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

1. Building sin and cos from the circle

WHY the range is $[-1,1]$ : the point lives on a circle of radius $1$ , so neither coordinate can ever exceed $1$ in size. Hence $-1 \le \sin x \le 1, \qquad -1 \le \cos x \le 1.$

WHY they repeat: going once round the circle is $360^\circ = 2\pi$ radians. After that you are at the same point, so the values repeat. This "repeat length" is the period.

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

2. Reading off key points

Track the moving point at the four "corners" of the circle:

Angle $x$	Point on circle	$\cos x$	$\sin x$
$0$	right $(1,0)$	$1$	$0$
$\pi/2$	top $(0,1)$	$0$	$1$
$\pi$	left $(-1,0)$	$-1$	$0$
$3\pi/2$	bottom $(0,-1)$	$0$	$-1$
$2\pi$	back to right	$1$	$0$

Key features to say out loud:

$\sin x$ : passes through origin, odd (rotational symmetry about $0$ : $\sin(-x)=-\sin x$ ), max at $\pi/2$ , min at $3\pi/2$ , zeros at $0,\pi,2\pi,\dots$
$\cos x$ : starts at max, even (mirror symmetry in $y$ -axis: $\cos(-x)=\cos x$ ), zeros at $\pi/2, 3\pi/2,\dots$

3. Building tan from sin and cos

WHY the vertical asymptotes: wherever $\cos x=0$ (at $x=\pm\tfrac{\pi}{2},\ \pm\tfrac{3\pi}{2},\dots$ ) we divide by zero, so $\tan x$ shoots to $\pm\infty$ . These are vertical asymptotes at $x=\tfrac{\pi}{2}+n\pi$ .

WHY the period is $\pi$ , not $2\pi$ (derivation): Move half a turn round the circle ( $x\to x+\pi$ ). The point flips to the exact opposite side, so both coordinates negate: $\sin(x+\pi)=-\sin x,\qquad \cos(x+\pi)=-\cos x.$ Therefore $\tan(x+\pi)=\frac{-\sin x}{-\cos x}=\frac{\sin x}{\cos x}=\tan x.$ The two minus signs cancel, so $\tan$ repeats twice as fast: period $=\pi$ .

4. Transformations (WHAT each number does)

For $y=A\sin(Bx+C)+D$ :

5. Worked examples

6. Common mistakes (Steel-manned)

7. Active recall

Recall Test yourself (hide the answers)

What are the coordinates of a unit-circle point at angle $x$ ? → $(\cos x,\sin x)$
Why is the range of $\sin$ exactly $[-1,1]$ ? → radius is 1, coordinate can't exceed it
Why is tan's period $\pi$ ? → both sin, cos negate at $+\pi$ , signs cancel in ratio
Period of $y=5\cos(\tfrac{x}{2})$ ? → $2\pi/(1/2)=4\pi$ ; amplitude $5$

Recall Feynman: explain to a 12-year-old

Imagine a kid on a Ferris wheel going round and round. How high they are above the middle, drawn against time, makes a smooth up-and-down wave — that's sine. How far left/right they are makes the same wave but starting from the top — that's cosine. Both waves repeat every full circle. Tan is a sneaky one: it's "height divided by sideways", and every time the kid passes straight up or straight down, the "sideways" becomes zero, so dividing by it makes the number blow up to a cliff. That's why tan has those steep walls (asymptotes) and repeats twice as often.

Connections

Unit Circle and Radian Measure — the source of all three graphs
Exact Values of Trig Ratios — supplies the key points $0,\tfrac{\pi}{2},\pi\dots$
Trigonometric Identities — $\tan=\sin/\cos$ , $\sin^2+\cos^2=1$
Transformations of Graphs — how $A,B,C,D$ move any curve
Solving Trigonometric Equations — uses periodicity to list all solutions
Simple Harmonic Motion — physics application of $\sin/\cos$ waves

Coordinates of a unit-circle point at angle x

(\cos x, \sin x)

Period of sin x and cos x

2\pi

Period of tan x

\pi

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

Zeros of sin x

x=n\pi

Zeros of cos x

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

cos x in terms of sin

\cos x = \sin(x+\tfrac{\pi}{2})

Period of y=A sin(Bx)

2\pi/|B|

Period of y=tan(Bx)

\pi/|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

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Socho ek point unit circle (radius 1) par anticlockwise ghoom raha hai, aur $x$ uska angle hai. Us point ki height yani y-coordinate hoti hai $\sin x$ , aur uski horizontal distance yani x-coordinate hoti hai $\cos x$ . Bas isi ek picture se saari cheezein nikal aati hain — koi ratta nahi maarna. Radius 1 hai isliye height ya width kabhi 1 se zyada nahi ho sakti, tabhi range $[-1,1]$ aata hai. Ek poora chakkar $2\pi$ ka hota hai, isliye graph har $2\pi$ baad repeat karta hai — yahi hai period, aur half-swing $=1$ hai isliye amplitude $=1$ .

$\tan x = \sin x / \cos x$ — yani height bata width. Jahan bhi $\cos x = 0$ hota hai (top ya bottom point, $x=\pi/2$ type), wahan width zero ho jaati hai, aur zero se divide karne par value $\pm\infty$ tak udd jaati hai — yeh hote hain vertical asymptotes. Aur ek mast baat: half-turn ( $x+\pi$ ) par sin aur cos dono ka sign ulta ho jaata hai, lekin ratio mein dono minus cancel ho jaate hain, isliye $\tan$ ka period sirf $\pi$ hota hai, na ki $2\pi$ . Yeh point exam mein bahut students galat karte hain.

Transformation yaad rakho $y=A\sin(Bx+C)+D$ mein: $A$ height stretch karta hai (amplitude $=|A|$ ), $B$ speed badhata hai isliye period chhota ho jaata hai $=2\pi/|B|$ (multiply nahi, divide!), $C$ left-right shift, $D$ upar-neeche shift. Yeh chhoti si samajh 80% questions cover kar deti hai. Ferris wheel wali picture dimaag mein rakho — sab kuch apne aap click ho jayega.

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

1. Building sin and cos from the circle

2. Reading off key points

3. Building tan from sin and cos

4. Transformations (WHAT each number does)

5. Worked examples

6. Common mistakes (Steel-manned)

7. Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Advanced Trigonometry

Connections

Function	Period	Amplitude	Range	Zeros
$\sin x$	$2\pi$	$1$	$[-1,1]$	$n\pi$
$\cos x$	$2\pi$	$1$	$[-1,1]$	$\tfrac{\pi}{2}+n\pi$
$\tan x$	$\pi$	— (none)	$(-\infty,\infty)$	$n\pi$