Algebraic formulas are the building blocks of mathematics, providing powerful tools for solving equations and manipulating expressions. These formulas help you transform complex problems into simpler ones, find unknown values, and recognize patterns across different mathematical contexts. Mastering these formulas will strengthen your problem-solving abilities in all areas of mathematics.
Basic Identities
These fundamental algebraic patterns appear frequently across mathematics. They help you expand expressions (like finding the square of a binomial), simplify complex terms, and recognize hidden structures in equations. When you memorize these identities, you'll solve problems more quickly and see connections between different mathematical concepts.
\((x + a)(x + b)(x + c) = x^3 + (a + b + c)x^2 + (ab + bc + ca)x + abc \)
Quadratic Formula
The quadratic formula provides a direct method for solving any quadratic equation (\(ax^2 + bx + c = 0\)). The discriminant (\( \Delta = b^2 -4ac \)) indicates the nature of the solutions: when \( \Delta > 0 \), there are two distinct real solutions; when \( \Delta = 0\), there is one repeated real solution; and when \( \Delta \lt 0 \), there are two complex solutions.
\( ax^2 + bx + c = 0 \)
\( x = {-b \pm \sqrt{b^2-4ac} \over 2a} \)
\( Discriminant: \Delta = b^2-4ac \)
Factorization
Factorization is like breaking down a complex expression into simpler building blocks. These formulas help you recognize common patterns that can be factored, making equations easier to solve. For example, the difference of squares pattern (a² - b²) appears frequently in mathematics and can always be factored as (a + b)(a - b). Recognizing these patterns will save you time and provide insights into the structure of expressions.
Logarithms transform multiplication and division into addition and subtraction, making complex calculations simpler. These properties are essential for solving exponential equations, working with compound interest, analyzing growth patterns, and understanding scales like pH or decibels. When you understand logarithm rules, you can solve problems involving exponential growth and decay that appear in science, finance, and data analysis.
Exponential rules govern how powers behave in mathematical expressions. These properties help simplify calculations involving repeated multiplication, scientific notation, and growth models. Understanding these rules is crucial for working with compound interest, population growth, radioactive decay, and many other real-world applications. Mastering these properties will help you manipulate expressions with exponents efficiently and solve exponential equations.
Area and perimeter formulas help us measure and compare two-dimensional shapes. Area represents the space inside a shape (measured in square units like \(cm^2\)), while perimeter measures the distance around the shape (in linear units like \(cm\)). These formulas are essential for practical applications in construction, design, landscaping, and everyday measurements. Understanding these relationships helps you solve real-world problems involving shapes and spaces.
Three-dimensional shapes have both volume (the space they occupy, measured in cubic units like \(cm^3\)) and surface area (the total area of all faces, measured in square units like \(cm^2\)). These formulas help us understand and work with objects in the physical world. Volume calculations are essential for determining capacity, weight, and material requirements, while surface area helps calculate material needed to cover an object. These concepts are fundamental in architecture, engineering, manufacturing, and everyday life.
Name of the Solid
Figure
Volume
Lateral/Curved Surface Area
Total Surface Area
Cuboid (Rectangular Prism)
$$ lbh $$
$$ 2h(l + b) $$
$$ 2(lb + bh + hl) $$
Cube
$$ a^3 $$
$$ 4a^2 $$
$$ 6a^2 $$
Right Circular Cylinder
$$ \pi r^2h $$
$$ 2\pi rh $$
$$ 2\pi r(r + h) $$
Right Circular Cone
$$ \frac{1}{3} \pi r^2h $$
$$ \pi rl $$
$$ \pi r(l + r) $$
Sphere
$$ \frac{4}{3} \pi r^3 $$
$$ 4\pi r^2 $$
$$ 4\pi r^2 $$
Hemisphere
$$ \frac{2}{3} \pi r^3 $$
$$ 2\pi r^2 $$
$$ 3\pi r^2 $$
Key Notes:
\(l = length,\; b = \frac{breadth}{width},\; h = height \)
Use these formulas with our related area calculations for complete geometry solutions.
