Half angle formulas — derivations from double angle

3.1.14 · Maths › Advanced Trigonometry

1. Beej: double-angle cosine (teen roopein)

Ye dono kyun derive karein? Kyunki har ek sirf ek single squared function isolate karta hai — exactly wohi jo hum baad mein $cos^{2}$ ya $sin^{2}$ ke liye solve karne ke liye chahiye.

Doosri form ki Derivation (Ye step kyun?): $cos 2 θ = cos^{2} θ - sin^{2} θ = (1 - sin^{2} θ) - sin^{2} θ = 1 - 2 sin^{2} θ .$ Yahaan humne $cos^{2} θ$ ko $1 - sin^{2} θ$ se replace kiya kyunki hum sirf $sin$ mein expression chahte the.

2. Half-angle formulas derive karna (scratch se)

Trick: $A = 2 θ$ maano, toh $θ = \frac{A}{2}$ . Do ek-function forms mein substitute karo.

(a) Sine ke liye half-angle

$cos 2 θ = 1 - 2 sin^{2} θ$ se shuru karo. $2 θ \to A$ replace karo: $cos A = 1 - 2 sin^{2} \frac{A}{2}$ $sin^{2} \frac{A}{2}$ ke liye solve karo (Kyun? hum half angle akela chahte hain): $2 sin^{2} \frac{A}{2} = 1 - cos A \Rightarrow sin^{2} \frac{A}{2} = \frac{1 - cos A}{2}$

(b) Cosine ke liye half-angle

$cos 2 θ = 2 cos^{2} θ - 1$ se shuru karo. $2 θ \to A$ replace karo: $cos A = 2 cos^{2} \frac{A}{2} - 1 \Rightarrow cos^{2} \frac{A}{2} = \frac{1 + cos A}{2}$

(c) Tangent ke liye half-angle

(a) ko (b) se divide karo — Kyun? $tan = sin / cos$ , toh $tan^{2} = sin^{2} / cos^{2}$ : $tan^{2} \frac{A}{2} = \frac{1 - c o s A}{1 + c o s A}$

"No-radical" tangent forms derive karna

$\frac{1 - cos A}{1 + cos A}$ ke upar aur neeche ko square-root ke andar se ek conjugate se multiply karo taaki sign ambiguity door ho: $tan^{2} \frac{A}{2} = \frac{( 1 - c o s A )}{( 1 + c o s A )} \cdot \frac{( 1 + c o s A )}{( 1 + c o s A )} = \frac{1 - c o s ^{2} A}{( 1 + c o s A ) ^{2}} = \frac{s i n ^{2} A}{( 1 + c o s A ) ^{2}} .$ Root lete hain: $tan \frac{A}{2} = \frac{sin A}{1 + cos A}$ — $\pm$ ki zaroorat nahi kyunki $sin A$ automatically sahi sign carry karta hai. Isi tarah $(1 - cos A)$ se multiply karne par $tan \frac{A}{2} = \frac{1 - cos A}{sin A}$ milta hai.

3. Worked examples

4. Common mistakes

5. Active recall

Recall Bina dekhe reproduce kar sakte ho?

$cos 2 θ$ ke teen roopein.
Woh substitution jo double ko half mein convert karta hai.
$tan \frac{A}{2}$ ke dono radical-free forms aur kyun unhe $\pm$ ki zaroorat nahi.
$sin \frac{A}{2}$ mein sign decide kaise karein.

Kaun si double-angle form

sin^{2} \frac{A}{2}

deti hai?

cos A = 1 - 2 sin^{2} θ

mein

2 θ = A

rakh kar, milta hai

sin^{2} \frac{A}{2} = \frac{1 - c o s A}{2}

cos^{2} \frac{A}{2}

ke liye half-angle formula?

\frac{1 + cos A}{2}

sin^{2} \frac{A}{2}

ke liye half-angle formula?

\frac{1 - cos A}{2}

tan \frac{A}{2}

ke do radical-free forms?

\frac{1 - cos A}{sin A}

aur

\frac{sin A}{1 + cos A}

Half-angle formulas mein

\pm

sign kis angle ke quadrant par depend karta hai?

A /2

par (na ki

A

par).

cos A = 2 cos^{2} \frac{A}{2} - 1

kahan se derive karte hain?

cos 2 θ = 2 cos^{2} θ - 1

se,

2 θ = A

set karke.

cos 1 5^{\circ}

ki exact value?

\frac{2 + 3}{2} \approx 0.9659

tan \frac{A}{2} = \frac{s i n A}{1 + c o s A}

\pm

ki zaroorat kyun nahi?

sin A

pehle se sahi sign carry karta hai; denominator

1 + cos A \geq 0

hota hai.

tan 22. 5^{\circ}

ki value?

2 - 1 \approx 0.4142

Recall Feynman: 12-saal ke bacche ko samjhao

Socho ek machine hai jo ek angle ke baare mein facts batati hai. Humare paas pehle se ek machine hai jo, jab tum usse koi angle do, toh woh tumhe double us angle ka cosine batati hai. Ab maano kisi ne tumhe ek poori pizza slice ka cosine diya aur aadhi slice ke baare mein poocha. Hum machine ko ulta chalate hain: uski equation ko arrange karo jab tak "aadhi slice" akeli ek taraf na ho. Do pakad: (1) aadha karne se tum "map ke kaun se corner" (quadrant) mein ho yeh badal sakta hai, toh check karo ki answer plus hona chahiye ya minus; (2) tum angle ki tarah sine ko bhi aadha nahi kar sakte — trig cake kaatne ki tarah kaam nahi karta.

Connections

Double angle formulas — ye seedha parent identities hain.
Weierstrass substitution — $t = tan \frac{x}{2}$ poori tarah inhi par based hai.
Pythagorean identity — $cos 2 θ$ ko ek-function forms mein split karne ke liye use hua.
Exact trig values — $1 5^{\circ}, 22. 5^{\circ}, 7 5^{\circ}$ half-angles se aate hain.
Product-to-sum formulas — inhi seed identities ki bhai jaisi manipulations.