LaTeX - A Crash Course

Jai Ganesh · 2022-11-23 15:10:41

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

.

Jai Ganesh · 2022-11-23 15:40:22

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

.

Jai Ganesh · 2022-11-23 19:02:16

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

.

Jai Ganesh · 2022-11-23 20:03:39

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

.

Jai Ganesh · 2022-11-24 00:30:17

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

.

Jai Ganesh · 2022-11-24 18:47:21

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

.

Jai Ganesh · 2022-11-24 23:00:32

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

Jai Ganesh · 2022-11-25 00:33:42

Coordinates of the mid-point

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

gives

Jai Ganesh · 2022-11-25 01:20:16

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

Jai Ganesh · 2022-11-25 02:11:09

Slope

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

gives

m = tan\theta

gives

.

Jai Ganesh · 2022-11-25 02:57:53

Equation of Straight Line : Two Point Form

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

gives

.

Jai Ganesh · 2022-11-25 20:52:41

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

.

Jai Ganesh · 2022-11-25 21:57:56

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

.

Jai Ganesh · 2022-11-26 02:04:12

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

.

Jai Ganesh · 2022-11-26 14:57:02

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

Jai Ganesh · 2022-11-26 22:01:15

Collinearity of three points

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

gives

Jai Ganesh · 2022-11-28 00:22:35

Quadrilateral

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

.

Jai Ganesh · 2022-11-30 21:10:34

Angle Between Straight Lines

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

gives .

Jai Ganesh · 2022-12-01 21:55:19

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

.

Jai Ganesh · 2022-12-01 22:37:44

Angle between two straight lines

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

gives

.

Jai Ganesh · 2022-12-02 19:41:00

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

Jai Ganesh · 2022-12-03 22:07:27

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

.

Jai Ganesh · 2022-12-04 21:57:50

Similar Triangles Formulas

In the previous section, we saw there are two conditions using which we can verify if the given set of triangles are similar or not. These conditions state that two triangles can be said similar if either their corresponding angles are equal or congruent or if their corresponding sides are in proportion. Therefore, two triangles △ABC and △EFG can be proved similar(△ABC ∼ △EFG) using either condition among the following set of similar triangles formulas,

Formula for Similar Triangles in Geometry:

\angle \ A = \angle \ E, \angle \  B = \angle \  F \ and \angle \ C = \ G

gives

AB/EF = BC/FG = AC/EG

gives

Similar Triangles Theorems

We can find out or prove whether two triangles are similar or not using the similarity theorems. We use these similarity criteria when we do not have the measure of all the sides of the triangle or measure of all the angles of the triangle. These similar triangle theorems help us quickly find out whether two triangles are similar or not. There are three major types of similarity rules, as given below,

*

AA \ (or \ AAA) \ or \ Angle-Angle \  Similarity \ Theorem

gives

*

SAS \ or \ Side-Angle-Side \ Similarity \ Theorem

gives

*

SSS \ or \ Side-Side-Side \ Similarity \ Theorem

gives

.

Jai Ganesh · 2022-12-07 21:40:46

Direction Ratios

Direction Ratios of a Line

The directional ratios of a line are the numbers that are proportional to the direct cosines of the line. If l, m, n are the direction cosines, and a,b c are the direction ratios, then

l = \dfrac{a}{\sqrt{a^2 + b^2 + c^2}}

gives

m = \dfrac{b}{\sqrt{a^2 + b^2 + c^2}}

gives

n = \dfrac{c}{\sqrt{a^2 + b^2 + c^2}}

gives

Direction ratios of line joining the points

P(x_1,y_1,z_1) \ and \ Q(x_2,y_2,z_2)

gives

are

(x_2 - x_1, y_2 - y_1, z_2 - z_1) \ or \ (x_1 - x_2, y_1 - y_2, z_1 - z_2)

gives.

.

Jai Ganesh · 2022-12-08 03:15:25

Skew lines in Geometry

The skew lines are the lines in space that are neither parallel nor intersecting, and they lie in different planes. The angle between two lines is

cos \ \theta  = |{|_1}{l_2} + {m_1}{m_2} + {n_1}{n_2}|

gives

\dfrac{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}{\sqrt{{a_1}^2 + {b_1}^2 + {c_1}^2}   + \sqrt{{a_1}^2 + {b_1}^2 + {c_1}^2}}

gives

.

Math Is Fun Forum

#176 2022-11-23 15:10:41

Re: LaTeX - A Crash Course

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