Mathematics

Math Laws, Identities, and Definitions (Pre-Calculus)

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Sets and Types of Numbers

  • Prime numbers = postive integer that are only divisible by themselves and 1
    • The lowest prime number is 2 (this is also the only even prime number)
    • Note: by convention, 1 is NOT considered a prime number
    • Cannot be factorized ⇒ prime numbers are considered the fundamental building blocks of positive integers
      • Important concept for coding
    • The Riemann hypothesis involves the distribution of prime numbers among the positive integers
  • Composite numbers = positive integers that are neither 1 nor a prime
    • Every composite number can be written as a unique product of prime factors
      • Examples:
        • 12 = 2^2 \times 3
        • 21 = 3 \times 7
        • 270 = 2 \times 3^3 \times 5
    • Currently, there is no genral algorithm that can determine the uniqe combination of prime factors for any given positive integer
      • This process is an ideal basis fo encryption systems

Trigonometry

Tangent Identities

  • \tan \theta = \frac{\sin \theta}{\cos \theta}
  • \cot \theta = \frac{\cos \theta}{\sin \theta}

Pythagorean Identities

  • \sin ^2 \theta  + \cos ^2 \theta = 1
  • \tan ^2 \theta + 1 = \sec ^2 \theta
  • \cot ^2 \theta + 1 = \csc ^2 \theta

Periodic Identities

  • \sin (\theta + 2 \pi n) = \sin \theta
  • \cos (\theta + 2 \pi n) = \cos \theta
  • \tan (\theta + \pi n) = \tan \theta
  • \csc (\theta + 2 \pi n) = \csc \theta
  • \sec (\theta + 2 \pi n) = \sec \theta
  • \cot (\theta + \pi n) = \cot \theta

Sources:

  • Glendinning, P. (2013). Math in minutes. New York, NY: Quercus.

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