Trigonometric Formulas & Identities
Trigonometry connects angles and sides of triangles, providing powerful tools for analyzing periodic phenomena and solving problems involving triangles. These formulas are essential in navigation, engineering, physics, computer graphics, and signal processing. Trigonometric functions describe repeating patterns like waves, oscillations, and circular motion. Understanding these relationships helps you solve complex problems in both pure and applied mathematics.
Basic Trigonometric Ratios (SOH-CAH-TOA)
These fundamental ratios relate the sides of a right triangle to its angles. The mnemonic SOH-CAH-TOA helps you remember that \(\text{Sine} = \frac{\text{Opposite}}{\text{Hypotenuse}} \), \(\text{Cosine} = \frac{\text{Adjacent}}{\text{Hypotenuse}} \), and \(\text{Tangent} = \frac{\text{Opposite}}{\text{Adjacent}} \). These relationships form the foundation of all trigonometry and allow you to find unknown sides or angles in right triangles. They're essential for surveying, navigation, engineering, and physics applications.
These commonly used angles \( (0^o, 30^o, 45^o, 60^o, and \; 90^o)\) have exact trigonometric values that can be expressed without a calculator. Memorizing these values helps you solve problems more efficiently and recognize patterns in trigonometric expressions. These special angles appear frequently in geometry, physics, and engineering applications. Understanding their exact values provides insight into the unit circle and the periodic nature of trigonometric functions.
Angle
0°
30°
45°
60°
90°
$$ \sin $$
$$ 0 $$
$$ \frac{1}{2} $$
$$ \frac{\sqrt{2}}{2} $$
$$ \frac{\sqrt{3}}{2} $$
$$ 1 $$
$$ \cos $$
$$ 1 $$
$$ \frac{\sqrt{3}}{2} $$
$$ \frac{\sqrt{2}}{2} $$
$$ \frac{1}{2} $$
$$ 0 $$
$$ \tan $$
$$ 0 $$
$$ \frac{1}{\sqrt{3}} $$
$$ 1 $$
$$ \sqrt{3} $$
$$ \text{undefined} $$
Trigonometric Identities
Trigonometric identities are equations that are always true for any angle value. They allow you to simplify expressions, solve equations, and transform complex trigonometric terms into more manageable forms. The Pythagorean identities connect squared trig functions, while sum and difference formulas help work with combined angles. These relationships are powerful tools for calculus, physics, signal processing, and analyzing periodic phenomena. Mastering these identities gives you flexibility in approaching trigonometric problems.
Identity Type
Formula
Description
Pythagorean
$$ \sin^2\theta + \cos^2\theta = 1 $$
\( \text{Fundamental identity} \)
Pythagorean
$$ 1 + \tan^2\theta = \sec^2\theta $$
\( \text{Divide by} \; cos^2\theta \)
Pythagorean
$$ 1 + \cot^2\theta = \csc^2\theta $$
\( \text{Divide by} \; sin^2\theta \)
Sum
$$ \sin(A + B) = \sin A \cos B + \cos A \sin B $$
\( \text{Sine addition} \)
Difference
$$ \sin(A - B) = \sin A \cos B - \cos A \sin B $$
\( \text{Sine subtraction} \)
Sum
$$ \cos(A + B) = \cos A \cos B - \sin A \sin B $$
\( \text{Cosine addition} \)
Difference
$$ \cos(A - B) = \cos A \cos B + \sin A \sin B $$
\( \text{Cosine subtraction} \)
Sum
$$ \tan(A + B) = \frac{\tan A + \tan B}{ 1 - \tan A \tan B} $$
\( \text{Tangent addition} \)
Difference
$$ \tan(A - B) = \frac{\tan A - \tan B}{ 1 + \tan A \tan B} $$
These powerful laws extend trigonometry beyond right triangles to any triangle.
Law of Sines relates the sides of a triangle to the sines of their opposite angles, making it ideal when you know two angles and one side (AAS or ASA) or two sides and an opposite angle (SSA).
