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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