Geometrical applications of the derivative (angle between curves, tangent and normal) - Increasing and decreasing functions - Maxima and Minima (single variable functions) using second order derivative only physical application - Rate Measure

Geometrical applications of the derivative:

The derivative can be used to find the slope of a tangent line to a curve at a given point.
The derivative can also be used to find the equation of a normal line to a curve at a given point, which is a line perpendicular to the tangent line at that point.
The angle between two curves at a point of intersection can be found using the slopes of the tangent lines to each curve at that point.

Increasing and decreasing functions, maxima and minima:

A function is increasing if its derivative is positive and decreasing if its derivative is negative.
A function has a local maximum at a point where the derivative changes sign from positive to negative, and a local minimum at a point where the derivative changes sign from negative to positive.
A function has a global maximum or minimum at the highest or lowest point, respectively, on the entire domain of the function.
The second order derivative can be used to determine whether a local maximum or minimum is a maximum or minimum, respectively. Specifically, a local maximum is a maximum if the second order derivative is negative at that point, and a local minimum is a minimum if the second order derivative is positive.

Physical applications of the derivative:

The derivative can be used to measure the rate of change of a physical quantity, such as velocity, acceleration, or temperature.
The derivative of displacement with respect to time gives velocity, the derivative of velocity with respect to time gives acceleration, and so on.
The integral of the derivative can be used to find the total change in a physical quantity over a given interval, such as the total distance traveled by an object or the total heat energy absorbed by a material.

Important Formulas to Remember in these topic

Derivative Formulas

Geometrical Applications of the Derivative

Angle between Two Curves: $$\tan\theta = \frac{dy_1/dx - dy_2/dx}{1 + dy_1/dx\cdot dy_2/dx}$$
Tangent and Normal to a Curve: $$y - y_0 = \frac{dy}{dx}(x - x_0) \quad \text{(Tangent)}$$ $$y - y_0 = -\frac{dx}{dy}(x - x_0) \quad \text{(Normal)}$$

Derivative of a function with respect to another function - Second order derivatives

Derivatives with respect to another function:

When taking the derivative of a function with respect to another function, we use the chain rule, treating the outer function as the variable and the inner function as the function.
This process is also known as implicit differentiation, and is often used to find the slope of a curve at a specific point or to solve equations where the dependent variable is not explicitly given in terms of the independent variable.
For example, if we have y = f(g(x)), then we can find dy/dx by applying the chain rule: dy/dx = (dy/dg) * (dg/dx).

Second order derivatives:

The second order derivative of a function is the derivative of its first order derivative. It measures how the rate of change of a function changes as the input value changes.
The second order derivative is denoted by f''(x) or d^2y/dx^2, and is calculated by taking the derivative of the first order derivative of the function.
If the second order derivative is positive at a certain point, then the function is said to be concave up at that point, whereas if it is negative, the function is said to be concave down.
A point where the second order derivative is zero is known as an inflection point, where the function transitions from being concave up to concave down, or vice versa.
Second order derivatives can be used to find the maximum or minimum points of a function, as a maximum or minimum occurs at a point where the second order derivative changes sign.
In some cases, the second order derivative can be used to determine the behavior of the function at extreme values, such as asymptotes or singularities.
The second order derivative can be extended to functions of multiple variables, where it measures the rate of change of the gradient of the function.

Important Formulas to Remember in these topic

Derivative Formulas

Derivative of a Function with Respect to Another Function

$$\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx} = f'(u)\cdot g'(x)$$

Second Order Derivatives

Second Derivative of a Function: $$\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)$$

Product Rule for Second Derivatives: $$\frac{d^2}{dx^2}(f(x)g(x)) = f''(x)g(x) + 2f'(x)g'(x) + f(x)g''(x)$$

Chain Rule for Second Derivatives: $$\frac{d^2y}{dx^2} = \frac{d^2y}{du^2}\cdot\left(\frac{du}{dx}\right)^2 + \frac{dy}{du}\cdot\frac{d^2u}{dx^2}$$

Differentiation: sum, product, quotient, trigonometric, inverse trigonometric, exponential, logarithmic, hyperbolic, implicit, explicit, and parametric functions.

