MISSION · Geometry
Secant, Cosecant and Cotangent
Three more ratios, each one over a ratio you already know
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The idea in one sentence
There are six ratios in a right triangle, not three, and the other three are simply \(\frac{1}{\sin}\), \(\frac{1}{\cos}\) and \(\frac{1}{\tan}\) under unfamiliar names.
The three names, and the lettering trap
\[\sec\theta = \frac{1}{\cos\theta} \qquad \csc\theta = \frac{1}{\sin\theta} \qquad \cot\theta = \frac{1}{\tan\theta}\]The pairing is deliberately awkward and catches everyone once: secant goes with cosine, and cosecant goes with sine. The “co” is on the opposite side from where you would guess.
The trick that fixes it: look at the third letter. Sec pairs with cosine. Cosec pairs with sine — third letter of “cosec” is s. Cotangent is the only sensible one, pairing with tangent.
Cosecant is written \(\csc\) in American texts and \(\operatorname{cosec}\) in British ones. Same function.
In terms of the triangle’s sides, they are the original ratios upside down:
\[\sec\theta = \frac{\text{hyp}}{\text{adj}} \qquad \csc\theta = \frac{\text{hyp}}{\text{opp}} \qquad \cot\theta = \frac{\text{adj}}{\text{opp}}\]Why bother having them
A fair question, since \(\frac{1}{\cos\theta}\) says the same thing in fewer symbols.
They keep expressions clean. Fractions inside fractions are hard to read and easy to mishandle. \(\sec^2\theta\) beats \(\frac{1}{\cos^2\theta}\) every time it appears in a chain of algebra.
They make the identities symmetric. The Pythagorean family below is memorable in these names and a mess without them.
Calculus insists on them. \(\frac{d}{dx}\tan x = \sec^2 x\) and \(\int \sec^2 x \, dx = \tan x + C\). Trigonometric substitution in integration is built on \(\sec\), and there is no way round it.
Historically they were also easier: before calculators, dividing by a table value was slow, so tables of secants were printed to turn a division into a lookup.
The other two Pythagorean identities
Start from the one you know:
\[\sin^2\theta + \cos^2\theta = 1\]Divide every term by \(\cos^2\theta\):
\[\tan^2\theta + 1 = \sec^2\theta\]Divide the original by \(\sin^2\theta\) instead:
\[1 + \cot^2\theta = \csc^2\theta\]Three identities, one source. Worth deriving once rather than memorising, because the derivation takes ten seconds and the memory of which goes with which does not survive a stressful exam.
Notice the pattern: whichever function is squared alone on the left, its co-partner’s reciprocal is squared on the right.
Their graphs are made of branches
Since \(\sec\theta = \frac{1}{\cos\theta}\), the secant blows up wherever cosine is zero — at \(90°, 270°\) and every \(180°\) after. So its graph is a series of U-shaped branches separated by vertical asymptotes, each branch sitting in the gap between two zeros of cosine.
It also never takes a value between \(-1\) and \(1\). Cosine never exceeds 1 in size, so its reciprocal is never smaller than 1 in size. The range of \(\sec\) is \(|\sec\theta| \ge 1\), and the same holds for \(\csc\).
\(\cot\) behaves differently, running smoothly downwards between asymptotes at the zeros of sine, and taking every real value on each branch.
Common mix-ups
- Pairing sec with sine. Check the third letter.
- Reading \(\sec^{-1}\) as \(\frac{1}{\sec}\). The \(-1\) means the inverse function, arcsec. The reciprocal of \(\sec\) is \(\cos\).
- Writing \(\cot\theta = \frac{\sin\theta}{\cos\theta}\). It is \(\frac{\cos\theta}{\sin\theta}\) — the reciprocal of tangent, so the fraction flips.
- Expecting \(\sec\) to take small values. It never lands strictly between \(-1\) and 1.
- Forgetting the asymptotes when sketching, or when checking whether a solution to an equation is valid.
Check yourself
Which ratio is \(\sec\theta\) the reciprocal of? If \(\cos\theta = 0.5\), what is \(\sec\theta\)? And what does \(1 + \tan^2\theta\) equal?
\(\sec\) pairs with cosine, so \(\sec\theta = \frac{1}{0.5} = 2\). And dividing \(\sin^2 + \cos^2 = 1\) through by \(\cos^2\) gives \(\tan^2\theta + 1 = \sec^2\theta\).
LEVEL 3 · REFLECT
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