Graphs of cosec x, sec x, cot x: Foundations — Maths deep dive

This page assumes nothing. Before we ever flip a graph, we must own every single symbol the parent note throws at you. We list them in build-order: each one only uses ideas defined above it.

1. The number line, $\infty$ , and the absolute value $∣ \cdot ∣$

Picture a number line with $0$ in the middle. Coming in from the right ( $0^{+}$ ) you tread on shrinking positive numbers; from the left ( $0^{-}$ ), shrinking negative numbers. This little distinction (which side of zero) is what decides whether a reciprocal graph shoots up or down, so we cannot skip it.

Figure — Graphs of cosec x, sec x, cot x

The word "limit" here just names the target of that sliding motion — see Vertical Asymptotes and Limits for the full treatment.

2. The fraction bar and "reciprocal"

Why does the whole topic live or die on this one idea? Because $csc$ , $sec$ , $cot$ are defined as reciprocals. So we must know exactly how $\frac{1}{u}$ behaves as $u$ moves:

3. What $sin x$ , $cos x$ , $tan x$ actually are (the raw material)

You cannot flip a graph you do not own. Here is the picture-level meaning of the three parent functions from Graphs of sin x, cos x, tan x. (The angle $x$ is measured in radians — a unit we define fully in Section 4; for now just read $x$ as "how far the point has spun.")

Because the point lives on a circle of radius $1$ , its coordinates can never escape $[- 1, 1]$ :

Hold this thought. When we take reciprocals in the parent note, " $sin x$ never exceeds $1$ " becomes " $csc x$ never dips below $1$ ." That is the entire reason for the forbidden band $(- 1, 1)$ .

The quadrant sign chart for $sin x$ (and therefore $csc x$ )

Since $csc x = \frac{1}{sin x}$ , the sign of $csc x$ always matches the sign of $sin x$ (dividing $1$ by a positive gives positive; by a negative gives negative). So if you know where $sin x$ is $+$ or $-$ , you know where each $csc x$ branch sits above or below the axis. The circle splits into four quadrants — quarter-turns numbered anticlockwise:

Formula Sign of

$sin x$ by quadrant (recall $sin x$ = height of the point)

Quadrant	Range of $x$	Point's height $sin x$	So $csc x$ is
I	$0 < x < \frac{π}{2}$	above axis, $+$	positive ( $\geq 1$ )
II	$\frac{π}{2} < x < π$	still above axis, $+$	positive ( $\geq 1$ )
III	$π < x < \frac{3 π}{2}$	below axis, $-$	negative ( $\leq - 1$ )
IV	$\frac{3 π}{2} < x < 2 π$	below axis, $-$	negative ( $\leq - 1$ )

Read it as one sentence: $sin x$ (and $csc x$ ) is positive on the upper half-turn $(0, π)$ and negative on the lower half-turn $(π, 2 π)$ . So between the walls at $0$ and $π$ the $csc$ branch is a U above $y = 1$ ; between $π$ and $2 π$ it is a U below $y = - 1$ .

This is the systematic tool: never guess whether a $csc$ branch is up or down — read the quadrant.

4. Radians, $π$ , and $nπ$

5. Vertical asymptotes — the "walls"

Combine the pieces you now own: $csc x = \frac{1}{sin x}$ . Wherever $sin x = 0$ (that is, $x = nπ$ ), the denominator is zero, so — by the reciprocal rule of Section 2 — the graph explodes. That vertical line is the asymptote. Full machinery: Vertical Asymptotes and Limits.

u \to 0^{+}

mean, and what does

\frac{1}{u}

do there?

u

shrinks toward

0

through positive numbers;

\frac{1}{u} \to + \infty

What does

∣ u ∣

mean?

The distance of

u

from

0

; always non-negative, e.g.

∣ - 3∣ = 3

Why does large

∣ u ∣

make

\frac{1}{u}

tiny (but never

0

One thing shared among many parts gives a small share each; a sliver always remains, so it never reaches

0

Why can

csc x

never equal

0

Because

\frac{1}{sin x}

has numerator

1

, and a fraction is zero only when its top is zero.

On the unit circle, which coordinate is

sin x

The vertical (height) coordinate of the spinning point.

Where is

tan x

undefined and why?

x = \frac{π}{2} + nπ

, because its denominator

cos x

is zero there.

On which half-turn is

csc x

positive, and why?

(0, π)

, because

sin x

is positive there and

csc x

shares its sign.

What bound does the unit circle force on

sin x

and

cos x

- 1 \leq sin x \leq 1

and

- 1 \leq cos x \leq 1

What list of values does

x = nπ

represent?

All integer multiples of

π

\dots, - π, 0, π, 2 π, \dots

State the precise periodicity property and why

csc

keeps

sin

's period.

f (x + P) = f (x)

for all

x

; taking

\frac{1}{\cdot}

sin (x + 2 π) = sin x

gives

csc (x + 2 π) = csc x

Which values must be excluded from

cot x

's domain?

x = nπ

, because its denominator

sin x

is zero there.

What does the notation

(- \infty, - 1] \cup [1, \infty)

describe?

The range of

csc x

sec x

— all

y

at most

- 1

or at least

1

, excluding the band

(- 1, 1)

Even vs odd: give the defining equation of each.

Even:

f (- x) = f (x)

. Odd:

f (- x) = - f (x)

Why is

cot x

's period

π

not

2 π

Because it inherits the period of

tan x

, which repeats every

π

Connections

Graphs of cosec x, sec x, cot x — the parent topic these foundations feed.
Graphs of sin x, cos x, tan x — the raw graphs we reciprocate.
Vertical Asymptotes and Limits — the wall-making machinery.
Domain and Range — allowed inputs, resulting outputs.
Odd and Even Functions — the symmetry vocabulary.
Trigonometric Identities — reciprocal and Pythagorean relations.
Solving Trigonometric Equations — where these tools are applied.

Foundations — Graphs of cosec x, sec x, cot x

1. The number line, $\infty$ , and the absolute value $∣ \cdot ∣$

2. The fraction bar and "reciprocal"

3. What $sin x$ , $cos x$ , $tan x$ actually are (the raw material)

The quadrant sign chart for $sin x$ (and therefore $csc x$ )

4. Radians, $π$ , and $nπ$

5. Vertical asymptotes — the "walls"

6. Period, domain, range, odd/even

7. The identities you will meet

Prerequisite map

Equipment checklist

Connections