Graphs of Secant and Cosecant

Definition The reciprocal of a function $g$ is the function $\frac{1}{g}$ with values $\frac{1}{g(x)}$, defined at exactly those $x$ in the domain of $g$ where $g(x) \ne 0$. Secant and cosecant are the reciprocals of cosine and sine,

Wherever both are defined, a function and its reciprocal have the same sign. They are equal exactly where the value is $1$ or $-1$, and where $0 < \lvert g(x) \rvert < 1$ the reciprocal satisfies

Definition The secant graph is the graph of $y = \sec x$ with $x$ in radians. Its domain is all real numbers except $\frac{\pi}{2} + k\pi$ for integers $k$, and its vertical asymptotes are the lines $x = \frac{\pi}{2} + k\pi$, where cosine is zero. Its range is $(-\infty, -1] \cup [1, \infty)$, so it never takes a value strictly between $-1$ and $1$ and it is never zero. It has period $2\pi$ and it is even, $\sec(-x) = \sec x$. It has a local minimum value $1$ at each $x = 2k\pi$ and a local maximum value $-1$ at each $x = \pi + 2k\pi$.

Definition The cosecant graph is the graph of $y = \csc x$ with $x$ in radians. Its domain is all real numbers except $k\pi$ for integers $k$, and its vertical asymptotes are the lines $x = k\pi$, where sine is zero. Its range is also $(-\infty, -1] \cup [1, \infty)$. It has period $2\pi$ and it is odd, $\csc(-x) = -\csc x$. It has a local minimum value $1$ at each $x = \frac{\pi}{2} + 2k\pi$ and a local maximum value $-1$ at each $x = \frac{3\pi}{2} + 2k\pi$. For every $x$ that is not a multiple of $\pi$

so the cosecant graph is the secant graph shifted $\frac{\pi}{2}$ units to the right.

Definition A branch of the secant or cosecant graph is the part of the graph over an open interval between two consecutive vertical asymptotes. On such an interval the related cosine or sine keeps one sign.

• If it is positive, the branch lies on or above the line $y = 1$, opens upward, and has its lowest point where the cosine or sine equals $1$.
• If it is negative, the branch lies on or below the line $y = -1$, opens downward, and has its highest point where the cosine or sine equals $-1$.

These turning points are the only points the branch shares with the cosine or sine graph, because a number $c \ne 0$ equals its reciprocal $\frac{1}{c}$ only when $c = 1$ or $c = -1$.

Definition For constants $A \ne 0$ and $B \ne 0$, the function

has period $\frac{2\pi}{\lvert B \rvert}$, vertical asymptotes wherever $\cos(Bx) = 0$, and range $(-\infty, -\lvert A \rvert] \cup [\lvert A \rvert, \infty)$. The turning points of its branches are exactly the highest and lowest points of the graph of $y = A\cos(Bx)$. The same holds for $y = A\csc(Bx)$ with $\sin(Bx)$ and $y = A\sin(Bx)$ in place of $\cos(Bx)$ and $y = A\cos(Bx)$.