Tangent and Secant Lines

Summary: §521. The curves $y=\arctan x$ (tangent line) and $y=\mathrm{arcsec}x$ (secant line) — siblings of the sine-line in the circular-arc genus. The tangent line has infinitely many parallel asymptotes; the secant line has infinitely many infinite branches.

Sources: chapter21 §521.

Last updated: 2026-05-12

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Tangent line (§521)

$y=\arctan x,\text{equivalently}x=\tan y=\frac{\sin y}{\cos y}.$

For each $x$ there are infinitely many arcs whose tangent is $x$, separated by $\pi$. So $y$ takes the values $\arctan x+n\pi$ for $n\in \mathbb{Z}$. The curve has infinitely many asymptotes parallel to the $x$-axis, one at each height $y=(2n+1)\pi /2$ (where $\tan y$ blows up), all parallel to each other.

Geometrically: the curve is made of infinitely many congruent “S-shaped” pieces, each running between two adjacent asymptotes from $y=(2n-1)\pi /2$ up to $y=(2n+1)\pi /2$, with $x$ swinging from $-\infty$ to $+\infty$ as $y$ traverses the piece.

Secant line (§521)

$y=\mathrm{arcsec}x,\text{equivalently}x=\sec y=\frac{1}{\cos y}.$

Each value of $x$ with $|x|\ge 1$ corresponds to infinitely many arcs $y$. For $|x|<1$, the equation has no real solution. The curve has infinitely many branches, each going to infinity — one for each interval between consecutive zeros of $\cos y$.

Place in the chapter

Euler dispatches these in a single short paragraph, treating them as obvious modifications of the sine-line:

The curve has an infinite number of asymptotes parallel to each other. Likewise we can draw the secant line from the equation $y=\mathrm{arcsec}x$, or $x=\sec y=1/\cos y$, which has an infinite number of branches, each of which goes to infinity. (source: chapter21, §521)

He immediately moves on to the cycloid as the genus’s most important example.

Related pages

• chapter-21-on-transcendental-curves
• sine-line — first cousin.
• cycloid — the genus’s prize specimen.
• transcendental-curves