Maths with Hamza - Tutorials in Urdu | English

Solve by using Matrix Method & Cramer's Rule | Matrices Ki Equation Ko Kaise Solve Kare ?

2020-09-14T13:32:00.062+05:00

MATRIX METHOD:

Use matrix method to solve the following system of linear equations.

` x + 2 y + z = 8 `

` 2 x - y + z = 3 `

` x + y - z = 0 `

Solution:

The given system can be written in the matrix form AX=B, as

` \begin{bmatrix}a & b \\c & d \end{bmatrix} `

` \begin{bmatrix}2&\;\;\;4\\3&-4\end{bmatrix} `

` \begin{bmatrix} 11 & 21 & 21\\ 212 & 2 & 12 \end{bmatrix} `

` \left(\begin{array}{c}54\\ 23\end{array}\right) `

` \begin{vmatrix}54&5\\3&3\end{vmatrix} `

` \begin{bmatrix}&\\&\end{bmatrix} `

CRAMER'S RULE:

Use cramer's rule to solve the following system of linear equations.

` x + 2 y + z = 8 `

` 2 x - y + z = 3 `

` x + y - z = 0 `

Solution:

` \ begin{ b matrix } 234 \ \ 23 \ end{ b matrix } `

Sequence and Series Formulas

2020-09-12T19:43:00.234+05:00

Below are the list of all sequence and series formulas;

ARITHMETIC SEQUENCE OR

ARITHMETIC PROGRESSION (A.P):

To find the nth term of Arithmetic Progressions are

` T_n = a + \ left( n - 1 \ right) d `

Where;

` a ` = first term ,

` d ` = difference ,

` n ` = number of terms ,

` T_n ` = nth term or last term or general term or rule of formation

To find the number of terms of Arithmetic Progressions are

` n = \ frac{ T_n - a } d+1 `

To find the difference of Arithmetic Progressions are

Difference = Second term - First term

` d = \ frac{ T_n - a } { n-1 } `

ARITHMETIC SERIES:

To find the sum of Arithmetic Progressions are

` S_n = \ frac{ n } { 2 } \ [ 2 a + (n - 1 ) d ]`

` S_n = \ frac{ n } { 2 } \ ( a + l )`

ARITHMETIC MEANS:

The formula to find the arithmetic mean are given as;

` A.M = \ frac{ a+b } { 2 } `

When you have more than one arithmetic mean, first you can find d by using this formula;

` d = \ frac{ b-a } { n+1 } `

Then, you can find n(number of) arithmetic means;

` A_1 = a + 1 d `

` A_2 = a + 2 d `

` A_3 = a + 3 d `

` A_n = a + n d `

WHEN TERMS ARE IN ARITHMETIC

PROGRESSION:

When three numbers are in Arithmetic Progression (A.P)

First number are in Arithmetic Progression (A.P) = ` a - d `

Second number are in Arithmetic Progression (A.P) = ` a `

Third number are in Arithmetic Progression (A.P) = ` a + d `

When four numbers are in Arithmetic Progression (A.P)

First number are in Arithmetic Progression (A.P) = ` a - 3 d `

Second number are in Arithmetic Progression (A.P) = ` a - d `

Third number are in Arithmetic Progression (A.P) = ` a + d `

Fourth number are in Arithmetic Progression (A.P) = ` a + 3 d `

When five numbers are in Arithmetic Progression (A.P)

First number are in Arithmetic Progression (A.P) = ` a - 2 d `

Second number are in Arithmetic Progression (A.P) = ` a - d `

Third number are in Arithmetic Progression (A.P) = ` a `

Fourth number are in Arithmetic Progression (A.P) = ` a + d `

Fifth number are in Arithmetic Progression (A.P) = ` a + 2 d `

GEOMETRIC SEQUENCE OR

GEOMETRIC PROGRESSION (G.P):

To find the nth term of Geometric Progressions are

` T_n = a r^{ n-1 } `

Where;

` a ` = first term ,

` r ` = ratio ,

` n ` = number of terms ,

` T_n ` = nth term or last term or general term or rule of formation

To find the ratio of Geometric Progressions are

ratio = Second term ` \div ` First term

GEOMETRIC SERIES:

