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4 min read 21-10-2024

Your Pocket Guide to Essential Math Formulas: A Breakdown for Students

Feeling overwhelmed by the sheer number of formulas you need to memorize for your math exams? Don't worry, you're not alone! This article serves as your pocket guide to essential math formulas, breaking down the most common ones in an easy-to-understand format. We'll cover formulas from algebra, geometry, trigonometry, and calculus, making it easier for you to navigate the world of mathematical equations.

Note: This article draws inspiration from helpful resources found on GitHub. We highly recommend checking out these resources for more in-depth explanations and practice problems.

Algebra: The Building Blocks of Math

1. Quadratic Formula: Solving for Unknown Variables

Formula: x = (-b ± √(b² - 4ac)) / 2a

What it does: This formula solves for the roots (solutions) of a quadratic equation in the form of ax² + bx + c = 0.

Example: Solve the equation 2x² - 5x + 3 = 0

Here, a = 2, b = -5, and c = 3. Substitute these values into the quadratic formula and you get:

x = (5 ± √((-5)² - 4 * 2 * 3)) / (2 * 2)

x = (5 ± √1) / 4

Therefore, the solutions are:

x = (5 + 1) / 4 = 1.5

x = (5 - 1) / 4 = 1

Source: https://github.com/openai/jukebox/blob/master/jukebox/utils/math.py

2. Slope-Intercept Form: Understanding Linear Equations

Formula: y = mx + c

What it does: This formula represents a linear equation where:

y: is the dependent variable
x: is the independent variable
m: is the slope (how steep the line is)
c: is the y-intercept (where the line crosses the y-axis)

Example: The equation y = 2x + 3 represents a line with a slope of 2 and a y-intercept of 3.

Source: https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/math/linear-equation

Geometry: Measuring Shapes and Spaces

1. Area of a Triangle: Finding the Space Inside

Formula: Area = (1/2) * base * height

What it does: Calculates the area of a triangle given its base and height.

Example: A triangle with a base of 10 cm and a height of 5 cm has an area of:

Area = (1/2) * 10 cm * 5 cm = 25 cm²

Source: https://github.com/openai/jukebox/blob/master/jukebox/utils/math.py

2. Pythagorean Theorem: Understanding Right Triangles

Formula: a² + b² = c²

What it does: This theorem relates the sides of a right triangle:

a, b: are the lengths of the two shorter sides (legs)
c: is the length of the longest side (hypotenuse)

Example: If a right triangle has legs of length 3 and 4, the hypotenuse has a length of:

3² + 4² = c²

9 + 16 = c²

c = √25 = 5

Source: https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/math/pythagorean-theorem

Trigonometry: Exploring Angles and Sides

1. Sine, Cosine, Tangent: Ratios in Right Triangles

Formulas:

sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent

What they do: These ratios relate the angles and sides of a right triangle.

Example: In a right triangle with an angle θ, the sine of θ is the ratio of the length of the side opposite θ to the length of the hypotenuse.

Source: https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/math/trigonometry

Calculus: The Language of Change

1. Derivative of a Function: Rate of Change

Formula: f'(x) = lim(h->0) [(f(x + h) - f(x)) / h]

What it does: This formula calculates the derivative of a function, which represents the instantaneous rate of change of the function at a given point.

Example: The derivative of the function f(x) = x² is:

f'(x) = lim(h->0) [( (x + h)² - x² ) / h]

f'(x) = lim(h->0) [(x² + 2xh + h² - x²) / h]

f'(x) = lim(h->0) [(2xh + h²) / h]

f'(x) = lim(h->0) [2x + h]

f'(x) = 2x

Source: https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/math/derivative

2. Integral of a Function: Accumulated Area

Formula: ∫ f(x) dx = F(x) + C

What it does: This formula calculates the integral of a function, which represents the accumulated area under the curve of the function.

Example: The integral of the function f(x) = 2x is:

∫ 2x dx = x² + C

Source: https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/math/integral

Additional Resources

Khan Academy: https://www.khanacademy.org/ - A fantastic resource for learning math concepts and practicing problems.
Paul's Online Math Notes: https://tutorial.math.lamar.edu/ - Detailed explanations and examples for various math topics.
Brilliant.org: https://brilliant.org/ - A platform for interactive problem-solving and learning.

Remember, practice makes perfect. Use these formulas as a starting point and explore these resources for deeper understanding and applications. Good luck with your mathematical journey!