The sum rule states that the derivative of the sum of two functions is equal to the sum of their derivatives.
The product rule states that the derivative of the product of two functions is equal to the first function times the derivative of the second, plus the second function times the derivative of the first.
The quotient rule states that the derivative of the quotient of two functions is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.
The chain rule, also known as function of function rule, is used to differentiate composite functions, or functions of functions, and states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
The derivative of trigonometric functions depends on the specific trigonometric function, but can generally be found using basic trigonometric identities.
The derivative of inverse trigonometric functions is found using differentiation by substitution and the inverse function theorem.
The derivative of exponential functions is equal to the function itself, multiplied by the natural logarithm of the base e.
The derivative of logarithmic functions is found using the logarithmic differentiation technique, which involves taking the natural logarithm of both sides of the equation and differentiating using the chain rule.
The derivative of hyperbolic functions is similar to that of trigonometric functions and can be found using basic identities.
The differentiation of implicit, explicit, and parametric functions involves different techniques, but typically involves applying the chain rule and solving for the derivative.

Important Formulas to Remember in these topic

Differentiation Formulas

Sum of Functions: $$\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$$

Product of Functions: $$\frac{d}{dx}(f(x)g(x)) = f(x)\frac{d}{dx}g(x) + g(x)\frac{d}{dx}f(x)$$

Quotient of Functions: $$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{g(x)^2}$$

Function of Function: $$\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$$

Trigonometric Functions: $$\frac{d}{dx}\sin(x) = \cos(x), \quad \frac{d}{dx}\cos(x) = -\sin(x), \quad \frac{d}{dx}\tan(x) = \sec^2(x)$$

Inverse Trigonometric Functions: $$\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\arccos(x) = -\frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\arctan(x) = \frac{1}{1+x^2}$$

Exponential Functions: $$\frac{d}{dx}e^x = e^x$$

Logarithmic Functions: $$\frac{d}{dx}\ln(x) = \frac{1}{x}, \quad \frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}$$

Hyperbolic Functions: $$\frac{d}{dx}\sinh(x) = \cosh(x), \quad \frac{d}{dx}\cosh(x) = \sinh(x), \quad \frac{d}{dx}\tanh(x) = \operatorname{sech}^2(x)$$

Implicit Functions: $$\frac{dy}{dx} = -\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}}$$

Explicit Functions: $$\frac{d}{dx}y = \frac{dy}{dx}$$

Parametric Functions: $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$

Functions and limits - Standard limits

Standard limits are specific values that help to determine the limits of functions as they approach certain values.
There are six standard limits, including the limit of a constant function, the limit of a linear function, the limit of a quadratic function, the limit of a square root function, the limit of a rational function, and the limit of an exponential function.
Knowing the standard limits can help you quickly determine the limit of more complex functions by applying them to smaller parts of the larger function.
The limit of a function is the value that the function approaches as the input approaches a certain value, and may or may not be equal to the actual value of the function at that point.
Understanding standard limits is a crucial part of calculus, as they form the foundation for more advanced topics such as derivatives and integrals.

Important Formulas to Remember in these Functions and limits

$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$

$$\lim_{x \to 0} \frac{\tan x}{x} = 1$$

$$\lim_{x \to 0} (1+x)^{\frac{1}{x}} = e$$

$$\lim_{x \to \infty} \left(1+\frac{1}{x}\right)^x = e$$

$$\lim_{x \to \infty} \left(1+\frac{k}{x}\right)^x = e^k$$

$$\lim_{x \to 0} \frac{e^x-1}{x} = 1$$

$$\lim_{x \to \infty} \frac{\ln x}{x} = 0$$

$$\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$$

$$\lim_{x \to \infty} \frac{a^x}{x!} = 0$$

$$\lim_{x \to \infty} \frac{x^k}{a^x} = 0$$

$$\lim_{x \to \infty} \sqrt[x]{x} = 1$$

Conic Section

Properties of parabola:

A parabola is a set of all points that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix).
The axis of symmetry of a parabola is a line that passes through the focus and is perpendicular to the directrix.
The vertex of the parabola is the point where the axis of symmetry intersects the parabola.
The standard form of a parabola with vertex at the origin is y^2 = 4px, where p is the distance from the vertex to the focus.