There are two formulas to find the sum of geometric series formulas;

When r > 1

` S_n = \ frac{ a \ left( r ^ n - 1 \ right) } { r - 1 } `

` S_n = \ frac{ r l - a } { r - 1 } `

When r < 1

` S_n = \ frac{ a \ left( 1 - r ^ n \ right) } { 1 - r } `

` S_n = \ frac{ a - r l } { 1 - r } `

INFINITE GEOMETRIC SERIES:

The formula to find the sum of infinite geometric series are given as;

` S = \ frac{ a } { 1-r } `

GEOMETRIC MEANS:

The formula to find the geometric mean are given as;

` G.M = \ pm \ sqrt { a b } `

When you have more than one geometric mean, first you can find r by using this formula;

` r = \ left( \ frac b a \ right) ^ \frac { 1 } { n+1 } `

Then, you can find n(number of) geometric means;

` G_1 = a r ^ 1 `

` G_2 = a r ^ 2 `

` G_3 = a r ^ 3 `

` G_n = a r ^ n `

WHEN TERMS ARE IN GEOMETRIC

PROGRESSION:

When three numbers are in Geometric Progression (G.P)

First number are in Geometric Progression (G.P) = ` \frac a r `

Second number are in Geometric Progression (G.P) = ` a `

Third number are in Geometric Progression (G.P) = ` a r `

When four numbers are in Geometric Progression (G.P)

First number are in Geometric Progression (G.P) = ` \ frac a { r^3 } `

Second number are in Geometric Progression (G.P) = ` \frac a r `

Third number are in Geometric Progression (G.P) = ` a r `

Fourth number are in Geometric Progression (G.P) = ` a r^3 `

When five numbers are in Geometric Progression (G.P)

First number are in Geometric Progression (G.P) = ` \ frac a { r^2 } `

Second number are in Geometric Progression (G.P) = ` \frac a r `

Third number are in Geometric Progression (G.P) = ` a `

Fourth number are in Geometric Progression (G.P) = ` a r `

Fifth number are in Geometric Progression (G.P) = ` a r^2 `

HARMONIC SEQUENCE OR

HARMONIC PROGRESSION (H.P):

To find the nth term of Harmonic Progressions are

` T_n = \ frac{ a b } { b + \ left( n - 1 \ right) \ left( a - b \ right) } `

HARMONIC MEANS:

The formula to find the harmonic mean are given as;

` H.M = \ frac{ 2 a b } { a+b } `

When you have more than one harmonic mean, first you can find d by using this formula;

` d = \ frac{ ( a - b ) } { left( n - 1 \ right) \ a b } `

Then, you can find n(number of) harmonic means;

` H_1 = \ frac{ \ left( n + 1 \ right) ab } { a + nb } `

` H_2 = \ frac{ \ left( n + 1 \ right) ab } { 2a + \ left( n - 1 \ right) b } `

` H_3 = \ frac{ \ left( n + 1 \ right) ab } { 3a + \ left( n - 2 \ right) b } `

` H_n = \ frac{ \ left( n + 1 \ right) ab } { na + b } `

Binomial Theorem

2020-09-12T12:53:00.015+05:00

BINOMIAL THEOREM FORMULAS:

Below are the list of some important formulas;

BINOMIAL THEOREM:

` \ left( a + b \ right) ^ n = a^n + \ ^ n C_1 \ a ^ { n-1 } b^1 + \ ^ n C_2 \ a ^ { n-2 } b^2 + \ ^ n C_3 \ a ^ { n-3 } b^3 + \ cdot \ cdot \ cdot + \ ^ n C_n \ a ^ { n-n } b^n `

` \ left( a - b \ right) ^ n = a^n - \ ^ n C_1 \ a ^ { n-1 } b^1 + \ ^ n C_2 \ a ^ { n-2 } b^2 - \ ^ n C_3 \ a ^ { n-3 } b^3 + \ cdot \ cdot \ cdot + \ ^ n C_n \ a ^ { n-n } b^n `

COMBINATION:

` \ ^ n C_r = \ frac { n! } { r! ( n - r) ! } `

PERMUTATION:

` \ ^ n P_r = \ frac { n! } { ( n - r) ! } `

THE GENERAL TERM:

The term shows below is called the general term in the expansion of ` \ left( a + b \ right) ^ n ` for a positive integral index n.

