5.4.1: Resources and Key Concepts

Math 373: Trigonometry for Calculus · LibreTexts

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    Key Concepts

    Definitions

    • Trigonometric Functions (Coordinate Definition): For an angle \(\theta\) in standard position, a point \(P(x,y)\) on its terminal side at a distance \(r\) from the origin has coordinates given by \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\).
    • Maximum Displacement: In the context of a sinusoidal function, the maximum displacement is the amplitude, which measures the wave's maximum change in position from its midline.
    • Ordinary Frequency (f): The number of cycles a sinusoid completes in a given unit of time. It is the reciprocal of the period (\(f = \frac{1}{T}\)).
    • Hertz (Hz): The unit for ordinary frequency, equivalent to one cycle per second.
    • Angular Frequency (\(\omega\) or B): The value of the parameter \(B\) in a sinusoidal function. It is related to the ordinary frequency by the formula \(\omega = 2\pi f\).
    • Phase Angle: The value of the parameter \(C\) in the sinusoidal function \(f(t) = A\sin(Bt+C)+D\). It corresponds to the angle at \(t=0\).

    Theorems

    • Sine Curve Properties: For a function of the form \(f(t) = A \sin(Bt + C) + D\):
      • The amplitude is \(|A|\).
      • The angular frequency is \(B\).
      • The ordinary frequency is \(f = \frac{B}{2\pi}\).
      • The period is \(T = \frac{2\pi}{B} = \frac{1}{f}\).
      • The phase angle is \(C\).
      • The phase shift is \(-\frac{C}{B}\).
      • The vertical shift, or midline, is \(D\).

    Common Mistakes

    • Confusing Period and Frequency: The period is the time it takes to complete one cycle (e.g., seconds per cycle), while frequency is the number of cycles completed in a unit of time (e.g., cycles per second). They are reciprocals of each other.
    • Confusing Angular Frequency and Ordinary Frequency: The angular frequency is the parameter \(B\) in the sinusoidal model, while the ordinary frequency \(f\) must be calculated using \(f = B / 2\pi\).
    • Confusing Phase Angle and Phase Shift: The phase angle is the parameter \(C\), while the phase shift is a horizontal translation of the graph calculated as \(-C/B\).
    • Assuming Only One Correct Model: For a given sinusoidal graph, there are infinitely many correct equations that can model it. A cosine function can be used instead of a sine function, and different phase shifts can produce the same graph.

    This page titled 5.4.1: Resources and Key Concepts was last modified on Thu, 03 Jul 2025 07:15:05 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.