Turn every height upside down.

$\sec x$, $\operatorname{cosec} x$ and $\cot x$ are one over $\cos x$, $\sin x$ and $\tan x$. Their graphs look strange until you build one, height by height.

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$\sec x = \dfrac{1}{\cos x}$. Take each height of the cosine curve and turn it upside down: $1$ stays $1$, $\tfrac{1}{2}$ becomes $2$ and $\tfrac{1}{4}$ becomes $4$. Small heights make big reciprocals.

As $\cos x$ shrinks towards $0$, its reciprocal grows without limit. Where $\cos x = 0$, at $x = \tfrac{\pi}{2}$ and $\tfrac{3\pi}{2}$, $\sec x$ is not defined: the graph has vertical asymptotes there.

A reciprocal keeps the sign of the height underneath, so between the asymptotes $\sec x$ hangs below $-1$. The two graphs touch where $\cos x = \pm 1$, and $\sec x$ never takes a value between $-1$ and $1$.

To solve $\sec x = 2$ for $0 \le x \le 2\pi$, turn both sides over: $\cos x = \tfrac{1}{2}$. The lines $y = 2$ and $y = \tfrac{1}{2}$ meet the two graphs at the same places:

Upside-down heights, asymptotes where the curve meets zero.

$\operatorname{cosec}$ and $\cot$, the identities $1 + \tan^2 x = \sec^2 x$ and its partner, arcsin, arccos and arctan, and twelve graded exam questions with full worked solutions are waiting.

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