Chapter 8 — Periodic Functions

$y = \sin x$ has these properties:

| Property | Value | |———-|——-| | Domain | $(-\infty, \infty)$ | | Range | $[-1, 1]$ | | Period | $2\pi$ | | Amplitude | $1$ | | $x$-intercepts | $x = n\pi$ for integer $n$ | | Maximum | $y = 1$ at $x = \dfrac{\pi}{2} + 2n\pi$ | | Minimum | $y = -1$ at $x = \dfrac{3\pi}{2} + 2n\pi$ | | Symmetry | Odd ($\sin(-x) = -\sin x$); origin symmetry |

$y = \cos x$ has these properties:

| Property | Value | |———-|——-| | Domain | $(-\infty, \infty)$ | | Range | $[-1, 1]$ | | Period | $2\pi$ | | Amplitude | $1$ | | Maximum | $y = 1$ at $x = 2n\pi$ | | Minimum | $y = -1$ at $x = \pi + 2n\pi$ | | Symmetry | Even ($\cos(-x) = \cos x$); $y$-axis symmetry |

It measures the height from the midline to the peak (or trough).

• $

  a

  > 1$: vertical stretch (taller waves)

• $0 <

  a

  < 1$: vertical compression (shorter waves)

• $a < 0$: reflection over the $x$-axis (flipped)

• $

  b

  > 1$: horizontal compression (shorter period, faster oscillation)

• $0 <

  b

  < 1$: horizontal stretch (longer period, slower oscillation)

| Parameter | Effect | Formula | |———–|——–|———| | $a$ | Amplitude | $|a|$ | | $b$ | Period | $P = \dfrac{2\pi}{|b|}$ | | $c$ | Phase shift | $\dfrac{c}{b}$ (right if positive) | | $d$ | Vertical shift / Midline | $y = d$ |

Phase Shift $= \dfrac{c}{b}$ in $y = a\sin(bx - c) + d$.

Factor out $b$: $y = a\sin!\left[b!\left(x - \dfrac{c}{b}\right)\right] + d$

The graph shifts right by $\dfrac{c}{b}$ if $c > 0$, or left if $c < 0$.

$y = \tan x = \dfrac{\sin x}{\cos x}$

| Property | Value | |———-|——-| | Domain | All reals except $x = \dfrac{\pi}{2} + n\pi$ | | Range | $(-\infty, \infty)$ | | Period | $\pi$ | | Vertical asymptotes | $x = \dfrac{\pi}{2} + n\pi$ | | $x$-intercepts | $x = n\pi$ | | Symmetry | Odd (origin symmetry) | | Behavior | Increasing on each branch |

$y = \cot x = \dfrac{\cos x}{\sin x}$

| Property | Value | |———-|——-| | Domain | All reals except $x = n\pi$ | | Range | $(-\infty, \infty)$ | | Period | $\pi$ | | Vertical asymptotes | $x = n\pi$ | | $x$-intercepts | $x = \dfrac{\pi}{2} + n\pi$ | | Symmetry | Odd | | Behavior | Decreasing on each branch |

$y = \sec x = \dfrac{1}{\cos x}$

| Property | Value | |———-|——-| | Domain | All reals except $x = \dfrac{\pi}{2} + n\pi$ | | Range | $(-\infty, -1] \cup [1, \infty)$ | | Period | $2\pi$ | | Vertical asymptotes | $x = \dfrac{\pi}{2} + n\pi$ (where $\cos x = 0$) | | No $x$-intercepts | (secant is always $\ge 1$ or $\le -1$) | | Symmetry | Even |

$y = \csc x = \dfrac{1}{\sin x}$

| Property | Value | |———-|——-| | Domain | All reals except $x = n\pi$ | | Range | $(-\infty, -1] \cup [1, \infty)$ | | Period | $2\pi$ | | Vertical asymptotes | $x = n\pi$ (where $\sin x = 0$) | | Symmetry | Odd |

Inverse Sine: $y = \sin^{-1}(x) = \arcsin(x)$

“What angle in $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ has sine equal to $x$?”

\(\sin^{-1}(x) = \theta \iff \sin\theta = x \text{ and } -\frac{\pi}{2} \le \theta \le \frac{\pi}{2}\)

Inverse Cosine: $y = \cos^{-1}(x) = \arccos(x)$

“What angle in $[0, \pi]$ has cosine equal to $x$?”

\(\cos^{-1}(x) = \theta \iff \cos\theta = x \text{ and } 0 \le \theta \le \pi\)

Inverse Tangent: $y = \tan^{-1}(x) = \arctan(x)$

“What angle in $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ has tangent equal to $x$?”

\(\tan^{-1}(x) = \theta \iff \tan\theta = x \text{ and } -\frac{\pi}{2} < \theta < \frac{\pi}{2}\)

Cancellation Properties:

\(\sin(\sin^{-1}(x)) = x \quad \text{for } x \in [-1, 1]\) \(\sin^{-1}(\sin(x)) = x \quad \text{for } x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)

Same pattern for cosine and tangent with their respective restricted ranges.