Notes on Functions of One Variable

Introduction

• Topic: Functions of one variable
• Focus: Trigonometric functions and general properties of functions

Trigonometric Functions

Sine Function

• Defined as: ( f(x) = ext{sin}(x) )
• Properties:
  • Periodic function with period ( 2\pi )
  • Values range between -1 and 1
• Graph:
  • Sinusoidal shape; repeats every ( 2\pi )
  • Calculation:
    • Defined between ( 0 ) and ( 2\pi )
    • For ( x ) in ( [0, \frac{\pi}{2}] ):
      • ( ext{sin}(x) = \frac{\text{opposite}}{\text{hypotenuse}} )
    • Extend to other intervals using symmetry and periodicity

Cosine Function

• Defined as: ( f(x) = ext{cos}(x) )
• Properties:
  • Similar to sine: periodic with period ( 2\pi ), values between -1 and 1
• Calculation:
  • Defined between ( 0 ) and ( \frac{\pi}{2} )
  • ( ext{cos}(x) = \frac{\text{adjacent}}{\text{hypotenuse}} )
• Relation to sine:
  • ( ext{cos}(x) ) is a phase shift of ( ext{sin}(x) ) by ( \frac{\pi}{2} )

Tangent Function

• Defined as: ( f(x) = ext{tan}(x) )
• Properties:
  • Not defined at odd multiples of ( \frac{\pi}{2} ) (undefined at ( -\frac{\pi}{2}, \frac{\pi}{2}, \ldots ))
  • Periodic function with period ( \pi )
• Calculation:
  • Defined by:
    • ( ext{tan}(x) = \frac{\text{opposite}}{\text{adjacent}} )
    • Defined between ( -\frac{\pi}{2} ) and ( \frac{\pi}{2} )

Other Trigonometric Functions

• Cotangent: ( ext{cot}(x) = \frac{1}{\text{tan}(x)} )
• Secant: ( ext{sec}(x) = \frac{1}{\text{cos}(x)} )
• Cosecant: ( ext{csc}(x) = \frac{1}{\text{sin}(x)} )

Trigonometric Identities

• ( \text{sin}(-x) = -\text{sin}(x) )
• ( \text{cos}(-x) = \text{cos}(x) )
• ( \text{tan}(-x) = -\text{tan}(x) )
• Pythagorean identity: ( \text{sin}^2(x) + \text{cos}^2(x) = 1 )
• ( \text{tan}(x) = \frac{\text{sin}(x)}{\text{cos}(x)} )
• ( \text{sec}(x) = \frac{1}{\text{cos}(x)} )
• ( \text{csc}(x) = \frac{1}{\text{sin}(x)} )

General Properties of Functions of One Variable

• Arithmetic Operations:
  • Sum: ( (f+g)(x) = f(x) + g(x) )
  • Product: ( (fg)(x) = f(x)\cdot g(x) )
  • Scalar multiplication: ( (Cf)(x) = C\cdot f(x) )
  • Quotient: ( \frac{f}{g}(x) = \frac{f(x)}{g(x)} ) (if ( g(x) \neq 0 ))
• Composition of Functions:
  • If ( g: E \to R ) and the range of ( f: D \to E ), then define:
    • ( (g \circ f)(x) = g(f(x)) )
  • Example:
    • If ( f(x) = x^2 + 1 ) and ( g(x) = \sqrt{x} ) then ( (g \circ f)(x) = \sqrt{x^2 + 1} )

Conclusion

• Overview of functions of one variable, focusing on trigonometric functions and their properties.
• Mathematical operations and composition of functions are essential tools in understanding functions of one variable.