trig lesson 4.1 & 4.2

  • Lines Through the Origin

    • Any diagonal line that goes through the origin can be expressed in the form y = mx, where m is the slope.

    • Examples include:

      • y = x (m = 1)

      • y = 2x (m = 2)

      • y = -1/3x (m = -1/3)

    • Lines represented must not have a y-intercept other than the origin to be considered odd functions in this context.

  • Introduction of the Function f(x) = 1/x

    • This function is not defined for x = 0; hence, zero does not belong to the domain of f(x).

    • When a function has a domain exclusion, it can either:

      • Have a hole in the graph (removable discontinuity)

      • Have a vertical asymptote (essential discontinuity)

    • Table of values for f(x) = 1/x:

      • f(1) = 1 → Point (1,1)

      • f(2) = 1/2 → Point (2,0.5)

      • f(100) = 1/100 → Approaching 0 but never equal to 0 (horizontal asymptote behavior)

  • Asymptotic Behavior

    • Horizontal Asymptote: y = 0 (x-axis) as x approaches ±∞.

    • Vertical Asymptote: x = 0, as f(x) approaches ±∞ when x approaches 0 from either side.

    • The function f(x) = 1/x exhibits odd symmetry about the origin.

  • Odd Functions and Their Symmetry

    • Odd functions satisfy f(-x) = -f(x).

    • For the function f(x) = 1/x:

      • f(2) = 1/2

      • f(-2) = -1/2

    • Graphs of odd functions are symmetric about the origin, meaning if you reflect any point (a,b) over the origin, you obtain (-a,-b).

  • Negative Angle Identities in Trigonometry

    • Sine and Cosine Functions:

      • Cosine is even: cos(-θ) = cos(θ)

      • Sine is odd: sin(-θ) = -sin(θ)

    • Periodicity in Trigonometric Functions:

      • For sine and cosine, periodicity means they repeat their values, specifically:

        • sin(θ + 2kπ) = sin(θ) for any integer k.

        • cos(θ + 2kπ) = cos(θ) for any integer k.

  • Cosecant and Secant

    • Cosecant and Secant are derived from sine and cosine, respectively:

      • Csc(-θ) = -csc(θ) (odd function)

      • Sec(-θ) = sec(θ) (even function)

  • Period and Amplitude of Sine Functions

    • General form: y = A sin(Bx)

      • Amplitude (|A|): the height from the center line to the peak of the wave (absolute value of the coefficient of sine).

      • Period (2π/|B|): the length of one complete cycle of the wave.

    • Example of effect:

      • If B = 2, the period is π.

  • Graphing Sine Functions

    • A typical sine function starts at (0,0), peaks at (π/2, A), returns to the axis (π,0), troughs at (3π/2, -A), and completes the cycle at (2π,0).

    • Each segment should be proportionate and respect the amplitude and period adjustments made by coefficients.

  • Reflection and Inversion

    • A negative coefficient in front of sine results in reflection over the x-axis.

    • Adjustments in the sine function graph will maintain symmetrical properties.

    • Example: y = -sin(x) results in an inverted sine graph.

  • Final Notes

    • Define amplitude and period accordingly for sine and cosine:

      • Amplitude: max value minus min value over 2.

      • Period: determined by the coefficient of x in input,

    • Practicing graphing with these adjustments is essential.