Fundamental Mathematical Formulas to Know for General Mathematics

Fundamental mathematical formulas are essential tools that help us understand and solve various problems in geometry, algebra, and beyond. These formulas, like the Pythagorean Theorem and the Quadratic Formula, are widely used in fields such as engineering, physics, and finance.

1. Pythagorean Theorem

   • States that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): ( c^2 = a^2 + b^2 ).
   • Used to determine the length of a side in a right triangle when the lengths of the other two sides are known.
   • Fundamental in various fields such as architecture, engineering, and physics.

2. Quadratic Formula

   • Provides the solutions to the quadratic equation ( ax^2 + bx + c = 0 ) as ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
   • Useful for finding the roots of any quadratic equation, regardless of whether they are real or complex.
   • The discriminant (( b^2 - 4ac )) indicates the nature of the roots (two real, one real, or two complex).

3. Area of a Circle

   • The area (A) of a circle is calculated using the formula ( A = \pi r^2 ), where r is the radius.
   • Essential for problems involving circular shapes in geometry, physics, and engineering.
   • The constant ( \pi ) (approximately 3.14) is crucial in calculations involving circles.

4. Slope-Intercept Form of a Line

   • The equation of a line in slope-intercept form is ( y = mx + b ), where m is the slope and b is the y-intercept.
   • Allows for easy graphing of linear equations and understanding the relationship between variables.
   • Slope indicates the steepness of the line, while the y-intercept shows where the line crosses the y-axis.

5. Distance Formula

   • The distance (d) between two points ((x_1, y_1)) and ((x_2, y_2)) in a Cartesian plane is given by ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ).
   • Essential for calculating the straight-line distance in coordinate geometry.
   • Used in various applications, including navigation and computer graphics.

6. Midpoint Formula

   • The midpoint (M) between two points ((x_1, y_1)) and ((x_2, y_2)) is calculated as ( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ).
   • Useful for finding the center point of a line segment in coordinate geometry.
   • Important in various fields, including design and construction.

7. Trigonometric Ratios (sine, cosine, tangent)

   • Sine, cosine, and tangent are ratios derived from the sides of a right triangle relative to its angles.
   • Defined as:
     • ( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} )
     • ( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} )
     • ( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} )
   • Fundamental in solving problems involving angles and distances in trigonometry.

8. Exponential Growth/Decay Formula

   • Describes processes that increase or decrease at a rate proportional to their current value, expressed as ( N(t) = N_0 e^{kt} ).
   • ( N_0 ) is the initial quantity, k is the growth (k > 0) or decay (k < 0) constant, and t is time.
   • Widely used in fields such as biology, finance, and physics to model population growth or radioactive decay.

9. Logarithmic Properties

   • Logarithms are the inverse operations of exponentiation, with key properties including:
     • ( \log_b(mn) = \log_b(m) + \log_b(n) )
     • ( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) )
     • ( \log_b(m^k) = k \cdot \log_b(m) )
   • Essential for solving exponential equations and understanding growth rates.

10. Arithmetic Sequence Formula

    • An arithmetic sequence is defined by the formula ( a_n = a_1 + (n-1)d ), where ( a_1 ) is the first term, d is the common difference, and n is the term number.
    • Useful for finding any term in a sequence of numbers that increases or decreases by a constant amount.
    • Applications include finance, computer science, and statistics.

11. Geometric Sequence Formula

    • A geometric sequence is defined by the formula ( a_n = a_1 \cdot r^{(n-1)} ), where ( a_1 ) is the first term, r is the common ratio, and n is the term number.
    • Useful for finding any term in a sequence where each term is multiplied by a constant factor.
    • Commonly used in finance for compound interest calculations.

12. Binomial Theorem

    • Describes the expansion of powers of a binomial ( (a + b)^n ) as ( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k ).
    • Provides a way to calculate coefficients in polynomial expansions without direct multiplication.
    • Important in combinatorics, probability, and algebra.

13. Law of Sines

    • Relates the lengths of the sides of a triangle to the sines of its angles: ( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ).
    • Useful for solving triangles when given two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).
    • Important in navigation, physics, and engineering.

14. Law of Cosines

    • Relates the lengths of the sides of a triangle to the cosine of one of its angles: ( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) ).
    • Useful for solving triangles when given two sides and the included angle (SAS) or all three sides (SSS).
    • Important in various applications, including physics and computer graphics.

15. Volume of a Sphere

    • The volume (V) of a sphere is calculated using the formula ( V = \frac{4}{3} \pi r^3 ), where r is the radius.
    • Essential for problems involving three-dimensional shapes in geometry and physics.
    • The constant ( \pi ) plays a crucial role in calculations involving circular and spherical objects.