Functions Involving Trigonometric Ratios

Rules and Formulas for Functions Involving Trigonometric Ratios
Periodicity
  • Rule: A function can be expressed as f(x+2p)=sin(x+2p)+cos(2(x+2p))f(x + 2p) = \text{sin}(x + 2p) + \text{cos}(2(x + 2p)).
  • Rule: This indicates that the function is periodic with a period equal to 2p2p.
  • Rule: To analyze the function behavior, it's sufficient to examine it over one period ([0,2p][0, 2p]).
Finding X-Intercepts
  • Rule: To find x-intercepts, set the function equal to zero.
  • Equation: For a function of the form sin(x)+cos(2x)=0\text{sin}(x) + \text{cos}(2x) = 0, it leads to:
    • 2sin2(x)−sin(x)−1=02 \text{sin}^2(x) - \text{sin}(x) - 1 = 0
    • Factoring gives: (2sin(x)+1)(sin(x)−1)=0(2 \text{sin}(x) + 1)(\text{sin}(x) - 1) = 0
Derivative and Critical Points
  • Rule: The first derivative helps identify critical points where the function's slope is zero or undefined.
  • Formula: The first derivative is given by: y′=cos(x)−2sin(2x)y' = \text{cos}(x) - 2 \text{sin}(2x)
  • Rule: Critical points occur when y′=0y' = 0, specifically when:
    • cos(x)=0\text{cos}(x) = 0
    • or 1−4sin(x)=01 - 4 \text{sin}(x) = 0
  • Related Formula: The derivative can also be expressed as: y′=cos(x)(1−4sin(x))y' = \text{cos}(x)(1 - 4 \text{sin}(x))
Sign Analysis of Derivative
  • Rule: To determine increasing/decreasing behavior, test intervals around critical points where y′=0y' = 0.
  • Rule: Evaluate changes in the sign of y′y' across intervals.
Vertical Asymptotes and Undefined Points
  • Rule: The function may not be defined at specific points, leading to vertical asymptotes.
  • Rule: As xx approaches these points, yy heads towards $- \text{∞}$ or $+ \text{∞}$.
Second Derivative Test
  • Rule: The second derivative, y′′y'', provides information about concavity and points of inflection.
Examples
Example 1: Function Analysis for Graphing
  • Function: Analyze the function: y=sin(2x)+cos(x)y = \text{sin}(2x) + \text{cos}(x)
  • Solution Strategy (Implied):
    • Identify its periodicity (period 2p).
    • Study its behavior over one period, such as [0,2p][0, 2p].
    • Use derivatives to find turning points and concavity.
    • Determine x-intercepts and any undefined points/asymptotes.
Example 2: Trigonometric Identity Simplification
  • Equation: An example of simplification or identity is: 4sin2(x+p)=4sin(2x+2p)4 \text{sin}^2(x + p) = 4 \text{sin}(2x + 2p)