Straight Lines

Straight Lines:

A straight line is a line that extends infinitely in both directions and has no curves or bends.
The slope-intercept form of a straight line is y = mx + b, where m is the slope of the line and b is the y-intercept.
The point-slope form of a straight line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
The standard form of a straight line is Ax + By = C, where A, B, and C are constants and A and B are not both zero.
Two lines are parallel if and only if they have the same slope and never intersect.

Important Formulas to Remember in these Straight lines topic

Slope-intercept form: $$y = mx + b$$
Point-slope form: $$(y - y_1) = m(x - x_1)$$
Two-point form: $$\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}$$

Distance of a point from a line:

The distance between a point and a line is the length of the perpendicular line segment from the point to the line.
To find the distance between a point and a line, you can use the formula d = |Ax1 + By1 + C| / sqrt(A^2 + B^2), where (x1, y1) is the point and Ax + By + C = 0 is the equation of the line.
The sign of the expression Ax1 + By1 + C indicates which side of the line the point is on.
If the line passes through the origin (0,0), the distance between the point and the line is simply |Ax1 + By1| / sqrt(A^2 + B^2).
The shortest distance between two parallel lines is the perpendicular distance between them, which can be found using the same formula as above.

Important Formulas to Remember in these distance of a point from line topic

Cartesian form: $$\frac{|ax + by + c|}{\sqrt{a^2 + b^2}}$$
Vector form: $$\frac{|\mathbf{n} \cdot \mathbf{p} + d|}{\lVert \mathbf{n} \rVert}$$

Angle between two lines:

The angle between two lines is the angle formed by their intersection.
The slope of a line is related to the angle it makes with the positive x-axis by the formula theta = arctan(m), where m is the slope.
The angle between two lines with slopes m1 and m2 is given by the formula theta = |arctan((m1-m2) / (1+m1m2))|.
The angle between two perpendicular lines is 90 degrees or pi/2 radians.
The angle between two parallel lines is either 0 degrees or pi radians.

Important Formulas to Remember in these angle between two points topic

Formula: $$\theta = \cos^{-1}\left(\frac{m_1m_2 - 1}{\sqrt{1 + m_1^2}\sqrt{1 + m_2^2}}\right)$$

Intersection of two non-parallel lines:

Two non-parallel lines will intersect at a single point, unless they are coincident (i.e., the same line).
The point of intersection can be found by solving the system of equations Ax + By = C and Dx + Ey = F simultaneously.
If the lines are given in slope-intercept form, you can solve for the intersection point by setting their y-values equal to each other and solving for x, then plugging that value back into either equation to find the corresponding y-value.
If the lines are given in point-slope form, you can find the intersection point by plugging the coordinates of one of the points into the equation for the other line and solving for the parameter.
If the lines are given in standard form, you can solve for the intersection point by finding the determinant of a 2x2 matrix formed from the coefficients of x and y in the two equations.

Important Formulas to Remember in these intersection of two non parallel lines topic

Formula: $$\begin{aligned} x &= \frac{b_2c_1 - b_1c_2}{a_1b_2 - a_2b_1} \\ y &= \frac{c_2a_1 - c_1a_2}{a_1b_2 - a_2b_1} \end{aligned}$$

Distance between two parallel lines:

Two parallel lines have the same slope and will never intersect.
The distance between two parallel lines is the length of the shortest line segment that is perpendicular to both lines.
To find the distance between two parallel lines, you can find the distance between one of the lines and any point on the other line.
Alternatively, you can use the distance formula for a point and a line to find the distance between a point on one line and the other line.
The distance between two parallel lines is the absolute value of the difference between their y-intercepts, divided by the square root of the sum of the squares of their slopes.

Important Formulas to Remember in these distance between two parallel lines topic

Formula: $$\frac{|\mathbf{d} \cdot \mathbf{n}|}{\lVert \mathbf{n} \rVert}$$

Modulus-Amplitude form (Polar form) and Euler form (exponential form)

Modulus-Amplitude form (Polar form):

The modulus-amplitude form, also known as the polar form, of a complex number is given by z = r(cos θ + i sin θ), where r is the modulus and θ is the amplitude.
The modulus and amplitude can be calculated using the formulas r = |z| = √(x² + y²) and θ = arg(z) = tan⁻¹(y/x), where x and y are the real and imaginary parts of the complex number.
The polar form of a complex number is useful for performing operations such as multiplication and division, which are simpler in polar form than in rectangular form.
The polar form of a complex number is also useful for representing complex numbers graphically in the complex plane.
The polar form of a complex number can be converted to rectangular form using the formulas x = r cos θ and y = r sin θ, while the rectangular form can be converted to polar form using the formulas r = √(x² + y²) and θ = tan⁻¹(y/x).

