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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
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**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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
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**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
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**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**Coordinates of the mid-point**

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

gives

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**Slope**

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

gives

`m = tan\theta`

gives

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**Equation of Straight Line : Two Point Form**

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

gives

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
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**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**Collinearity of three points**

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

gives

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**Angle Between Straight Lines**

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

gives .

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
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**Angle between two straight lines**

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

gives

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
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**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.

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 48,320

**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

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

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