26.17: Base Graphs of the Trigonometric Functions
- Page ID
- 174421
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Definitions and Theorems
When a body repeats its movement along a path in a regular and predictable manner over a specific time interval, its movement is called periodic motion.
A mathematical function modeling periodic motion is called a periodic function. A periodic function repeats its values at regular intervals, meaning its graph has a pattern that repeats itself over a specific horizontal distance or time interval.
For any function \( f(x) \), the smallest positive number \( k \) for which\[ f(x + k) = f(x) \nonumber \]for all \( x \) in the domain of \( f \) is called the period of \( f(x) \).
A smooth, wave-like curve that oscillates (moves up and down) periodically and which can be derived from either the sine or cosine function is called a sine curve. A curve of this nature is often called sinusoidal,
The domain of \( f(x) = \sin\left( x \right) \) is \( \left( -\infty,\infty \right) \) and the range is \( \left[ -1,1 \right] \).
The period of \( f(x) = \sin\left( x \right) \) is \( 2 \pi \). This is also known as the natural period of the sine function.
The principal cycle of a trigonometric function is the interval starting at \( x = 0 \) and terminating at \( x =p \), where \( p \) is the period of the trigonometric function.
The principal cycle of the sine function is \( \left[ 0, 2\pi \right] \).
Let \( M \) and \( m \) be a function's greatest and least values, respectively. The amplitude of the graph of the function is defined to be\[ \text{Amplitude} = \dfrac{1}{2}\left| M - m \right|. \nonumber \]
The amplitude of \( f(x) = \sin\left( x \right) \) is 1.
- Proof
- The maximum value of \( f(x) = \sin\left( x \right) \) is \( M = 1 \) and the minimum value is \( m = -1 \). Hence,\[ \text{Amplitude} = \dfrac{1}{2}\left| 1 - (-1) \right| = 1. \nonumber \]
The zeros of \( f(x) = \sin\left( x \right) \) occur at \( x = \pi k \), where \( k \in \mathbb{Z} \).
The domain of \( f(x) = \cos\left( x \right) \) is \( \left( -\infty,\infty \right) \) and the range is \( \left[ -1,1 \right] \).
The period of \( f(x) = \cos\left( x \right) \) is \( 2 \pi \). Thus, the natural period of the cosine function is \( 2 \pi \).
The principal cycle of the cosine function is \( \left[ 0, 2\pi \right] \).
The amplitude of \( f(x) = \cos\left( x \right) \) is 1.
The zeros of \( f(x) = \cos\left( x \right) \) occur at \( x = \frac{\pi}{2} + \pi k \), where \( k \in \mathbb{Z} \).
The domain of the tangent function is \( \left\{x \mid x \neq \frac{\pi}{2} + \pi k \right\} \), where \( k \in \mathbb{Z} \). The range is \( \left( -\infty,\infty \right) \).
The period of \( f(x) = \tan\left( x \right) \) is \( \pi \). Thus, the natural period of the tangent function is \( \pi \).
The principal cycle of the tangent function is \( \left[ 0, \pi \right] \).
The zeros of the tangent function occur at \( x = \pi k \), where \( k \in \mathbb{Z} \).
The key numbers of a cycle for a trigonometric function are the \( x \)-values of the of the initial, final, middle, and quarter-points in the principal cycle. These can be found by partitioning the cycle into four equally-spaced subintervals.
The distance between any two consecutive key numbers is called the step size.
The step size for a trigonometric function is \( \frac{\text{Period}}{4} \).
- Domain: \( \left( -\infty, \infty \right) \)
- Range: \( \left[ -1,1 \right] \)
- Symmetry: Odd (symmetric about the origin)
- Principal Cycle: \( \left[ 0,2\pi \right] \)
- Natural Period: \( 2 \pi \)
- Amplitude: 1
- Zeros: \( x = k \pi \), where \( k \in \mathbb{Z} \)
- Domain: \( \left( -\infty, \infty \right) \)
- Range: \( \left[ -1,1 \right] \)
- Symmetry: Even (symmetric about the \( y \)-axis)
- Principal Cycle: \( \left[ 0,2\pi \right] \)
- Natural Period: \( 2 \pi \)
- Amplitude: 1
- Zeros: \( x = \frac{\pi}{2} + k \pi \), where \( k \in \mathbb{Z} \)
- Domain: All \( x \ne \frac{\pi}{2} + k \pi \), where \( k \in \mathbb{Z} \)
- Range: \( \left( -\infty,\infty \right) \)
- Symmetry: Odd (symmetric about the origin)
- Principal Cycle: \( \left[ 0,\pi \right] \)
- Natural Period: \( \pi \)
- Amplitude: N/A
- Zeros: \( x = k \pi \), where \( k \in \mathbb{Z} \)
- Domain: All \( x \ne k \pi \), where \( k \in \mathbb{Z} \)
- Range: \( \left( -\infty, -1 \right] \cup \left[ 1, \infty \right) \)
- Symmetry: Odd (symmetric about the origin)
- Principal Cycle: \( \left[ 0,2\pi \right] \)
- Natural Period: \( 2 \pi \)
- Amplitude: N/A
- Zeros: None
- Domain: All \( x \ne \frac{\pi}{2} + k \pi \), where \( k \in \mathbb{Z} \)
- Range: \( \left( -\infty, -1 \right] \cup \left[ 1, \infty \right) \)
- Symmetry: Even (symmetric about the \( y \)-axis)
- Principal Cycle: \( \left[ 0,2\pi \right] \)
- Natural Period: \( 2 \pi \)
- Amplitude: N/A
- Zeros: None
- Domain: All \( x \ne k \pi \), where \( k \in \mathbb{Z} \)
- Range: \( \left( -\infty, \infty \right) \)
- Symmetry: Odd (symmetric about the origin)
- Principal Cycle: \( \left[ 0,\pi \right] \)
- Natural Period: \( \pi \)
- Amplitude: N/A
- Zeros: \( x = \frac{\pi}{2} + k \pi \), where \( k \in \mathbb{Z} \)