Euler form (exponential form):

The Euler form, also known as the exponential form, of a complex number is given by z = re^(iθ), where r is the modulus and θ is the amplitude.
The exponential form of a complex number can be obtained from its polar form using the formula e^(iθ) = cos θ + i sin θ, known as Euler's formula.
The exponential form of a complex number is useful for performing operations such as powers and roots, which are simpler in exponential form than in rectangular or polar form.
The exponential form of a complex number is also useful for representing complex numbers graphically in the complex plane, as it provides a natural way to interpret rotations and magnifications.
The exponential form of a complex number can be converted to polar form using the formulas r = |z| = e^(ln(r)) and θ = arg(z) = Im(ln(z)/i), while the polar form can be converted to exponential form using the formula z = re^(iθ).

Arithmetic operations on complex numbers

Addition and subtraction of complex numbers are performed by adding or subtracting their real and imaginary parts separately.
Multiplication of complex numbers is performed by using the distributive property and the fact that i² = -1.
Division of complex numbers is performed by multiplying the numerator and denominator by the conjugate of the denominator, and then simplifying.
The absolute value or modulus of a complex number is obtained by taking the square root of the sum of the squares of its real and imaginary parts.
The complex conjugate of a complex number is obtained by changing the sign of its imaginary part.

Properties of Modulus, amplitude and conjugate of complex numbers

The modulus of a complex number is always non-negative, and is zero if and only if the complex number is zero.
The modulus of a complex number satisfies the triangle inequality, which states that the modulus of the sum of two complex numbers is less than or equal to the sum of their individual moduli.
The amplitude of a complex number is not unique, as adding or subtracting a multiple of 2π to the amplitude produces an equivalent angle.
The amplitude of the product of two complex numbers is the sum of their individual amplitudes, while the amplitude of the quotient of two complex numbers is the difference of their individual amplitudes.
The conjugate of a complex number is obtained by reflecting the number about the real axis, and has the same modulus as the original number.

Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined by i² = -1.
Complex numbers can be added, subtracted, multiplied, and divided using the rules of algebra.
The modulus of a complex number is the distance from the origin to the point in the complex plane that represents the number.
The amplitude of a complex number is the angle that the line from the origin to the point in the complex plane that represents the number makes with the positive real axis.
The conjugate of a complex number is obtained by changing the sign of the imaginary part.

Important Formulas to Remember in these Complex Numbers

$ z=a+bi $, where $ a,b\in\mathbb{R} $ and $ i^2=-1 $.
$ |z|=\sqrt{a^2+b^2} $ is the modulus of the complex number.
$ \text{Arg}(z)=\theta $, where $ \theta $ is the argument of the complex number, i.e. the angle it makes with the positive real axis in the complex plane.
$ \text{Re}(z)=a $ and $ \text{Im}(z)=b $ are the real and imaginary parts of the complex number, respectively.
$ \overline{z}=a-bi $ is the complex conjugate

Hyperbolic functions

Hyperbolic functions are analogues of trigonometric functions that are defined in terms of the exponential function.
The hyperbolic functions are denoted by sinh, cosh, tanh, csch, sech, and coth.
Hyperbolic functions have many of the same properties as trigonometric functions, such as being periodic and having certain symmetries and identities.
Hyperbolic functions are commonly used in areas of mathematics such as calculus, differential equations, and complex analysis.
The hyperbolic functions can be used to solve problems involving hyperbolic geometry and hyperbolic trigonometry.

Important Formulas to Remember in these Hyperbolic functions

$ \sinh x=\frac{1}{2}(e^x-e^{-x}) $
$ \cosh x=\frac{1}{2}(e^x+e^{-x}) $
$ \tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}} $

Inverse Trigonometric functions

Inverse trigonometric functions are used to find the angle that corresponds to a given value of a trigonometric function.
The inverse trigonometric functions are denoted by arcsin, arccos, arctan, arccsc, arcsec, and arccot.
The domain and range of the inverse trigonometric functions depend on the choice of branch, which is determined by the sign of the argument.
The inverse trigonometric functions are commonly used to solve trigonometric equations and to find the angles of right triangles given their sides.
The inverse trigonometric functions can also be used to convert between rectangular and polar coordinates.

