MTH111 Elementary Algebra I Summary Notes

Course Outline

  • Trigonometric Functions: Radian measure, Trigonometric identities, Laws of Sine and Cosine, Inverse trigonometric functions, Trigonometric equations.
  • Exponential Functions: Definition of $a^x$ for $a > 0$, graphs, Laws of Exponents, Natural exponential function.
  • Logarithmic Functions: Definition of $ ext{log}_a x$, graphs, Laws of Logarithms, Natural logarithmic function.
  • Algebraic Functions: Polynomials, Long division, Factor theorem, Remainder theorem.
  • Rational Functions: Asymptotes, Partial fraction decomposition, Roots, Domain.
  • Complex Numbers: Representation, Operations, Modulus and argument, Polar representation, De Moivre’s Theorem, Roots of polynomials,
  • Quadratic Formula.

Trigonometric Functions

  • Measurement of Angles:

    • Degree: Angle with vertex at circle center, arc length $= rac{1}{360}$ of circumference.
    • Radian: Angle with vertex at center, arc length $=r$.
    • Relationship:
      • $S = rac{ heta}{360} imes 2 ext{π}r$
      • $ heta = rac{S}{r}$
      • $1 ext{°} = rac{ ext{π}}{180} rad$
      • $1 rad = rac{180}{ ext{π}} ext{°}$

  • Trigonometric Ratios:

    • For special angles (e.g., $0°, 30°, 45°, 60°, 90°$)
    • Common uses include $ ext{sin, cos, tan,}$ and their special angle values in surd form.

Algebraic Functions

  • Polynomials: Operations and properties (degree, leading coefficient, monic).
  • Rational Functions: Expressing as sums of fractions, Partial fraction decomposition rules.

Exponential and Logarithmic Functions

  • Exponential Functions: Base $a$, increase/decrease behaviors, and Laws.
  • Logarithmic Laws: Product, quotient, and power rules for simplification.

Complex Numbers

  • Definition: $z = x + iy$
  • Operations: Addition, subtraction, multiplication, division, with Modulus $|z| = ext{√}(x^2 + y^2)$.
  • Polar Form: $z = r [ ext{cos } heta + i ext{sin } heta]$.
  • De Moivre’s Theorem for Roots.

Practical Applications

  • Graphing Rational Functions: Identifying intercepts, vertical/horizontal asymptotes, and shape.
  • Solving Polynomial Equations: Roots and factors, both over real and complex fields.