4.3.1: Resources and Key Concepts

Math 373: Trigonometry for Calculus · LibreTexts

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    From LibreTexts, under not declared, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Key Concepts

    Definitions

    • Trigonometric Functions (Unit Circle Definition): Let \(t\) be a real number representing the arc length from the point (1, 0) to a point \(P(x, y)\) on the unit circle. The trigonometric functions of \(t\), also called circular functions, are defined by the coordinates of \(P\). For example, \(\cos(t) = x\) and \(\sin(t) = y\).

    Theorems

    • Lemma: Let \(\theta\) be an angle in standard position, and suppose the terminal side of \(\theta\) intersects the unit circle at the point \(P(x, y)\). Then \(\cos(\theta) = x\) and \(\sin(\theta) = y\).
    • Domains of the Circular Functions: Let \(t\) be a real number and \(k\) be an integer. The domains of the circular functions are:
      • \(\sin(t)\) and \(\cos(t)\): All real numbers, or \((-\infty, \infty)\)
      • \(\tan(t)\) and \(\sec(t)\): All real numbers except \(t = \frac{\pi}{2} + k\pi\)
      • \(\cot(t)\) and \(\csc(t)\): All real numbers except \(t = k\pi\)
    • Ranges of the Circular Functions: Let \(t\) be a real number. The ranges of the circular functions are:
      • \(\sin(t)\) and \(\cos(t)\): \([-1, 1]\)
      • \(\tan(t)\) and \(\cot(t)\): \((-\infty, \infty)\)
      • \(\sec(t)\) and \(\csc(t)\): \((-\infty, -1] \cup [1, \infty)\)

    Common Mistakes

    • Using the Wrong Calculator Mode: When working with the unit circle definition, angles and arc lengths are measured in radians. You must ensure your calculator is set to Radian mode, not Degree mode, to get correct approximations.
    • Confusing Angle Measure with Arc Length: While the numerical value of an angle in radians is equal to the arc length it subtends on a unit circle, it is important to remember they are conceptually different. An angle is an amount of rotation, while an arc length is a distance.

    This page titled 4.3.1: Resources and Key Concepts was last modified on Tue, 01 Jul 2025 23:53:59 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.