Important Formulas to Remember in these Inverse Trigonometric functions

$ \sin^{-1}x+y=\sin^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2}) $
$ \cos^{-1}x+y=\cos^{-1}(x\sqrt{1-y^2}-y\sqrt{1-x^2}) $
$ \tan^{-1}x+y=\tan^{-1}(\frac{x+y}{1-xy}) $

Solving a triangle

To solve a triangle means to find the lengths of all its sides and the measures of all its angles.\
The SSS (side-side-side) condition is used when the lengths of all three sides of a triangle are given.
The SAS (side-angle-side) condition is used when the lengths of two sides and the measure of the included angle of a triangle are given.
The SAA (side-angle-angle) condition is used when the length of one side and the measures of two angles of a triangle are given.
To solve a triangle, one can use a combination of the sine, cosine, and tangent rules, as well as the Pythagorean theorem and basic algebra.

Important Terms to Remember in these Solving a triangle

When three sides (SSS) are given: use the cosine rule to find an angle, then the sine rule to find the other two angles.
When two sides and an included angle (SAS) are given: use the cosine rule to find the third side, then the sine rule to find the other two angles.
When one side and two angles are given (SAA): use the sine rule to find the other two sides, then the cosine rule to find the remaining angle.

Properties of triangles

The sine, cosine, and tangent rules are used to solve problems involving triangles.
The sine rule relates the length of a side of a triangle to the sine of the opposite angle.
The cosine rule relates the length of a side of a triangle to the lengths of the other two sides and the cosine of the included angle.
The tangent rule relates the length of a side of a triangle to the tangent of the opposite angle and the lengths of the other two sides.
The projection rule, also known as the law of sines, can be used to find the height of a triangle given the length of the opposite side and the angle opposite to it.

Important Formulas to Remember in these Properties of triangles

Sine rule: $ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} $
Cosine rule: $ a^2=b^2+c^2-2bc\cos A $
Tangent rule: $ \frac{a-b}{a+b}=\frac{\tan\frac{1}{2}(A-B)}{\tan\frac{1}{2}(A+B)} $
Projection rule: $ a=b\cos C+c\cos B $

Transformations of Products into sum or difference and vice versa

The product-to-sum and sum-to-product identities can be used to transform products of trigonometric functions into sums or differences, and vice versa.
The product-to-sum identity states that the product of two trigonometric functions can be expressed as the sum or difference of two other trigonometric functions.
The sum-to-product identity states that the sum or difference of two trigonometric functions can be expressed as the product of two other trigonometric functions.
The product-to-sum and sum-to-product identities are commonly used to simplify trigonometric expressions and solve trigonometric equations.
The product-to-sum and sum-to-product identities can also be used to transform products or powers of trigonometric functions into sums or differences of trigonometric functions.

Important Formulas to Remember in these topic

$ \sin x\sin y=\frac{1}{2}[\cos(x-y)-\cos(x+y)] $
$ \cos x\cos y=\frac{1}{2}[\cos(x-y)+\cos(x+y)] $
$ \sin x\cos y=\frac{1}{2}[\sin(x+y)+\sin(x-y)] $
$ \tan x\pm\tan y=\frac{\sin(x\pm y)}{\cos x\cos y} $

Sub-multiple angles

Sub-multiple angles are formed by dividing an angle by a positive integer.
The trigonometric functions of sub-multiple angles can be expressed in terms of the trigonometric functions of the original angle.
The sub-multiple angle formulas are used to simplify trigonometric expressions and solve trigonometric equations.
The most commonly used sub-multiple angle formulas are the half-angle and quarter-angle formulas, which can be used to find the values of trigonometric functions for angles that are half or a quarter the size of known angles.
The sub-multiple angle formulas can also be used to find the values of trigonometric functions for angles that are one-third or one-fifth the size of known angles.

Important Formulas to Remember in these Sub-multiple angles

$ \sin\frac{x}{2}=\pm\sqrt{\frac{1-\cos x}{2}} $
$ \cos\frac{x}{2}=\pm\sqrt{\frac{1+\cos x}{2}} $
$ \tan\frac{x}{2}=\frac{\sin x}{1+\cos x}=\frac{1-\cos x}{\sin x} $

Multiple angles

Multiple angles are formed by multiplying an angle by a positive integer.
The trigonometric functions of multiple angles can be expressed in terms of the trigonometric functions of the original angle.
The multiple angle formulas are used to simplify trigonometric expressions and solve trigonometric equations.
The most commonly used multiple angle formulas are the double angle formulas, which can be used to find the values of trigonometric functions for angles that are twice the size of known angles.
The triple angle and half-angle formulas can also be used to find the values of trigonometric functions for angles that are three times or half the size of known angles.

