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7.3: The Trigonometric Functions - Unit Circle Definition

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    Connecting the Sine and Cosine Functions to the Unit Circle

    Lemma

    Let \( \theta \) be an angle in standard position, and suppose the terminal side of \( \theta \) intersects the unit circle at the point \( P\left( x,y \right) \). Then\[ \cos\left( \theta \right) = x \quad \text{and} \quad \sin\left( \theta \right) = y. \nonumber \]

    Example \( \PageIndex{ 1 } \)

    Find the coordinates of point \(P\) on the unit circle below.

    Trigonometric Functions of Real Numbers and the Unit Circle Definition

    Definition: Trigonometric Functions (Unit Circle Definition)

    Let \( P\left( x,y \right) \) be a point on the unit circle, and let \( t \) be the arc length from the point \( \left( 1,0 \right) \) to \( P \) along the circumference of the unit circle. The trigonometric functions of the real number \( t \) (also called the circular functions in this context) are defined as follows: \[ \begin{array}{|ccc|ccc|}
    \hline
    \text{Function} & & \text{Ratio} & \text{Function} & & \text{Ratio} \\[6pt] \hline
    \sin\left( t \right) & = & y & \csc\left( t \right) & = & \dfrac{1}{y} \\[6pt] \cos\left( t \right) & = & x & \sec\left( t \right) & = & \dfrac{1}{x} \\[6pt] \tan\left( t \right) & = & \dfrac{y}{x} & \cot\left( t \right) & = & \dfrac{x}{y} \\[6pt] \hline \end{array} \nonumber \]

    Example \(\PageIndex{2}\)

    Let \( t = \frac{5 \pi}{6} \) be the length of the arc along the unit circle from the point \( \left( 1,0 \right) \) to \( P\left( x,y \right) \). Evaluate each of the following functions.

    1. \( \sin\left( t \right) \)
    2. \( \cot\left( t \right) \)
    Example \( \PageIndex{ 3} \)

    Find the coordinates of the terminal point, \(P\), of an arc of length \(t\) starting at \((1,0)\) on a unit circle.

    1. \(t=\frac{3\pi}{2}\)
    2. \(t=\frac{4 \pi}{3}\)

    Using a Calculator to Compute the Values of Trigonometric Functions

    Note: Change to Radian Mode

    If you use a scientific or graphing calculator, refer to your instruction manual on setting it to radian mode. The scientific calculator in Desmos is set to degree mode by default, so if you are using that technology, you need to click the "DEG" toggle so that "RAD" is showing; however, the graphing calculator in Desmos defaults to radian mode.

    Example \( \PageIndex{ 4} \)

    Use your calculator to find the sine and cosine of the following angles in radians. Round your answers to four decimal places...

    Domain and Range of the Circular Functions

    Interactive Element: Geometric Interpretations of the Circular Functions

    Interact: Move the point \(D\) to help you investigate the circular functions geometrically.

    Theorem: Domains of the Circular Functions

    Let \( t \) be a real number and \( k \) be an integer. The domains of the circular functions are\[ \begin{array}{rcl}
    \sin\left( t \right) \text{ and } \cos\left( t \right) & \quad & \text{All real numbers, or } \left( -\infty, \infty \right) \\[6pt] \tan\left( t \right) \text{ and } \sec\left( t \right) & \quad & \text{All real numbers except }t = \dfrac{\pi}{2} + k \pi \\[6pt] \cot\left( t \right) \text{ and } \csc\left( t \right) & \quad & \text{All real numbers except }t = k \pi \\[6pt] \end{array} \nonumber \]

    Theorem: Ranges of the Circular Functions

    Let \( t \) be a real number. The ranges of the circular functions are\[ \begin{array}{rcl}
    \sin\left( t \right) \text{ and } \cos\left( t \right) & \quad & \left[ -1,1 \right] \\[6pt] \tan\left( t \right) \text{ and } \cot\left( t \right) & \quad & \left( -\infty, \infty \right) \\[6pt] \sec\left( t \right) \text{ and } \csc\left( t \right) & \quad & \left( -\infty,-1 \right] \cup \left[ 1, \infty \right) \\[6pt] \end{array} \nonumber \]

    Example \(\PageIndex{8}\)

    Is it possible for \( \csc\left( t \right) = -\frac{1}{2} \)?

    This page titled 7.3: The Trigonometric Functions - Unit Circle Definition is shared under a CC BY-SA 12 license and was authored, remixed, and/or curated by Roy Simpson.