MHF4U Reciprocal Trigonometric Functions: Notes and Practice

There is nothing new to learn here in the sense of new values. Cosecant, secant, and cotangent are simply the reciprocals of sine, cosine, and tangent, and every single one of their properties — asymptotes, turning points, domain, range, period — can be read off the base function. Learn to look at $sin x$ and predict $csc x$ and this topic takes care of itself.

The three definitions

csc θ = \frac{1}{sin θ} sec θ = \frac{1}{cos θ} cot θ = \frac{1}{tan θ} = \frac{cos θ}{sin θ}

The pairings are deliberately counterintuitive: coSecant goes with Sine and Secant goes with Cosine. Match the THIRD letter of the reciprocal name to the function it inverts — co-Se-cant and Si-ne, Se-cant and Co-si-ne.
Cotangent has two equally useful forms. $\frac{1}{tan θ}$ is quickest for evaluating; $\frac{cos θ}{sin θ}$ is the one that correctly explains where the graph is zero and where it is undefined.
A reciprocal always carries the same sign as what it inverts, so CAST applies unchanged: $csc$ is positive exactly where $sin$ is, $sec$ exactly where $cos$ is, and $cot$ exactly where $tan$ is.

What the graphs must look like

Three rules generate the whole picture. Wherever the base function is ZERO, the reciprocal divides by zero, so there is a VERTICAL ASYMPTOTE. Wherever the base function reaches a peak of $1$ or a trough of $- 1$ , the reciprocal is also $1$ or $- 1$ — the two graphs touch at those points, and because the reciprocal cannot get any closer to the axis there, that touching point is a local MINIMUM of an upper branch or a local MAXIMUM of a lower branch. And wherever the base function is small (near zero), the reciprocal is huge, which is why every branch flares away from the axis.

Fig 1Each reciprocal curve drawn over its base function, showing where the asymptotes and the touching points land.

Function	Period	Asymptotes	Domain	Range
$y = csc x$	$2 π$	$x = nπ$ (where $sin x = 0$ )	$x \neq = nπ$	$y \leq - 1$ or $y \geq 1$
$y = sec x$	$2 π$	$x = \frac{π}{2} + nπ$ (where $cos x = 0$ )	$x \neq = \frac{π}{2} + nπ$	$y \leq - 1$ or $y \geq 1$
$y = cot x$	$π$	$x = nπ$ (where $sin x = 0$ )	$x \neq = nπ$	all real numbers

Here n is any integer. Notice cotangent behaves quite differently from the other two.

Why cosecant and secant can never land between minus one and one

Sine and cosine are trapped in the interval from minus one to one, so their size is never bigger than one. Taking a reciprocal of a number no bigger than one gives a number no smaller than one. That is the whole proof that the range of cosecant and secant is everything outside the open interval from minus one to one, with nothing in the gap.

Cotangent is the odd one out, and the $\frac{cos θ}{sin θ}$ form explains why. Its asymptotes sit where $sin x = 0$ (at $x = nπ$ ), but its ZEROS sit where $cos x = 0$ (at $x = \frac{π}{2} + nπ$ ) — exactly the reverse of tangent, whose asymptotes and zeros are swapped. It also inherits tangent's period of $π$ and tangent's unrestricted range, so cotangent has no amplitude and no gap in its outputs.

VideoA short Ontario Grade 12 lesson that explores the cosecant, secant and cotangent graphs alongside the sine, cosine and tangent graphs they come from.Watch on YouTube

Evaluating exactly

To evaluate a reciprocal function exactly, evaluate the base function first with the usual quadrant-and-related-angle routine, then flip the answer and rationalize the denominator. For example, $sec \frac{2 π}{3} = \frac{1}{cos ( 2 π /3 )} = \frac{1}{- 1/2} = - 2 .$ When the flipped result contains a radical, as in $\frac{1}{3 /2} = \frac{2}{3}$ , multiply top and bottom by $3$ to finish with $\frac{2 3}{3}$ .

Fig 2Reciprocal values come straight from the coordinates: cosecant is one over the y-coordinate and secant is one over the x-coordinate.

$θ$	$csc θ$	$sec θ$	$cot θ$
$0$	undefined	$1$	undefined
$\frac{π}{6}$	$2$	$\frac{2 3}{3}$	$3$
$\frac{π}{4}$	$2$	$2$	$1$
$\frac{π}{3}$	$\frac{2 3}{3}$	$2$	$\frac{3}{3}$
$\frac{π}{2}$	$1$	undefined	$0$

Each column is the reciprocal of the matching column in the sine, cosine, and tangent table, with denominators rationalized.

Cosecant is not inverse sine

The notation looks dangerously similar but the meanings are unrelated. Cosecant of theta is one divided by the sine of theta, a number. Inverse sine of a value is the ANGLE whose sine is that value. Writing sine to the power negative one means the inverse function, never the reciprocal.

Reciprocal Trigonometric Functions

Key things to remember

The three definitions

What the graphs must look like

Evaluating exactly

Worked examples

Where marks get lost

Flashcards 10

Practice