Trig functions for angles beyond 90° — ASTC rule (All, Sin, Tan, Cos)

3.1.4 · Maths › Advanced Trigonometry

HUM ISKO KYUN SEEKHTE HAIN?

Original definitions ("opposite/hypotenuse") sirf right triangle ke acute angles ke liye kaam karti hain. Lekin hum constantly $120°$ , $210°$ , $- 45°$ , $750°$ jaise angles se milte hain. Isliye hum definition ko unit circle ki madad se upgrade karte hain, jahan har angle allowed hai.

Figure — Trig functions for angles beyond 90° — ASTC rule (All, Sin, Tan, Cos)

REFERENCE ANGLE KYA HOTA HAI?

Definition Reference angle

Reference angle $θ^{'}$ wo acute angle hai jo terminal ray aur $x$ -axis ke beech hota hai (kabhi $y$ -axis nahi). Yeh hamesha $0 \leq θ^{'} \leq 90°$ hota hai.

Quadrant	$θ$ range	reference $θ^{'}$
I	$0$ – $90°$	$θ$
II	$90$ – $180°$	$180° - θ$
III	$180$ – $270°$	$θ - 180°$
IV	$270$ – $360°$	$360° - θ$

YEH FORMULAS KYUN? Har quadrant mein terminal ray $x$ -axis ke najdeeki adhe hisse se ek angle banati hai. Jaise QII mein ray $90°$ aur $180°$ ke beech hoti hai, aur negative $x$ -axis ( $180°$ ) se uska gap $180° - θ$ hai.

SIGNS KAISE AATE HAIN — ASTC ko scratch se derive karna

Har ratio ka sign bas us quadrant mein $x$ aur $y$ ka sign hai. $(x, y)$ signs draw karo:

Quadrant	$x = cos$	$y = sin$	$tan = y / x$	Positive wale
I	$+$	$+$	$+$	All
II	$-$	$+$	$-$	Sin
III	$-$	$-$	$+$	Tan
IV	$+$	$-$	$-$	Cos

QI se anticlockwise jaate hue "positive" column padho: A, S, T, C. Bas yahi poora rule hai — yeh derive hota hai, andha yaad nahi kiya jaata.

Worked examples

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek ghadi ki sui 3 baje ki position se spin ho rahi hai, anticlockwise ja rahi hai. Uski tip ki left-right position ( $x$ ) hai aur up-down position ( $y$ ) hai. Cos left-right hai, sin up-down hai. Jab sui upar-left taraf point kare, left-right negative hai lekin up-down positive hai — isliye us region mein sirf sin positive rehta hai. Number ka size wahi hota hai jaise sui friendly bottom-right corner mein ho; sirf plus/minus badalta hai. "All Students Take Coffee" bas yaad dilata hai ki kaunsa positive rehta hai jab hum spin karte hain.

Active recall

Kis quadrant mein SAARI trig ratios positive hoti hain?

Quadrant I (0°–90°).

Quadrant II mein kaun si single ratio positive hoti hai?

Sine (aur uska reciprocal cosec).

Quadrant III mein kaun si positive hoti hai?

Tangent (aur cot).

Quadrant IV mein kaun si positive hoti hai?

Cosine (aur sec).

QII mein reference angle ka formula?

180° - θ

QIII mein reference angle ka formula?

θ - 180°

QIV mein reference angle ka formula?

360° - θ

Reference angle kis axis se measure hota hai?

Hamesha

x

-axis se (kabhi

y

-axis se nahi).

cos θ

aur

sin θ

ki unit-circle definition?

cos θ = x

sin θ = y

of the point

P

on the unit circle.

QIII mein

tan

positive kyun hota hai?

x

aur

y

dono negative hain, isliye

y / x > 0

sin 120°

evaluate karo.

+ sin 60° = \frac{3}{2}

cos 210°

evaluate karo.

- cos 30° = - \frac{3}{2}

tan 315°

evaluate karo.

- tan 45° = - 1

ASTC sign deta hai ya magnitude?

Sirf sign; magnitude reference angle se aati hai.

360° se zyada angles ko kaise handle karo?

360° ke multiples subtract karo (periodicity) taaki

[0°, 360°)

mein aa jao.

Connections

Unit Circle — wo definition jis par ASTC build hota hai.
Reference Angles — magnitude provide karta hai.
Trigonometric Identities — $90°$ shifts aur co-function rules.
Radians and Degrees — same rule jisme $π$ , $180°$ ki jagah leta hai.
Graphs of Sin Cos Tan — quadrant ke hisaab se signs = graph par axis ke upar/neeche.

HUM ISKO KYUN SEEKHTE HAIN?

REFERENCE ANGLE KYA HOTA HAI?

SIGNS KAISE AATE HAIN — ASTC ko scratch se derive karna

Worked examples

Active recall

Connections

Concept Map