Graphs of sin x, cos x, tan x — key features, period, amplitude

3.1.6 · Maths › Advanced Trigonometry

1. Circle se sin aur cos banana

WHY range $[- 1, 1]$ hai: point radius $1$ ke circle pe rehta hai, isliye koi bhi coordinate $1$ se bada nahi ho sakta. Isliye $- 1 \leq sin x \leq 1, - 1 \leq cos x \leq 1.$

WHY ye repeat karte hain: circle ke ek chakkar ka matlab hai $36 0^{\circ} = 2 π$ radians. Uske baad tum same point pe hote ho, isliye values repeat hoti hain. Is "repeat length" ko period kehte hain.

Figure — Graphs of sin x, cos x, tan x — key features, period, amplitude

2. Key points padhna

Circle ke chaar "corners" pe moving point ko track karo:

Angle $x$	Circle pe point	$cos x$	$sin x$
$0$	right $(1, 0)$	$1$	$0$
$π /2$	top $(0, 1)$	$0$	$1$
$π$	left $(- 1, 0)$	$- 1$	$0$
$3 π /2$	bottom $(0, - 1)$	$0$	$- 1$
$2 π$	wapas right	$1$	$0$

Key features jo loud bolne chahiye:

$sin x$ : origin se guzarta hai, odd hai (rotational symmetry about $0$ : $sin (- x) = - sin x$ ), $π /2$ pe max, $3 π /2$ pe min, zeros $0, π, 2 π, \dots$ pe
$cos x$ : max se start hota hai, even hai (mirror symmetry in $y$ -axis: $cos (- x) = cos x$ ), zeros $π /2, 3 π /2, \dots$ pe

3. Sin aur cos se tan banana

WHY vertical asymptotes hote hain: jahan bhi $cos x = 0$ ho (yaani $x = \pm \frac{π}{2}, \pm \frac{3 π}{2}, \dots$ pe) hum zero se divide kar rahe hain, isliye $tan x$ $\pm \infty$ tak jaata hai. Ye hain vertical asymptotes $x = \frac{π}{2} + nπ$ pe.

WHY period $π$ hai, $2 π$ nahi (derivation): Circle ke aadhe chakkar ke baad ( $x \to x + π$ ), point bilkul opposite side pe flip ho jaata hai, isliye dono coordinates negate ho jaate hain: $sin (x + π) = - sin x, cos (x + π) = - cos x .$ Isliye $tan (x + π) = \frac{- s i n x}{- c o s x} = \frac{s i n x}{c o s x} = tan x .$ Dono minus signs cancel ho jaate hain, isliye $tan$ do baar teezi se repeat karta hai: period $= π$ .

4. Transformations (har number KYA karta hai)

$y = A sin (B x + C) + D$ ke liye:

5. Worked examples

6. Common mistakes (Steel-manned)

7. Active recall

Recall Khud test karo (answers chhupao)

Angle $x$ pe unit-circle point ke coordinates kya hain? → $(cos x, sin x)$
$sin$ ki range exactly $[- 1, 1]$ kyun hai? → radius 1 hai, coordinate isse zyada nahi ho sakta
Tan ka period $π$ kyun hai? → $+ π$ pe sin, cos dono negate hote hain, ratio mein signs cancel ho jaate hain
$y = 5 cos (\frac{x}{2})$ ka period? → $2 π / (1/2) = 4 π$ ; amplitude $5$

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek bachcha Ferris wheel pe chadhke ghoom raha hai. Wo middle ke upar kitna ooncha hai, wo time ke against draw karo, toh ek smooth upar-neeche ki wave banti hai — wo hai sine. Wo left/right mein kitna door hai wo bhi same wave hai lekin top se start hoti hai — wo hai cosine. Dono waves har poore circle ke baad repeat karti hain. Tan ek chalak wali hai: ye "height divided by sideways" hai, aur har baar jab bachcha seedha upar ya seedha neeche se guzarta hai, "sideways" zero ho jaata hai, toh ussse divide karna number ko ek cliff tak blast kar deta hai. Isliye tan ke wo steep walls (asymptotes) hain aur ye do baar utni tezi se repeat hoti hai.

Connections

Unit Circle and Radian Measure — teeno graphs ka source
Exact Values of Trig Ratios — key points $0, \frac{π}{2}, π \dots$ deta hai
Trigonometric Identities — $tan = sin / cos$ , $sin^{2} + cos^{2} = 1$
Transformations of Graphs — $A, B, C, D$ kisi bhi curve ko kaise move karte hain
Solving Trigonometric Equations — saare solutions list karne ke liye periodicity use karta hai
Simple Harmonic Motion — $sin / cos$ waves ka physics application

Coordinates of a unit-circle point at angle x

(cos x, sin x)

Period of sin x and cos x

2 π

Period of tan x

π

Amplitude of sin x and cos x

1

Amplitude of tan x

undefined (unbounded)

Range of sin x and cos x

[- 1, 1]

Range of tan x

all real numbers

Why does tan x have period π

sin and cos both negate at x+π, so the two minus signs cancel in sin/cos

Where are tan x's asymptotes

where cos x = 0, i.e.

x = \frac{π}{2} + nπ

Zeros of sin x

x = nπ

Zeros of cos x

x = \frac{π}{2} + nπ

cos x in terms of sin

cos x = sin (x + \frac{π}{2})

Period of y=A sin(Bx)

2 π /∣ B ∣

Period of y=tan(Bx)

π /∣ B ∣

Amplitude of y=A sin(Bx)+D

∣ A ∣

Is sin x odd or even

odd,

sin (- x) = - sin x

Is cos x odd or even

even,

cos (- x) = cos x

Function	Period	Amplitude	Range	Zeros
$sin x$	$2 π$	$1$	$[- 1, 1]$	$nπ$
$cos x$	$2 π$	$1$	$[- 1, 1]$	$\frac{π}{2} + nπ$
$tan x$	$π$	— (nahi hai)	$(- \infty, \infty)$	$nπ$