Law of Cosines generalizes the Pythagorean theorem and works best when you know three sides (SSS) or two sides and the included angle (SAS). These laws are essential for surveying, navigation, physics problems, and any application involving non-right triangles.
Law of Tangents relates the tangents of half the difference and half the sum of two angles to the ratio of the lengths of the sides opposite those angles. This law is particularly useful for solving triangles when you know two sides and the included angle (SAS) or two angles and one side (ASA).
\(\text{with 3 sides and a, b, and c and opposite angles}\)
Note: When \( C = 90^\circ,\;\) Law of Cosines becomes Pythagorean theorem \( (c^2 = a^2 + b^2) \)
Unit Circle & Angle Conversions
The unit circle (a circle with radius 1) provides a geometric interpretation of trigonometric functions for any angle. It helps visualize how sine and cosine relate to coordinates on the circle, with sine being the y-coordinate and cosine being the x-coordinate. Understanding angle conversions between degrees and radians is essential because radians are the natural unit for calculus and advanced mathematics. Arc length and sector area formulas connect angle measure to the geometry of circles, with applications in physics, engineering, and calculus.
\( Arc Length: s = r\theta \;(\text{where} \;\theta\; \text{is in radians}) \)
\( Sector Area: A = \frac{1}{2}r^2\theta \;(\text{where} \;\theta\; \text{is in radians}) \)
Product-to-Sum & Sum-to-Product Formulas
These formulas allow you to convert between products and sums of trigonometric functions. Product-to-sum formulas transform expressions like \( \sin(A)\sin(B) \) into sums or differences, while sum-to-product formulas convert expressions like \( \sin(A) + \sin(B)\) into products. These transformations are valuable for simplifying complex expressions, solving certain types of integrals in calculus, and analyzing wave patterns in physics and signal processing. They demonstrate the rich interconnections between different trigonometric expressions.
Product-to-Sum
Sum-to-Product
$$ \sin A \sin B = \frac{1}{2}[\cos (A-B) - \cos (A+B)] $$
$$ \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) $$
$$ \cos A \cos B = \frac{1}{2}[\cos (A-B) + \cos (A+B)] $$
$$ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) $$
$$ \sin A \cos B = \frac{1}{2} [\sin (A+B) + \sin (A-B)] $$
$$ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) $$
$$ \cos A \sin B = \frac{1}{2} [\sin (A+B) - \sin (A-B)]$$
$$ \cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) $$
Triple Angle Formulas
Triple angle formulas express trigonometric functions of \( 3\theta \) in terms of functions of \(\theta\). These formulas extend the concept of double angle formulas and are useful for simplifying expressions with triple angles, solving certain trigonometric equations, and analyzing wave patterns with specific frequencies. They demonstrate how complex trigonometric expressions can be rewritten in terms of simpler ones, revealing the elegant patterns within trigonometry.
Inverse trigonometric functions (arcsin, arccos, arctan) find the angle when you know the trigonometric ratio. They're essential for solving equations where the unknown is an angle rather than a side length. Each function has specific domain and range restrictions to ensure they produce unique values. These functions appear in physics when analyzing projectile motion, in engineering when determining angles from measurements, and in calculus when working with certain integrals. Understanding their properties helps you solve problems involving angles and trigonometric equations.
Hyperbolic functions (sinh, cosh, tanh) are related to the regular trigonometric functions but are based on the hyperbola rather than the circle. They appear naturally in many physical phenomena including the shape of hanging cables (catenary curves), electric and magnetic fields, and special relativity. These functions have important applications in engineering, physics, and differential equations. Their definitions using exponential functions connect them to growth and decay processes, while their identities parallel those of circular trigonometric functions.
These formulas provide general solutions to basic trigonometric equations. Unlike algebraic equations that typically have a finite number of solutions, trigonometric equations have infinitely many solutions due to the periodic nature of trigonometric functions. These general solution formulas help you find all possible angles that satisfy a given equation. Understanding these patterns is essential for solving more complex trigonometric equations in physics, engineering, and advanced mathematics where periodic phenomena are modeled.