` T_{r+1} = \ ^ n C_r \ a ^{ n-r } \ b^r `

MIDDLE TERMS:

We shall find the middle term or terms in the expansion of ` \ left( a + b \ right) ^ n ` , n.

If n = even, we have find only one terms by using this formula

` \ frac { n+2 } 2 `

If n = odd, we have find only two terms by using this formula

` \ frac { n+1 } 2 `

` \ frac{ n+3 } 2 `

BINOMIAL SERIES:

Statement 1:

If n is a positive integer and | x | < 1 then

` \ left( 1 + x \ right) ^ n = 1 + nx + \ frac{ n ( n-1 ) } { 2 ! } x ^ 2 + \ cdot \ cdot \ cdot + \ frac{ n ( n-1 ) ( n-2 ) \ cdot \ cdot \ cdot ( n-r+1 ) x^r } { r ! } `

Statement 2:

If n is a positive integer and replacing x by -x then

` \ left( 1 - x \ right) ^ { n } = 1 - nx + \ frac{ n ( n-1 ) } { 2 ! } x ^ 2 + \ cdot \ cdot \ cdot + ( -1 ) ^r \ \ frac{ n ( n-1 ) ( n-2 ) \ cdot \ cdot \ cdot ( n-r+1 ) } { r ! } x^r `

Statement 3:

Changing the sign of n in statement 1 then

` \ left( 1 + x \ right) ^ { -n } = 1 - nx + \ frac{ n ( n+1 ) } { 2 ! } x ^ 2 - \ cdot \ cdot \ cdot + ( -1 ) ^r \ \ frac{ n ( n+1 ) ( n+2 ) \ cdot \ cdot \ cdot ( n+r-1 ) } { r ! } x^r `

Statement 4:

Changing the sign of n in statement 2 then

` \ left( 1 - x \ right) ^ { -n } = 1 + nx + \ frac{ n ( n+1 ) }{ 2 ! } x ^ 2 + \ cdot \ cdot \ cdot + ( -1 ) ^r \ \ frac{ n ( n+1) ( n+2 ) \ cdot \ cdot \ cdot ( n+r-1 ) } { r ! } x^r `

Here, for | x | < 1,

Statement 5:

` ( 1 + x ) ^ { -1 } = 1 - x + x ^ 2 - x ^ 3 + \ cdot \ cdot \ cdot + ( -1 ) ^ r x ^ r + \ cdot \ cdot \ cdot `

Statement 6:

` ( 1 - x ) ^ { -1 } = 1 + x + x ^ 2 + x ^ 3 + \ cdot \ cdot \ cdot + x ^ r + \ cdot \ cdot \ cdot `

Statement 7:

` ( 1 + x ) ^ { -2 } = 1 - 2 x + 3 x ^ 2 - 4 x ^ 3 + \ cdot \ cdot \ cdot +( -1 ) ^ r ( r + 1 ) x ^ r + \ cdot \ cdot \ cdot `

Statement 8:

` ( 1 - x ) ^ { -2 } = 1 + 2 x + 3 x ^ 2 + 4 x ^ 3 + \ cdot \ cdot \ cdot +( r + 1 ) x ^ r + \ cdot \ cdot \ cdot `

Mathematical Induction

2020-09-12T12:52:00.004+05:00

Complex Number

2020-08-25T16:49:00.005+05:00

What is Complex Number ?

کمپلیکس نمبر کیا ہے؟

How to Solve Quadratic Equation ? | Quadratic Equation Ko 3 Methods Se Kaise Solve Kare

2020-08-20T17:23:00.022+05:00

How to Solve Quadratic Equation ?

QUADRATIC EQUATION ?

   A quadratic equation in x is an equation that can be written in the standard form
` a x^2 + b x + c = 0 `

where a, b, and c are real numbers and a ` \neq ` 0.

a represents the numerical coefficient of ` x^2 ` ,
b represents the numerical coefficient of ` x ` , and
c represents the constant numerical term.


