Trig Equations

House Domain Restriction

  • House domain restriction typically refers to restricting the domain of a function.
  • For certain functions, this means identifying asymptotes.
  • Asymptotes: Lines that a graph approaches but never touches.
  • Domain restriction: The numbers that cannot be included in the domain of a function are generally the asymptotes themselves.
  • Thus, the domain will be all real numbers except for the asymptotes.

Polynomial Function Example

  • Considering the function:
    f(x)=−3extcot(2extπx)+53−1f(x) = -3 ext{cot}(2 ext{π}x) + \frac{5}{3} - 1
  • Step 1: Identify the coefficients and constants.
      - Coefficients: -3 (amplitude), 2π (frequency), and other constants.
        - The amplitude represents the height of the function oscillation from the midline, and affects the overall height of the graph.
        - The frequency determines how many cycles occur within a given interval.
  • Step 2: Identify the period.
      - For cotangent, the period is generally determined by the formula:
    extPeriod=2extπbextwhereb=2ext{Period} = \frac{2 ext{π}}{b} ext{ where } b = 2
      - Thus, the period for this function will be
    extPeriod=2extπ2=extπext{Period} = \frac{2 ext{π}}{2} = ext{π}.
  • Step 3: Identify phase shift and vertical shifts.
      - The phase shift here is determined by the term within the cotangent, specifically adjusting for frequency and horizontal shifts:
      - The phase shift is given by −extshiftb=−5/32=−56-\frac{ ext{shift}}{b} = -\frac{5/3}{2} = -\frac{5}{6}.

Phase Shift and Vertical Shifts

  • Phase Shift: Horizontal movement of the graph, which can be calculated based on the function's transformation.
  • Vertical Shift: Determined by any additions or subtractions to the function (in this case, ( -1 ) wins from the vertical shift of (+\frac{5}{3}-1)).
  • Altitudes: Represent mid-level shifts of the function:
      - For cotangent, the vertical shift does not affect the range or amplitude significantly, as the oscillation can go to positive and negative infinity.
  • Identify bounds of cotangent normally ranging from 0 to π, here shifted between 0 and ( 0.5 ).

Finding Asymptotes

  • The standard asymptote locations for cotangent function: located at every integer multiple of ( ext{π} )
      - Adapted to this function due to phase shift, yielding vertical asymptotes at:
        - x=0x = 0 and x=extπx = ext{π},
  • However, adjustments for the phase shift will give us new asymptotes at:
      - x=−16x = -\frac{1}{6} and x=16x = \frac{1}{6}.
  • Asymptotes can also be cyclical, altering at every half period, hence extasymptotes=x=−16+kπ2ext{asymptotes} = x = -\frac{1}{6} + k \frac{π}{2}.

Domain Considerations

  • The resulting domain for the function would be:
      - x<br/>≠−16x <br />\neq -\frac{1}{6} and x<br/>≠16x <br />\neq \frac{1}{6}
  • This will also imply going from one asymptote to the next.

Range Considerations

  • Range for Cotangent: Generally, the full range of cotangent is from negative infinity to positive infinity, though vertical shifts can impact it.
  • Vertical shifts in this context do not affect the infinite bounds of the range.

Cosecant and Secant Functions

  • Reviewing the cosecant function:
    f(x)=−2extcosecant(extπx)−π2+12f(x) = -2 ext{cosecant}( ext{π}x) - \frac{π}{2} + \frac{1}{2}
  • Finding Asymptotes:
      - Step one: finding the period:
      - The period for cosecant (as it is the reciprocal of sine) is the same as sine:
      - Period is denoted as extPeriod=2πbext{Period} = \frac{2π}{b},
      - Wherein this function, b=1b = 1. Hence the period would remain 2π.
  • Adjustments for amplitude, phase shift, and vertical transformations accordingly lead to:
      - The phase shift derived from the transformation.

Midline and Range for a Function

  • The range will also consider the amplitude:
      - For the amplitude determined by the coefficient of cosecant, the range can be adjusted based on max and min relative to vertical shifts.

Procedures for Finding Trigonometric Values

  • When analyzing trigonometric angles that go beyond the normal bounds, cyclical properties come into play:
      - Using angles such as extsine(extalpha)−19πext{sine}( ext{alpha}) - 19π or extcosine(extbeta)+202πext{cosine}( ext{beta}) + 202π,
      - It’s crucial to adjust angles to find equivalence within principal bounds:
          1. For 19π,19π, convert by subtracting relevant multiples of 2π2π (resulting in -ππ or relevant positives).
          2. Match periodicity to help identify if sine or cosine brings values back into range.

Summary of Function Transformations

  • Basic transformations of sine and cosine functions include amplitude, phase shifts, vertical shifts, and their impacts on ranges:
      - Ensuring transformations do not carry over to include or exclude values from the considered domain. The ranges reflect transformations but not restrictions from horizontal or vertical shifts.

  • Each analysis leads to periodic functions characterized by their oscillation, amplitude, shifts, and crucial properties leading to specific intervals and asymptotes.