Equation
General Solution
$$ \sin\theta = 0 $$
$$ \theta = \frac{n}{pi}, \; where \; n \in \mathbb{Z} $$
$$ \cos\theta = 0 $$
$$ \theta = \frac{(2n+1)\pi}{2}, \; where \; n \in \mathbb{Z} $$
$$ \tan\theta = 0 $$
$$ \theta = n\pi, \; where \; n \in \mathbb{Z} $$
$$ \sin\theta = \sin\alpha $$
$$ \theta = n\pi + (-1)^n\alpha, \; where \; n \in \mathbb{Z} $$
$$ \cos\theta = \cos\alpha $$
$$ \theta = 2n\pi \pm \alpha, \; where \; n \in \mathbb{Z} $$
$$ \tan\theta = \tan\alpha $$
$$ \theta = n\pi + \alpha, \; where \; n \in \mathbb{Z} $$
Quick Reference Tips:
SOH-CAH-TOA helps remember basic ratios
All Students Take Calculus - signs in quadrants (All+, Sin+, Tan+, Cos+)
30-60-90 triangle: sides in ratio 1 : √3 : 2
45-45-90 triangle: sides in ratio 1 : 1 : √2
Unit circle: radius = 1, so \( \sin\theta \) = y-coordinate, \( \cos\theta \) = x-coordinate
Period: \(\sin \), \(\cos \) have period \(2\pi\); \(\tan \), \(\cot \) have period \(\pi\)
Even/Odd: \(\cos \) is even, \(\sin \) and \(\tan \) are odd functions
Calculus provides powerful tools for analyzing change and accumulation. It has two main branches: differential calculus (finding rates of change using derivatives) and integral calculus (finding accumulation or total change). These concepts are fundamental to physics, engineering, economics, statistics, and many other fields. Calculus helps us model dynamic systems, optimize functions, calculate areas and volumes, and understand how quantities change in relation to each other.
Basic Derivatives
Derivatives measure the rate of change of a function at any point. These fundamental derivative rules form the building blocks for differentiating more complex functions. The power rule handles expressions with variables raised to powers, while specific rules exist for common functions like exponentials, logarithms, and trigonometric functions. Mastering these basic derivatives is essential for analyzing motion, finding slopes of curves, optimizing functions, and solving problems involving rates of change in science, engineering, and economics.
These rules allow you to find derivatives of complex functions by breaking them down into simpler operations. The sum and difference rules show that derivatives distribute over addition and subtraction. The product and quotient rules handle multiplication and division of functions. The chain rule is particularly powerful, allowing you to differentiate composite functions (functions within functions). Together, these rules provide a systematic approach to finding derivatives of virtually any function, making them essential tools in calculus applications across science and engineering.
Integration is the reverse process of differentiation, finding the accumulation or total change of a function. These fundamental integral formulas are the building blocks for evaluating more complex integrals. The constant C (integration constant) appears because derivatives of functions that differ only by a constant are identical. Integrals are essential for calculating areas, volumes, work, probability, average values, and total change in various applications. These basic patterns help you recognize and solve integration problems across mathematics, physics, and engineering.
$$ \int c dx = cx + C$$
$$ \int x dx = \frac{x^2}{2} + C$$
$$ \int x^n dx = \frac{x^{n + 1}}{(n + 1)} + C$$
$$ \int e^x dx = e^x + C$$
$$ \int \frac{1}{x} dx = ln \vert x\vert + C$$
$$ \int sin(x) dx = -cos(x) + C$$
$$ \int cos(x) dx = sin(x) + C$$
Integration Techniques
Integration techniques help solve more complex integrals that don't match basic patterns. Integration by parts extends the product rule for derivatives, allowing you to transform products of functions into potentially simpler integrals. The Fundamental Theorem of Calculus connects definite integrals to antiderivatives, providing a powerful method for evaluating definite integrals. These techniques are essential for applications in physics (work, energy, center of mass), engineering (fluid flow, electrical circuits), and statistics (expected values, probability distributions).
$$ \text{Integration by Parts:} \int u dv = uv - \int v du $$
Linear algebra provides powerful tools for working with systems of linear equations, vectors, and matrices. These concepts are fundamental to many fields including computer graphics, machine learning, quantum mechanics, economics, and engineering. Linear algebra helps us understand transformations in space, solve systems of equations efficiently, and model complex relationships between variables.
