Graphs of cosec x, sec x, cot x — Hinglish notes

3.1.7 · Maths › Advanced Trigonometry

WHAT are these functions?

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

Core reasoning tool: ek number ke reciprocal ke baare mein socho jab woh change hota hai.

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

$sin x$ ko lightly guide ke taur par draw karo.
$sin x$ ka har zero ( $x = nπ$ ) $csc x$ ka vertical asymptote ban jaata hai. Kyun? $1/0$ blow up karta hai.
$sin x$ ka har maximum ( $sin x = 1$ ) $csc x$ ka minimum ban jaata hai $y = 1$ par. Kyun? $1/1 = 1$ , aur iske aas-paas reciprocal thoda bada hota hai, isliye yeh upper branch ka sabse neecha point hai.
$sin x$ ka har minimum ( $sin x = - 1$ ) $csc x$ ka maximum ban jaata hai $y = - 1$ par.
Har do asymptotes ke beech ek U-shaped branch hoti hai (upar $y = 1$ ke oopar khulti hai, neeche $y = - 1$ ke neeche khulti hai).

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

Same recipe $cos x$ ke saath. Kyunki $cos x = sin (x + \frac{π}{2})$ hai, $sec$ graph sirf $csc$ graph ka left shift hai $\frac{π}{2}$ se. Asymptotes $x = \frac{π}{2} + nπ$ par; touch hota hai $(0, 1)$ , $(π, - 1)$ , etc. par.

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

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

Asymptotes jahan $sin x = 0$ ho: $x = nπ$ (kyun? denominator zero).
Zeros jahan $cos x = 0$ ho: $x = \frac{π}{2} + nπ$ (kyun? numerator zero).
Yeh har interval $(nπ, (n + 1) π)$ par decreasing hai, $+ \infty$ se $- \infty$ tak jaata hai. (Yeh $tan x$ ke opposite hai, jo increase karta hai.)
Period $= π$ .

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

Key features at a glance

Worked examples

Common mistakes (Steel-man + fix)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek number aur uske "ulte twin" ( $1$ us par) ke baare mein socho. Agar number tiny hai, toh uska twin bahut bada hai. Agar number exactly $1$ hai, toh twin bhi $1$ hai — dono high-five karte hain. Ab $sin$ , $cos$ , $tan$ wavy numbers hain. Inke ulte twins ( $csc, sec, cot$ ) jahan bhi original zero ho wahan crazy-tall ho jaate hain (wahan ek "deewar" draw karo — asymptote), aur jahan original apne top aur bottom ko touch kare wahan $1$ aur $- 1$ par gently touch karte hain. Isliye tum koi fresh graph nahi banate — jo graph tumhe pata hai, usse hi flip kar do!

Flashcards

csc x

ke vertical asymptotes kahan hain?

x = nπ

par (jahan

sin x = 0

ho).

sec x

ke vertical asymptotes kahan hain?

x = \frac{π}{2} + nπ

par (jahan

cos x = 0

ho).

cot x

ke vertical asymptotes kahan hain?

x = nπ

par (jahan

sin x = 0

ho).

csc x

aur

sec x

ka range kya hai?

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

—

(- 1, 1)

mein koi value nahi.

cot x

ka range kya hai?

Saare real numbers,

(- \infty, \infty)

csc x

sec x

cot x

ka period?

Krama se

2 π

2 π

, aur

π

cot x

har interval par increasing hai ya decreasing?

Decreasing (from

+ \infty

- \infty

csc x

ke zeros kyun nahi hote?

Kyunki

1/ sin x

kabhi

0

nahi ho sakta; numerator

1

wala fraction kabhi zero nahi hota.

x = \frac{π}{2}

par

csc x

kya hai?

1

(upper branch ka minimum, kyunki

sin = 1

cot x = 0

kahan hota hai?

x = \frac{π}{2} + nπ

par (jahan

cos x = 0

ho).

Even ya odd:

sec x

Even, kyunki

sec (- x) = sec x

sec x

graph

csc x

se kaise related hai?

Yeh

csc x

\frac{π}{2}

left shift hai.

Connections

Graphs of sin x, cos x, tan x — parent functions jinke ye reciprocals hain.
Vertical Asymptotes and Limits — kyun $1/0$ ek deewar banata hai.
Trigonometric Identities — jaise $1 + cot^{2} x = csc^{2} x$ , $1 + tan^{2} x = sec^{2} x$ .
Solving Trigonometric Equations — in graphs se solutions padhna.
Domain and Range — forbidden band $(- 1, 1)$ .
Odd and Even Functions — har graph ki symmetry.

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