Example:

` 2 x^2 = 0 `
` 2 x^2 - 50 =0 `
` 2 x^2 - 7 x = 0 `
` 5 x^2 - 3 x + 3 = 0 `

Before you solve the quadratic equation, you need to check the quadratic equation must be in standard form.

FORMULA:

` \ frac { - b \ pm \ sqrt { b ^ 2 - 4 a c } } { 2 a } \ `

METHODS:

There are three methods to solve quadratic equation:

1. Quadratic formula

2. Middle term breaking

3. Factorization

EXAMPLE QUESTION:

Q) `6x^2-x-2=0`

1. QUADRATIC FORMULA:

Solution:

  `6x^2-x-2=0`

Using the quadratic formula;

   ` x = \ frac { - b \ pm \ sqrt { b ^ 2 - 4 a c } } { 2 a } \ `

  ` x = \ frac { - ( -1 ) \ pm \ sqrt { ( -1 )^2 -4 (6) (-2) } } { 2(6) } \ `

` x = \ frac { 1 \ pm \ sqrt { ( 1 ) + ( 48 ) } } { 12 } \ `

  ` x = \ frac { 1 \ pm \ sqrt { 49 } } { 12 } \ `

  ` x = \ frac { 1 \ pm \ 7 } { 12 } \ `

                                       either          ;           or

` x = \ frac { 1+7 } { 12 } `              ;       ` x = \ frac { 1-7 } { 12 }`

` x = \ frac 8 { 12 } `                         ;        ` x = \ frac { -6 } { 12 } `

` x = \ frac 2 \ 3 `                              ;      ` x = \ frac - 1 \ 2`

Hence,

  Solution Set = { ` \ frac 2 \ 3 ` , `  \ frac -1 \ 2 ` }

2. MIDDLE TERM BREAKING:

Solution:

3. FACTORIZATION:

Solution:

Harmonic Mean

2020-08-12T21:20:00.015+05:00

Harmonic Mean : ہم آہنگی کا مطلب

Harmonic Series

2020-08-12T21:20:00.013+05:00

Harmonic Series : ہارمونک سیریز

Infinite Geometric Series

2020-08-12T21:17:00.008+05:00

Infinite Geometric Series : لامحدود ہندسی سلسلہ

Geometric Mean

2020-08-12T21:17:00.007+05:00

Geometric Mean : ہندسی مطلب

Geometric Series

2020-08-12T21:12:00.007+05:00

Geometric Series : ہندسی سلسلہ

Arithmetic Mean

2020-08-12T21:10:00.010+05:00

Arithmetic Mean : حسابی اوسط

Geometry | Types of Geometry

2020-08-10T14:07:00.005+05:00

What is Geometry and its types ? : جیومیٹری کیا ہے اور اس کی اقسام کیا ہیں؟

Arithmetic Series

2020-08-09T22:36:00.007+05:00

Arithmetic Series : ریاضی سیریز

Matrices | Types of Matrices

2020-08-09T22:33:00.005+05:00

What is Matrices and its types ? : میٹرک کیا ہے اور اس کی اقسام کیا ہیں؟

Integrational Formulas

2020-07-27T22:55:00.005+05:00

Integration Formulas : انضمام کے فارمولے

Integration Topics

2020-07-27T22:51:00.004+05:00

Integration Topics: انٹیگریشن کے عنوانات

Straight Line Equation

2020-07-27T22:50:00.006+05:00

Straight Line Equation : سیدھی لائن مساوات

Algebra

2020-07-27T22:49:00.003+05:00

Algebra : الجبرا

Statistics

2020-07-27T22:48:00.008+05:00

Statistics : اعدادوشمار

Differential terms and Conditions

2020-07-27T22:45:00.004+05:00

Differential terms and Conditions : مختلف شرائط و ضوابط

Trigonometric Functions Formulas

2020-05-28T16:29:00.036+05:00

Trigonometric Formulas:

Below are the list of all trigonometric functions formulas;

Basic Formulas:

` \ sin \ left( \theta\right) \ = \ \ frac1 {\ csc( \theta)}\ `

OR

` \ csc \ left( \theta\right)\ = \ \ frac1 {\ sin\ left( \theta\right)}`

=====

` \ cos \ left( \theta\right) \ = \ \ frac1 {\ sec( \theta)}\ `

OR

` sec \ left( \theta\right) \ = \ \ frac1 {\ cos\ left( \theta\right)} `

=====

` \ tan\ left( \theta\right) \ = \ \frac {\ sin\ left( \theta\right)}{\ cos\ left( \theta\right)}\ `