Matrix Operations
Matrices are rectangular arrays of numbers that can represent linear transformations, systems of equations, and data structures. These operations allow you to combine and manipulate matrices to solve problems in various fields. Matrix addition and scalar multiplication follow intuitive patterns, while matrix multiplication represents composition of linear transformations and is fundamental to many applications.
$$ \text{Addition: } (A + B)_{ij} = A_{ij} + B_{ij} \text{ (matrices must be same size)} $$
$$ \text{Scalar Multiplication: } (cA)_{ij} = c \cdot A_{ij} $$
These properties help characterize matrices and understand their behavior in equations and transformations. The determinant indicates whether a matrix is invertible and helps find the inverse. The trace relates to eigenvalues, while the rank indicates the dimension of the matrix's image. Understanding these properties is essential for solving systems of equations, analyzing transformations, and working with linear models.
$$ \text{Determinant (2×2): } \det(A) = |A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc $$
$$ \text{Inverse (2×2): } A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \text{ where } A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$
Vector Operations
Vectors represent quantities with both magnitude and direction. These operations allow you to combine vectors and measure relationships between them. The dot product relates to projections and angles between vectors, while the cross product (in 3D) creates a vector perpendicular to two given vectors. These operations are fundamental in physics, computer graphics, and any field involving directional quantities.
Eigenvalues and eigenvectors reveal the fundamental behavior of linear transformations. An eigenvector is a non-zero vector whose direction remains unchanged when the transformation is applied (it only gets scaled). The corresponding eigenvalue is the scaling factor. These concepts are crucial in stability analysis, principal component analysis, quantum mechanics, and many other applications where understanding the core behavior of transformations is essential.
$$ A\vec{v} = \lambda\vec{v} \text{ where } \vec{v} \text{ is an eigenvector and } \lambda \text{ is an eigenvalue} $$
Statistics provides methods for collecting, analyzing, interpreting, and presenting data. These formulas help us understand patterns, make predictions, and draw conclusions from data sets. Statistical concepts are essential in research, business, economics, social sciences, and any field that involves data analysis and decision-making under uncertainty.
Descriptive Statistics
Descriptive statistics summarize and describe the main features of a data set. Measures of central tendency (mean, median, mode) indicate the "center" of the data, while measures of dispersion (variance, standard deviation) show how spread out the data is. These basic statistics provide a foundation for understanding data distributions and are essential first steps in any data analysis.
$$ \text{Standard Deviation: } \sigma = \sqrt{\sigma^2} \text{ or } s = \sqrt{s^2} $$
Probability Distributions
Probability distributions describe the likelihood of different outcomes in random processes. The normal distribution is particularly important due to the Central Limit Theorem, which states that the sum of many independent random variables tends toward a normal distribution. These distributions help model real-world phenomena and are fundamental to statistical inference, hypothesis testing, and predictive modeling.
$$ \text{Normal Distribution PDF: } f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2} $$
$$ \text{Standardization (Z-score): } z = \frac{x - \mu}{\sigma} $$
Correlation and Regression
Correlation measures the strength and direction of the relationship between two variables, while regression provides a mathematical model of this relationship. These tools help analyze how variables influence each other and make predictions based on observed data. They're essential in economics, social sciences, machine learning, and any field that involves understanding relationships between variables.
$$ \text{Simple Linear Regression: } y = mx + b \text{ where } m = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2} $$
Hypothesis testing is a method for making decisions based on data. It involves comparing a null hypothesis (typically representing "no effect" or "no difference") against an alternative hypothesis. These formulas help determine whether observed data provides sufficient evidence to reject the null hypothesis. Understanding these concepts is crucial for scientific research, quality control, and any field requiring evidence-based decision making.