Properties of ellipse:

An ellipse is a set of all points such that the sum of the distances from two fixed points (called foci) is constant.
The major axis of an ellipse is the longest diameter and contains the two foci. The minor axis is the perpendicular diameter.
The center of an ellipse is the midpoint of the major axis.
The standard form of an ellipse with the center at the origin is x^2/a^2 + y^2/b^2 = 1, where a and b are the lengths of the semi-major and semi-minor axes.

Properties of hyperbola:

A hyperbola is a set of all points such that the difference of the distances from two fixed points (called foci) is constant.
The transverse axis of a hyperbola is the longest diameter and contains the two vertices. The conjugate axis is the perpendicular diameter.
The center of a hyperbola is the midpoint of the transverse axis.
The standard form of a hyperbola with the center at the origin is x^2/a^2 - y^2/b^2 = 1, where a and b are the lengths of the semi-transverse and semi-conjugate axes.

Important Points to Remember in these Properties of parabola, ellipse and hyperbola:

A parabola is the set of all points that are equidistant from a fixed point (focus) and a fixed line (directrix).
An ellipse is the set of all points such that the sum of the distances from two fixed points (foci) is constant.
A hyperbola is the set of all points such that the difference of the distances from two fixed points (foci) is constant.

Standard forms with vertex at origin:

The standard form of a parabola with vertex at the origin is y^2 = 4px, where p is the distance from the vertex to the focus.
The standard form of an ellipse with the center at the origin is x^2/a^2 + y^2/b^2 = 1, where a and b are the lengths of the semi-major and semi-minor axes.
The standard form of a hyperbola with the center at the origin is x^2/a^2 - y^2/b^2 = 1, where a and b are the lengths of the semi-transverse and semi-conjugate axes.
For all three types of conic sections, if the vertex is not at the origin, the equations can be shifted by replacing x with (x - h) and y with (y - k), where (h, k) is the coordinates of the vertex.

Important Formulas to Remember in these Standard forms with vertex at origin:

Parabola: $$y^2 = 4px$$
Ellipse: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
Hyperbola: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

Additional properties of conic sections:

The eccentricity of an ellipse is the ratio of the distance between the foci to the length of the major axis. It is always less than 1.
The eccentricity of a hyperbola is the ratio of the distance between the foci to the length of the transverse axis. It is always greater than 1.
The directrix of a parabola is a line that is equidistant to the focus and perpendicular to the axis of symmetry.
The asymptotes of a hyp-erbola are two lines that the hyperbola approaches but never intersects. The slopes of the asymptotes are ±(b/a).

Circles

1) Equation of circle given center and radius:

The equation of a circle with center (a,b) and radius r is given by (x-a)² + (y-b)² = r².
This equation can be derived from the distance formula, by setting the distance between (x,y) and (a,b) equal to r.
The center of the circle is the point (a,b), and the radius is r.
The equation can be written in general form as x² + y² + Dx + Ey + F = 0, where D = -2a, E = -2b, and F = a² + b² - r².
The standard form of the equation is (x-h)² + (y-k)² = r², where (h,k) is the center of the circle.

Important Formulas to Remember in these Equation of circle given center and radius:

Standard form: $$(x - a)^2 + (y - b)^2 = r^2$$

2) General equation:

The general equation of a circle is given by Ax² + Ay² + Bx + Cy + D = 0, where A and B are not both zero.
This equation can be obtained by substituting x² and y² with their respective coefficients in the standard form of the equation.
The coefficients A, B, and C determine the position of the circle in the xy-plane, while D determines its size.
To convert the general equation to standard form, complete the square for both x and y, and then combine the resulting expressions.
The standard form of the equation can be used to find the center and radius of the circle.

Important Formulas to Remember in these General equation:

General form: $$ax^2 + by^2 + 2gx + 2fy + c = 0$$

3) Finding center and radius:

To find the center and radius of a circle given its equation in standard form, rewrite the equation in the form (x-h)² + (y-k)² = r².
Complete the square for both x and y, and then rearrange the terms to isolate the center (h,k) and the radius r.
The center of the circle is (h,k), and the radius is r.
To find the center and radius of a circle given its equation in general form, convert the equation to standard form first, and then use the method described above.
If the equation of the circle is given in another form, such as center and a point on the circumference or 3 non-collinear points, use the appropriate method to find the center and radius.

