Unit circle definition of trig functions — all 6 trig functions for any angle

3.1.1 · Maths › Advanced Trigonometry

WHAT: setup kya hai

Poora trick yahi hai: woh landing point hi sab kuch define karta hai.

WHY yeh right-triangle trig se match karta hai? $P$ se x-axis tak ek vertical line drop karo. Tumhe ek right triangle milega jiska hypotenuse = radius = $1$ hai, horizontal leg $= x$ hai, vertical leg $= y$ hai. Tab $cos θ = \frac{adjacent}{hyp} = \frac{x}{1} = x, sin θ = \frac{opposite}{hyp} = \frac{y}{1} = y .$ Toh $0° < θ < 90°$ ke liye purana SOH-CAH-TOA answer exactly coordinate ke barabar hai. Circle bas wahan se aage jaata rehta hai jahan triangle ruk jaata hai.

HOW: baaki chaar functions (derive kiye jaate hain, yaad nahi kiye)

Baki sab kuch $sin$ aur $cos$ se bana hai. Ratios se derive karo:

WHY yeh definitions? Right triangle mein, $tan = \frac{opp}{adj} = \frac{y}{x}$ . Secant/cosecant/cotangent bas unke reciprocals hain — "co-" partner hota hai "co-" ke saath, toh $sec$ pair karta hai $cos$ ke saath (dono ratio x-side se start karte hain) — actually reciprocal se yaad karo, mnemonic dekho.

Key consequence — yeh kahan undefined ho jaate hain:

$tan θ, sec θ$ undefined hain jab $x = cos θ = 0$ (yaani $θ = 90°, 270°, \dots$ ).
$cot θ, csc θ$ undefined hain jab $y = sin θ = 0$ (yaani $θ = 0°, 180°, \dots$ ).

Tum yeh dekh sakte ho: division by zero tab hota hai jab koi coordinate axis ko touch karta hai.

Figure — Unit circle definition of trig functions — all 6 trig functions for any angle

Pythagorean identity — free mein mil jaati hai

Quadrant ke hisaab se signs

Jaise-jaise tum rotate karte ho, $x$ aur $y$ ka sign badalta hai. Toh trig functions ka bhi.

Quadrant	angle range	$x = cos$	$y = sin$	positive functions
I	$0°$ – $90°$	$+$	$+$	All
II	$90°$ – $180°$	$-$	$+$	Sine (+csc)
III	$180°$ – $270°$	$-$	$-$	Tan (+cot)
IV	$270°$ – $360°$	$+$	$-$	Cos (+sec)

Negative aur bahut bade angles

Worked examples

Common mistakes

Recall Feynman: 12-saal ke bachche ko explain karo

Socho ek merry-go-round hai jiska radius exactly $1$ meter hai, aur tum "3 o'clock" wali jagah se shuru karte ho. Jab tum kisi angle tak spin karte ho, dekho tum kahan ho: kitna right ho tum (woh $cos$ hai) aur kitna upar ho tum (woh $sin$ hai). Agar tum top ke past jaao, "kitna right" negative ho jaata hai kyunki tum ab left side pe ho. Poora chakkar lagao aur tum wapas wahan ho jahan se shuru kiya tha — isliye numbers repeat hote hain. Baaki chaar functions ( $tan$ , etc.) bas inhi dono ko aapas mein divide karna hai. Jab tum exactly top ya side pe ho aur ek zero ho, usse divide karna "calculator tod deta hai" — isliye kuch undefined hote hain.

Flashcards

Unit circle pe

cos θ

kya hai?

Woh point ka x-coordinate jahan tum

(1, 0)

θ

rotate karne ke baad pahuNchte ho.

Unit circle pe

sin θ

kya hai?

Us point ka y-coordinate.

Coordinates se

tan θ

define karo.

tan θ = y / x = sin θ / cos θ

sin^{2} θ + cos^{2} θ = 1

kyun hai?

Kyunki

(x, y)

circle

x^{2} + y^{2} = 1

pe hai, aur

x = cos θ

y = sin θ

tan θ

undefined kab hota hai aur kyun?

Jab

cos θ = x = 0

(

θ = 90°, 270°, \dots

) — division by zero.

csc θ

undefined kab hota hai?

Jab

sin θ = y = 0

(

θ = 0°, 180°, \dots

Kin quadrants mein

sin θ

positive hota hai?

Q1 aur Q2 (upper half,

y > 0

Kis quadrant mein sirf

tan

aur

cot

positive hote hain?

Q3 (dono

x, y < 0

Cosine even hai ya odd?

Even:

cos (- θ) = cos θ

(clockwise rotate karne par same x milta hai).

Sine even hai ya odd?

Odd:

sin (- θ) = - sin θ

(y-coordinate mirror ho jaata hai).

sin 90°

aur

cos 90°

ki value?

sin 90° = 1

cos 90° = 0

(point

(0, 1)

sin (390°)

kaise handle karein?

360°

subtract karo (periodicity):

sin 390° = sin 30° = 1/2

Quadrant ke hisaab se positive functions ka mnemonic?

"All Students Take Calculus": All, Sine, Tan, Cos.

Coordinates mein

sec θ = ?

1/ x = 1/ cos θ

Connections

Right-triangle trigonometry (SOH-CAH-TOA) — special case $0° < θ < 90°$ .
Reference angles — kisi bhi angle ko acute angle tak kaise reduce karein.
Pythagorean identities — teeno $x^{2} + y^{2} = 1$ se aate hain.
Radian measure — unit circle pe arc length ke roop mein angles.
Graphs of trig functions — sine/cosine graphs y/x coordinates ko $θ$ ke against plot karna hai.
Even and odd functions — cosine even, sine odd.
Periodicity and $2\pi$ — full-loop invariance.