Graphs of cosec x, sec x, cot x

WHAT are these functions?

WHY the graphs look the way they do (derive, don't memorise)

Core reasoning tool: think about the reciprocal of a number as it changes.

Formula The reciprocal rule that generates everything

For a value $y = \dfrac{1}{u}$ :

When $u \to 0^{+}$ , $\;y \to +\infty$ .
When $u \to 0^{-}$ , $\;y \to -\infty$ .
When $u = \pm 1$ , $\;y = \pm 1$ (unchanged — the graphs touch).
When $|u|$ is large, $|y|$ is small — but $\sin,\cos$ never exceed $1$ , so $\csc,\sec$ never fall between $-1$ and $1$ .

Building $\csc x$ (from $\sin x$ ) — HOW step by step

Draw $\sin x$ lightly as a guide.
Every zero of $\sin x$ ( $x=n\pi$ ) becomes a vertical asymptote of $\csc x$ . Why? $1/0$ blows up.
Every maximum of $\sin x$ ( $\sin x = 1$ ) becomes a minimum of $\csc x$ at $y=1$ . Why? $1/1 = 1$ , and just around it the reciprocal is slightly bigger, so it's a lowest point of the upper branch.
Every minimum of $\sin x$ ( $\sin x = -1$ ) becomes a maximum of $\csc x$ at $y=-1$ .
Between each pair of asymptotes sits one U-shaped branch (opening up above $y=1$ , opening down below $y=-1$ ).

Building $\sec x$ (from $\cos x$ )

Same recipe with $\cos x$ . Because $\cos x = \sin(x+\tfrac{\pi}{2})$ , the $\sec$ graph is just the $\csc$ graph shifted left by $\tfrac{\pi}{2}$ . Asymptotes at $x = \tfrac{\pi}{2}+n\pi$ ; touches at $(0,1)$ , $(\pi,-1)$ , etc.

Building $\cot x$ (from $\tan x$ )

$\cot x = \dfrac{\cos x}{\sin x}$ .

Asymptotes where $\sin x = 0$ : $x = n\pi$ (why? denominator zero).
Zeros where $\cos x = 0$ : $x = \tfrac{\pi}{2}+n\pi$ (why? numerator zero).
It is decreasing on every interval $(n\pi,(n+1)\pi)$ , running from $+\infty$ down to $-\infty$ . (This is opposite to $\tan x$ , which increases.)
Period $=\pi$ .

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

Key features at a glance

Formula Summary table

Function	Period	Asymptotes	Range	Odd/Even
$\csc x$	$2\pi$	$x=n\pi$	$(-\infty,-1]\cup[1,\infty)$	odd
$\sec x$	$2\pi$	$x=\tfrac{\pi}{2}+n\pi$	$(-\infty,-1]\cup[1,\infty)$	even
$\cot x$	$\pi$	$x=n\pi$	$(-\infty,\infty)$	odd

Why odd/even? $\csc(-x)=1/\sin(-x)=-1/\sin x=-\csc x$ (odd). $\sec(-x)=1/\cos(-x)=1/\cos x=\sec x$ (even). $\cot(-x)=-\cot x$ (odd).

Worked examples

Common mistakes (Steel-man + fix)

Recall Feynman: explain to a 12-year-old

Imagine a number and its "upside-down twin" ( $1$ over it). If the number is tiny, its twin is huge. If the number is exactly $1$ , the twin is also $1$ — they high-five. Now $\sin$ , $\cos$ , $\tan$ are wavy numbers. Their upside-down twins ( $\csc,\sec,\cot$ ) go crazy-tall wherever the original hits zero (you draw a "wall" there — an asymptote), and they gently touch at $1$ and $-1$ where the original touches its own top and bottom. So you never draw a fresh graph — you flip the one you know!

Flashcards

Where are the vertical asymptotes of

\csc x

x=n\pi

(where

\sin x=0

Where are the vertical asymptotes of

\sec x

x=\tfrac{\pi}{2}+n\pi

(where

\cos x=0

Where are the vertical asymptotes of

\cot x

x=n\pi

(where

\sin x=0

What is the range of

\csc x

and

\sec x

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

— no values in

(-1,1)

What is the range of

\cot x

All real numbers,

(-\infty,\infty)

Period of

\csc x

\sec x

\cot x

2\pi

2\pi

, and

\pi

respectively.

\cot x

increasing or decreasing on each interval?

Decreasing (from

+\infty

-\infty

Why has

\csc x

no zeros?

Because

1/\sin x

can never equal

0

; a fraction with numerator

1

is never zero.

x=\tfrac{\pi}{2}

, what is

\csc x

1

(a minimum of the upper branch, since

\sin=1

Where does

\cot x=0

x=\tfrac{\pi}{2}+n\pi

(where

\cos x=0

Even or odd:

\sec x

Even, because

\sec(-x)=\sec x

How is the

\sec x

graph related to

\csc x

It is

\csc x

shifted left by

\tfrac{\pi}{2}

Connections

Graphs of sin x, cos x, tan x — the parents these are reciprocals of.
Vertical Asymptotes and Limits — why $1/0$ produces a wall.
Trigonometric Identities — e.g. $1+\cot^2x=\csc^2x$ , $1+\tan^2x=\sec^2x$ .
Solving Trigonometric Equations — reading solutions off these graphs.
Domain and Range — the forbidden band $(-1,1)$ .
Odd and Even Functions — symmetry of each graph.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, yeh teen functions — $\csc x$ , $\sec x$ , $\cot x$ — koi naye alien graph nahi hai. Ye bas $\sin$ , $\cos$ , $\tan$ ke reciprocal (ulta, $1$ divided by) hai. Toh inhe alag se ratne ki zaroorat hi nahi. Rule simple hai: jahan neeche wala part (denominator) zero hota hai, wahan graph phat jaata hai — infinity ki taraf bhaagta hai, aur wahan hum ek vertical asymptote (khadi deewar) khinch dete hai.

Jahan original function ka value $1$ ya $-1$ hota hai, wahan reciprocal bhi $1$ ya $-1$ hi rehta hai — matlab dono curve wahan "touch" karte hai. Aur ek important baat: kyunki $\sin$ aur $\cos$ kabhi $1$ se zyada nahi hote, unke reciprocal $\csc$ aur $\sec$ kabhi bhi $-1$ aur $1$ ke beech nahi aa sakte. Yeh "forbidden band $(-1,1)$ " bahut common exam trap hai.

$\cot x$ thoda special hai: iski asymptotes wahan hai jahan $\sin=0$ ( $x=n\pi$ ), aur zeros wahan jahan $\cos=0$ . Aur yaad rakho — $\tan$ badhta hai par $\cot$ har interval me ghatata hai. Mnemonic: "co-sec aur co-tangent, dono sine ke saath mrte hai; se-C cosine ke saath." Bas graph flip karo, ratna band karo — exam me time bachega aur galti kam hogi.

WHAT are these functions?

WHY the graphs look the way they do (derive, don't memorise)

Building $\csc x$ (from $\sin x$ ) — HOW step by step

Building $\sec x$ (from $\cos x$ )

Building $\cot x$ (from $\tan x$ )

Key features at a glance

Worked examples

Common mistakes (Steel-man + fix)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Advanced Trigonometry

Connections