Important Formulas to Remember in these Finding center and radius:

Center: $$(a,b)$$
Radius: $$r = \sqrt{(x - a)^2 + (y - b)^2}$$

4) Center and a point on the circumference:

To find the center and radius of a circle given its center (h,k) and a point on the circumference (x1,y1), use the distance formula.
The distance between the center and the point on the circumference is equal to the radius of the circle.
The center of the circle is the midpoint of the line segment joining (h,k) and (x1,y1), and the radius is the distance between the center and (x1,y1).
Alternatively, use the equation of the circle to solve for the radius, and then substitute the coordinates of the point on the circumference to solve for the center.
There are two possible circles that can pass through a given center and point on the circumference, one with a positive radius and one with a negative radius.

Important Formulas to Remember in these Center and a point on the circumference:

Center: $$(a,b)$$
Point on circumference: $$(x_1,y_1)$$
Equation: $$(x - a)^2 + (y - b)^2 = (x_1 - a)^2 + (y_1 - b)^2$$

5) 3 non-collinear points:

To find the center and radius of a circle given three non-collinear points (x1,y1), (x2,y2), and (x3,y3), use the circumcenter formula.
The circumcenter is the point of intersection of the perpendicular bisectors of the sides of the triangle formed by the three points.
Find the slope and midpoint of each side of the triangle, and then find the equations of the perpendicular bisectors.
The point of intersection of the perpendicular bisectors is the center of the circle.
The radius of the circle is the distance between the center and any of the three points.

Important Formulas to Remember in these 3 non-collinear points:

Center: $$(a,b)$$
Points: $$(x_1,y_1), (x_2,y_2), (x_3,y_3)$$
Equation: $$\begin{aligned} a &= \frac{1}{2}\begin{vmatrix} x_1^2 + y_1^2 & y_1 & 1 \\ x_2^2 + y_2^2 & y_2 & 1 \\ x_3^2 + y_3^2 & y_3 & 1 \end{vmatrix} \\ b &= -\frac{1}{2}\begin{vmatrix} x_1^2 + y_1^2 & x_1 & 1 \\ x_2^2 + y_2^2 & x_2 & 1 \\ x_3^2 + y_3^2 & x_3 & 1 \end{vmatrix} \\ r &= \sqrt{(x_1 - a)^2 + (y_1 - b)^2} \end{aligned}$$

6) Center and tangent:

To find the center and radius of a circle given its center (h,k) and a tangent line at a point (x1,y1) on the circumference, use the perpendicular bisector of the tangent line.
The perpendicular bisector of the tangent line passes through the center of the circle.
Find the slope of the tangent line and the midpoint of the line segment joining (h,k) and (x1,y1), and then find the equation of the perpendicular bisector.
The point of intersection of the perpendicular bisector and the line passing through (h,k) and (x1,y1) is the center of the circle.
The radius of the circle is the distance between the center and (x1,y1).

Important Formulas to Remember in these Center and tangent:

Center: $$(a,b)$$
Point on tangent: $$(x_1,y_1)$$
Equation: $$(x - a)(x_1 - a) + (y - b)(y_1 - b) = r^2$$

7) Equation of tangent and normal at a point on the circle:

The tangent line to a circle at a point (x1,y1) on the circumference is perpendicular to the radius passing through the point.
The slope of the tangent line is equal to the negative reciprocal of the slope of the radius passing through the point.
The equation of the tangent line is y - y1 = m(x - x1), where m is the slope of the tangent line.
The normal line to the circle at the same point is perpendicular to the tangent line and passes through the point (x1,y1).
The slope of the normal line is equal to the negative of the slope of the tangent line, and the equation of the normal line is y - y1 = (-1/m)(x - x1), where m is the slope of the tangent line.

Important Formulas to Remember in these Equation of tangent and normal at a point on the circle:

Center: $$(a,b)$$
Point on circle: $$(x_1,y_1)$$
Equation of tangent: $$\frac{x(x_1 - a) + y(y_1 - b)}{r^2} = \frac{x_1 - a}{r} = \frac{y_1 - b}{r}$$
Equation of normal: $$\frac{x(a - x_1) + y(b - y_1)}{r^2} = \frac{a - x_1}{r} = \frac{b - y_1}{r}$$

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)} $