Important Formulas to Remember in these Multiple angles

$ \sin2x=2\sin x\cos x $
$ \cos2x=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x $
$ \tan2x=\frac{2\tan x}{1-\tan^2x} $

Ratios of Compound angles

Compound angles are formed by adding, subtracting, multiplying, or dividing two or more angles.
The trigonometric functions of a sum or difference of two angles can be expressed in terms of the trigonometric functions of the individual angles.
The sum and difference formulas for sine and cosine are commonly used to simplify trigonometric expressions.
The product-to-sum and sum-to-product identities can be used to simplify expressions involving products or powers of trigonometric functions.
The half-angle formulas can be used to find the values of trigonometric functions for angles that are half the size of known angles.

Important Formulas to Remember in these Ratios of Compound angles

$ \sin(A\pm B)=\sin A\cos B\pm\cos A\sin B $
$ \cos(A\pm B)=\cos A\cos B\mp\sin A\sin B $
$ \tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B} $

Properties of Trigonometric functions

Trigonometric functions are ratios of the sides of a right triangle and are used to solve problems involving angles and distances.
The six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent.
Trigonometric functions are periodic, with a period of 360 degrees or 2π radians.
Trigonometric functions have certain symmetries, such as sine being an odd function and cosine being an even function.
Trigonometric functions have a number of identities that relate them to one another, such as the Pythagorean identity and the sum and difference formulas.

Important Formulas to Remember in these Properties of Trigonometric functions

$ \sin(-x)=-\sin(x) $
$ \cos(-x)=\cos(x) $
$ \sin(x\pm y)=\sin(x)\cos(y)\pm\cos(x)\sin(y) $
$ \cos(x\pm y)=\cos(x)\cos(y)\mp\sin(x)\sin(y) $
$ \tan(x\pm y)=\frac{\tan(x)\pm\tan(y)}{1\mp\tan(x)\tan(y)} $

Exponential function & Logarithmic function

Exponential function:

An exponential function is a mathematical function of the form f(x) = ab^x, where a and b are constants and b is the base of the exponential function.
Exponential functions exhibit exponential growth or decay, depending on whether b is greater than 1 or less than 1.
The exponential function is used in many areas of mathematics and science, including calculus, finance, physics, and biology.
The natural exponential function, denoted by e^x, is the exponential function with the base 'e'.
The exponential function is the inverse of the logarithmic function, and they can be used to solve equations involving logarithmic and exponential functions.

Logarithmic function:

A logarithmic function is a mathematical function of the form f(x) = logb(x), where b is the base of the logarithmic function and x is the argument.
Logarithmic functions are used to express the relationship between two quantities that are being multiplied or divided.
The natural logarithmic function, denoted by ln(x), is the logarithmic function with the base 'e'.
Logarithmic functions have several properties, including the product rule, quotient rule, power rule, and change of base rule.
Logarithmic functions are the inverse of exponential functions, and they can be used to solve equations involving exponential and logarithmic functions.

Important Formulas to Remember in these Exponential function & Logarithmic function

Exponential Function:

Definition: $$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

Properties:
- $$e^{a+b} = e^ae^b$$
- $$\frac{d}{dx}e^x = e^x$$
- $$\int e^x\,dx = e^x + C$$

Logarithmic Function:

Definition: $$\ln x = \log_e x$$

Properties:
- $$\ln 1 = 0$$
- $$\ln e = 1$$
- $$\ln(xy) = \ln x + \ln y$$
- $$\ln\left(\frac{x}{y}\right) = \ln x - \ln y$$
- $$\ln(x^c) = c\ln x$$
- $$\ln e^x = x$$
- $$\frac{d}{dx}\ln x = \frac{1}{x}$$
- $$\int \frac{1}{x}\,dx = \ln |x| + C$$

Meaning of 'e':

'e' is a mathematical constant that is approximately equal to 2.71828.
The constant 'e' is used in many areas of mathematics, including calculus, number theory, and probability theory.
The value of 'e' is irrational and cannot be expressed as a finite decimal or a fraction.
'e' is the base of the natural logarithm function, denoted by ln(x), and is used extensively in solving problems involving exponential and logarithmic functions.
The constant 'e' is also used in the exponential growth and decay models, and in the formula for calculating compound interest.

Important Formulas to Remember in these Meaning of 'e':

Meaning of 'e': $$e = \lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^n$$

Definition of logarithm and its properties

Definition of logarithm

The logarithm of a positive number x to the base b is the exponent y to which b must be raised to obtain x.
The logarithm function is denoted by logb(x), where b is the base and x is the argument.

properties:

The logarithm function has several properties, including the product rule, quotient rule, power rule, and change of base rule.
The product rule states that the logarithm of the product of two numbers is equal to the sum of their logarithms.
The quotient rule states that the logarithm of the quotient of two numbers is equal to the difference of their logarithms.