OR

` \ tan\ left( \theta\right) \ = \ \frac1 {\ cot\ left( \theta\right)} `

=====

` \ cot\ left( \theta\right)\ = \ \ frac {\ cos\ left( \theta\right)}{\ sin \ left( \theta\right)}`

OR

` \ cot \ left( \theta\right) \ = \ \ frac1 {\ tan\ left( \theta\right)} `

=====

` \ sin^2\ left( \theta\right) \ + \ \ cos^2 \ left( \theta\right)\ = \ 1 \ `

OR

` \ sin^2\ left( \theta\right) \ = \ 1 \ - \ \ cos^2 \ left( \theta\right) `

OR

` \ cos^2 \ left( \theta\right) \ = \ 1 \ - \ \ sin^2 \ left( \theta\right) `

=====

` 1 \ + \ \ tan^2 \ left( \theta\right) \ = \ \ sec^2 \ left( \theta\right) `

OR

` \ tan^2 \ left( \theta\right) \ = \ \ sec^2 \ left(\theta\right) \ - \ 1 `

=====

` 1 \ + \ \cot^2\ left( \theta\right) \ = \ \ csc^2 \ left(\theta\right) `

OR

` \ cot^2 \ left( \theta\right) \ = \ \ csc^2 \ left( \theta\right) \ - \ 1 `

=====

NEGATIVE(-ve) ANGLE FORMULAS:

` \ sin \ left( -\theta\right) \ =\ - \ sin\ left( \theta\right) `

` \ cos \ left( -\theta\right) \ = \ \ cos\ left( \theta\right) `

` \ tan \ left( -\theta\right) \ = \ - \ tan\ left( \theta\right) `

` \ cot \ left( -\theta\right) \ = \ - \ cot\ left( \theta\right) `

` \ sec \ left( -\theta\right) \ =\ \ sec\ left( \theta\right) `

` \ csc \ left( -\theta\right) \ = \ - \ csc\ left( \theta\right) `

DOUBLE ANGLE FORMULAS:

` \ sin \ left( 2\theta\right) \ = \ 2 \ sin\ left( \theta\right) \ cos\ left( \theta\right) `

OR

` \ sin \ left( 2\theta\right)\ = \ \ frac { 2 \tan\left( \theta\right)}{ 1\ + \ \ tan^2\ left( \theta\right)} `

=====

` \ cos \ left( 2\theta\right) \ = \ \ cos^2\ left( \theta\right)\ -\ \ sin^2\ left( \theta\right) `

OR

` \ cos \ left( 2\theta\right) \ = \ 2 \ cos^2 \ left( \theta\right)\ - \ 1 `

OR

` \ cos \ left( 2\theta\right) \ = \ 1\ - \ 2 \ sin^2\ left( \theta\right) `

OR

` \ cos \ left( 2\theta\right) \ = \ \ frac {1\ - \ \ tan^2\ left( \theta\right)}{1\ + \ \ tan^2\ left( \theta\right)} `

= = = = =

` \ tan \ left( 2\theta\right) \ = \ \ frac { 2 \ tan \ left( \theta\right)}{ 1-\ tan^2\ left( \theta\right)} `

HALF ANGLE FORMULAS:

` \ sin \ left( \ frac \alpha2\ right) \ = \ \ pm \ sqrt{ \ frac { 1- \ cos\ left( \ alpha \ right)} 2} `

` \ cos \ left( \ frac \alpha2\ right) \ = \ \ pm \ sqrt{ \ frac { 1+ \ cos\ left( \ alpha \ right)} 2} `

` \ tan \ left( \ frac \alpha2\ right) \ = \ pm \ sqrt{ \ frac{1 - \ cos\ left( \ alpha \ right)} {1 + \ cos\ left( \ alpha \ right)}} `

FUNDAMENTAL LAWS:

` \ sin \ left( \ alpha + \ beta \ right) = \ sin\ left( \ alpha \ right) \ cos\ left( \ beta \ right) + \ cos\ left( \ alpha \ right)\ sin\ left(\ beta \ right) `

