

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #200 2022-12-03 22:07:27

ganesh
Registered: 2005-06-28
Posts: 41,291

### Re: LaTeX - A Crash Course

Important Notes on Angles

0^o < \ Acute \ angle < \ {90}^o

gives

{90}^o < \ Obtuse \ angle < {180}^o

gives

{180}^o < \ Reflex \ angle < {360}^o

gives

A \ right \ angle \ is \ equal \ to \ {90}^o

gives

A \ straight \ angle \ is \ equal \ to {180}^o.

gives

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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