Important Formulas to Remember in these logarithm

Logarithms:

Definition of Logarithm: $$\log_{a}(b) = c \iff a^c = b$$

Properties of Logarithm:

$$\log_{a}(1) = 0$$

$$\log_{a}(a) = 1$$

$$\log_{a}(bc) = \log_{a}(b) + \log_{a}(c)$$

$$\log_{a}\left(\frac{b}{c}\right) = \log_{a}(b) - \log_{a}(c)$$

$$\log_{a}(b^c) = c\log_{a}(b)$$

$$\log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)}$$

Logarithms

Logarithm definition and their formulas:

Logarithms are mathematical functions that are used to express the relationship between two quantities that are being multiplied or divided.
The logarithm of a number is the exponent to which another fixed value called the base must be raised to produce that number.
Logarithms have a wide range of applications in mathematics, science, engineering, and finance.
The most commonly used logarithms are the base-10 logarithm (common logarithm) and the natural logarithm (base e).
Logarithmic functions are the inverse of exponential functions, and they can be used to solve equations involving exponential functions

Logarithms Formulas:

Definition of Logarithm: $$\log_{a}(b) = c \iff a^c = b$$

Properties of Logarithm:

$$\log_{a}(1) = 0$$

$$\log_{a}(a) = 1$$

$$\log_{a}(bc) = \log_{a}(b) + \log_{a}(c)$$

$$\log_{a}\left(\frac{b}{c}\right) = \log_{a}(b) - \log_{a}(c)$$

$$\log_{a}(b^c) = c\log_{a}(b)$$

$$\log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)}$$

Partial Fractions - Resolving a given rational function into partial fractions

Partial fractions is a technique used to decompose a rational function into simpler fractions.
The rational function is a fraction where the numerator and the denominator are both polynomials.
The partial fraction decomposition involves breaking the rational function into a sum of simpler fractions with denominators that are linear factors or irreducible quadratic factors.
The technique involves finding the unknown coefficients in the partial fraction expression by equating the numerator of the rational function to the sum of the numerators of the partial fractions.
Partial fraction decomposition is used in integration of rational functions, solving differential equations, and signal processing.

Resolving a given rational function into partial fractions:

To resolve a given rational function into partial fractions, first factorize the denominator of the rational function into linear or irreducible quadratic factors.
Express the partial fraction decomposition in the form of unknown coefficients over linear or irreducible quadratic factors.
Equate the numerator of the rational function to the sum of the numerators of the partial fractions, and solve for the unknown coefficients using algebraic methods.
If there are repeated factors in the denominator, then use the denominators raised to increasing powers as denominators in the partial fraction decomposition.
If the degree of the numerator is greater than or equal to the degree of the denominator, then perform long division to obtain a proper fraction, and then resolve the proper fraction into partial fractions.

Important Formulas to Remember in these Partial Fractions

Partial Fractions:

Let $R(x)$ and $Q(x)$ be two polynomials such that $deg(R(x)) < deg(Q(x))$. Then $R(x)/Q(x)$ can be expressed as a sum of partial fractions of the form: $$\\frac{A}{x-a} + \\frac{B}{(x-a)^2} + \\cdots + \\frac{L}{(x-a)^k} + \\frac{R(x)}{Q(x)}$$ where $a$ is a root of $Q(x)$ of multiplicity $k$ and $A, B, \\ldots, L$ are constants.

Resolving a Given Rational Function into Partial Fractions:

Let $R(x)$ and $Q(x)$ be two polynomials such that $deg(R(x)) < deg(Q(x))$. Then $R(x)/Q(x)$ can be expressed as a sum of partial fractions by following the steps given in the algorithm.

Gauss Jordan method

The Gauss-Jordan method is a technique used to solve systems of linear equations by manipulating an augmented matrix.
The augmented matrix is a matrix that contains the coefficients of the system of linear equations and the constants in the form of a matrix.
The method involves using row operations to transform the augmented matrix into a row echelon form or reduced row echelon form.
The row echelon form is a matrix where the leading coefficient of each row is to the right of the leading coefficient of the row above it.
The reduced row echelon form is a matrix where each leading coefficient is equal to 1, and all other entries in the column containing the leading coefficient are zero.
The reduced row echelon form is unique for each matrix, and it can be used to solve the system of linear equations.