` \ sin\ left( \ alpha - \ beta \ right) = \ sin\ left( \ alpha \ right) \ cos\ left( \ beta \ right) - \ cos\ left( \ alpha \ right)\ sin\ left(\ beta \ right) `

` \ cos\ left( \ alpha + \ beta \ right) = \ cos\ left( \ alpha \ right) \ cos\ left( \ beta \ right) - \ sin\ left( \ alpha \ right)\ sin\ left(\ beta \ right) `

` \ cos\ left( \ alpha - \ beta \ right) = \ cos\ left( \ alpha \ right) \ cos\ left( \ beta \ right) + \ sin\ left( \ alpha \ right)\ sin\ left(\ beta \ right) `

` \ tan\ left( \ alpha + \ beta \ right) = \ frac { \ tan \ left( \ alpha \ right) + \ tan\ left( \ beta \ right) } {1- \ tan\ left( \ alpha \ right) \ tan \ left( \ beta \ right) }`

`\ tan\ left( \ alpha -\ beta \ right) = \ frac { \ tan \ left( \ alpha \ right) -\ tan \ left( \ beta \ right) } {1 + \ tan\ left( \ alpha \ right) \ tan \ left( \ beta \ right) } `

` \ cot \ left( \ alpha + \ beta \ right) = \ frac { \ cot \ left( \ alpha \ right) \ cot\ left( \ beta \ right) - 1} { \ cot\ left( \ alpha \ right) + \ cot \ left( \ beta \ right)} `

` \ cot \ left( \ alpha -\ beta \ right) = \ frac { \ cot \ left( \ alpha \right) \ cot\ left( \ beta \ right) + 1} { \ cot\ left( \ beta \ right) - \ cot \ left( \ alpha \ right)} `

PRODUCT TO SUM FORMULAS:

` \ sin \ left( \ alpha \ right) \ \ cos \ left( \ beta \ right) = \ frac1\2 \ left[ \ sin \ left( \ alpha + \ beta \ right) + \ sin \ left( \ alpha - \ beta \ right) \ right] `

` \ cos \ left( \ alpha \ right) \ \ sin \ left( \ beta \ right) = \ frac1\2 \ left[ \ sin \ left( \ alpha + \ beta \ right) - \ sin \ left( \ alpha - \ beta \ right) \ right] `

` \ cos \ left( \ alpha \ right) \ \ cos \ left( \ beta \ right) = \ frac1\2 \ left[ \ cos \ left( \ alpha + \ beta \ right) + \ cos \ left( \ alpha - \ beta \ right) \ right] `

` \ sin \ left( \ alpha \ right) \ \ sin \ left( \ beta \ right) = - \ frac1\2 \ left[ \ cos \ left( \ alpha + \ beta \ right) - \ cos \ left( \ alpha - \ beta \ right) \ right] `

SUM TO PRODUCT FORMULAS:

` \ sin \ left( \ upsilon \ right) + \ sin \ left( \ nu \ right) = 2 \ sin \ left( \ frac { \ upsilon + \ nu} 2 \ right) \ cos \ left( \ frac { \ upsilon - \ nu} 2 \ right) `

` \ sin \ left( \ upsilon \ right) -\ sin \ left( \ nu \ right) = 2 \ cos \ left( \ frac { \ upsilon + \ nu} 2 \ right) \ sin \ left( \ frac { \ upsilon - \ nu} 2 \ right) `

` \ cos \ left( \ upsilon \ right) + \ cos \ left( \ nu \ right) = 2 \ cos \ left( \ frac { \ upsilon + \ nu} 2 \ right) \ cos \ left( \ frac { \ upsilon - \ nu} 2 \ right) `

` \ cos \ left( \ upsilon \ right) -\ cos \ left( \ nu \ right) = - 2 \ sin \ left( \ frac { \ upsilon + \ nu} 2 \ right) \ sin \ left( \ frac { \ upsilon - \ nu} 2 \ right) `

LAW OF SINE:

` \ frac { \ sin \ left( \ alpha \ right) } a = \ frac{ \ sin \ left( \ beta \ right) } b = \ frac{ \ sin \ left( \ gamma \ right) } c `

