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arrow_forwardFractions & Percentages
functionAlgebraic Formulas
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Algebraic Formulas
Difference of Squares
$$ a^2 - b^2 = (a-b)(a+b) $$
Perfect Square (+)
$$ (a+b)^2 = a^2 + 2ab + b^2 $$
Perfect Square (-)
$$ (a-b)^2 = a^2 - 2ab + b^2 $$
Sum of Cubes
$$ a^3 + b^3 = (a+b)(a^2 - ab + b^2) $$
Difference of Cubes
$$ a^3 - b^3 = (a-b)(a^2 + ab + b^2) $$
Cube of Sum
$$ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$
Cube of Difference
$$ (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 $$
functionsSeries & Sums
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Series & Sums
Sum of first n natural numbers
$$ \sum_{i=1}^{n} i = \frac{n(n+1)}{2} $$
Sum of first n squares
$$ \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} $$
Sum of first n cubes
$$ \sum_{i=1}^{n} i^3 = \left( \frac{n(n+1)}{2} \right)^2 $$
superscriptExponents & Roots
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Exponents & Roots
Product Rule
$$ a^m \cdot a^n = a^{m+n} $$
Quotient Rule
$$ \frac{a^m}{a^n} = a^{m-n} $$
Power of a Power
$$ (a^m)^n = a^{mn} $$
Power of a Product
$$ (ab)^n = a^n b^n $$
Zero Exponent
$$ a^0 = 1 $$
Negative Exponent
$$ a^{-n} = \frac{1}{a^n} $$
shapesGeometry Areas
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Geometry Areas
Rectangle Area
$$ A = l \times w $$
Triangle Area
$$ A = \frac{1}{2} bh $$
Circle Area
$$ A = \pi r^2 $$
Circle Circumference
$$ C = 2 \pi r $$
Trapezoid Area
$$ A = \frac{1}{2} (a+b)h $$
architectureTrigonometry
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Trigonometry
Sine Definition
$$ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} $$
Cosine Definition
$$ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} $$
Tangent Definition
$$ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} $$
Pythagorean Identity
$$ \sin^2 \theta + \cos^2 \theta = 1 $$
Sine Rule
Relates the sides of a triangle to the sines of its angles.
$$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R $$
Cosine Rule
Generalization of the Pythagorean theorem for any triangle.
$$ c^2 = a^2 + b^2 - 2ab \cos C $$
grid_onCoordinate Geometry
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Coordinate Geometry
Section Formula
$$ P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) $$
Slope of a Line
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Point-Slope Form
$$ y - y_1 = m(x - x_1) $$
Distance of Point from Line
$$ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} $$
Angle Between Lines
The angle θ between two lines with slopes m₁ and m₂.
$$ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right| $$
Area of Triangle
Area of a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃).
$$ \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| $$
Centroid of Triangle
The intersection of the three medians of the triangle.
$$ C = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) $$
Two-Point Form
$$ \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1} $$
Slope-Intercept Form
$$ y = mx + b $$
Intercept Form
$$ \frac{x}{a} + \frac{y}{b} = 1 $$
Dist. Between Parallel Lines
Distance between Ax + By + C₁ = 0 and Ax + By + C₂ = 0.
$$ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} $$