OR

` \ frac a { \ sin \ left( \ alpha \ right) } = \ frac b { \sin \ left( \ beta \ right) } = \ frac c { \ sin \ left( \ gamma \ right) } `

OR

` a \ : \ b \ : \ c \ = \ \ sin\ left( \ alpha \ right) \ : \ \ sin \ left( \ beta \ right) \ : \ \ sin \ left( \ gamma \ right) `

LAW OF COSINE:

` \ cos \ left( \ alpha \ right) = \ frac { b^2 + c^2 - a^2} { 2bc } `

OR

` a^2 = b^2 + c^2 -2bc \ cos \ left( \ alpha \ right) `

=====

` \ cos \ left( \ beta \ right) = \ frac { a^2 + c^2 - b^2} { 2ac } `

OR

` b^2 = a^2 + c^2 -2ac \ cos \ left( \ beta \ right) `

=====

` \ cos \ left( \ gamma \ right) = \ frac { a^2 + b^2 - c^2} { 2ab } `

OR

` c^2 = a^2 + b^2 -2ab \ cos \ left( \ gamma \ right) `

HALF ANGLE FORMULAS:

` \ sin \ left( \ frac \alpha2\ right) \ = \ sqrt { \ frac { (s-b) \ (s-c) } { bc } }`

` \ sin \ left( \ frac \beta2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-c) } { ac } }`

` \ sin \ left( \ frac \gamma2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-b) } { ab } }`

` \ cos \ left( \ frac \alpha2\ right) \ = \ sqrt { \ frac { s (s-a) } { bc } }`

` \ cos \ left( \ frac \beta2\ right) \ = \ sqrt { \ frac { s (s-b) } { ac } }`

` \ cos \ left( \ frac \gamma2\ right) \ = \ sqrt { \ frac { s (s-c) } { ab } }`

` \ tan \ left( \ frac \alpha2\ right) \ = \ sqrt { \ frac { (s-b) \ (s-c) } { s (s-a) } } `

` \ tan \ left( \ frac \beta2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-c) } { s (s-b) } } `

` \ tan \ left( \ frac \gamma2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-b) } { s (s-c) } } `

AREA OF A TRIANGLE:

` \ triangle = Area `

3 SIDES GIVEN:

` \ triangle = \ sqrt { s (s-a) (s-b) (s-c) } `

` s = \ frac { a + b + c } 2 `

2 SIDES INCLUDED ANGLE GIVEN:

` \ triangle = \ frac { bc \ sin \ left( \ alpha \ right) } 2 `

` \ triangle = \ frac { ac \ sin \ left( \ beta \ right) } 2 `

` \ triangle = \ frac { ab \ sin \ left( \ gamma \ right) } 2 `

1 SIDES INCLUDED 2 ANGLES GIVEN:

` \ triangle = \ frac { a^2 \ sin \ left( \ beta \ right) \ sin \ left( \ gamma \ right) } { 2 \ sin \ left( \ alpha \ right) } `

` \ triangle = \ frac { b^2 \ sin \ left( \ alpha \ right) \ sin \ left( \ gamma \ right) } { 2 \ sin \ left( \ beta \ right) } `

` \ triangle = \ frac { c^2 \ sin \ left( \ alpha \ right) \ sin \ left( \ beta \ right) } { 2 \ sin \ left( \ gamma \ right) } `

CIRCUM-RADIUS R IN TERMS OF THE MEASURE OF A THREE SIDES OF A TRIANGLE:

`R \ = \ frac{ \ abc } { 4\triangle}`

THE IN-RADIUS OF A TRIANGLE:

` r = \ frac \triangle s`

RADII OF E-CIRCLES OF A TRIANGLE:

` r_1 = \ frac \triangle {s-a} `

` r_2 = \ frac \triangle {s-b} `

` r_3 = \ frac \triangle {s-c} `

Trigonometric Identities

2020-05-28T16:09:00.003+05:00

Trigonometric Identities : سہ رخی شناخت

Is post main aap ko poore trigonometric formula shows honge aur umeed hai k aap ko yeh sab formula bohat asaani se samaj aega.

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A = All

S = sinθ

T = tanθ

C = cosθ